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The prime number theorem can be expressed as the assertion

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + o(x) \ \ \ \ \ (1)

as {x \rightarrow \infty}, where

\displaystyle  \Lambda(n) := \sum_{d|n} \mu(d) \log \frac{n}{d}

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + O(x) \ \ \ \ \ (2)

where the second von Mangoldt function {\Lambda_2} is defined by the formula

\displaystyle  \Lambda_2(n) := \sum_{d|n} \mu(d) \log^2 \frac{n}{d} \ \ \ \ \ (3)

or equivalently

\displaystyle  \Lambda_2(n) = \Lambda(n) \log n + \sum_{d|n} \Lambda(d) \Lambda(\frac{n}{d}). \ \ \ \ \ (4)

(We are avoiding the use of the {*} symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of the reader, we give a proof of the Selberg symmetry formula below the fold. Actually, for the purposes of proving the prime number theorem, the weaker estimate

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + o(x \log x) \ \ \ \ \ (5)

suffices.

In this post I would like to record a somewhat “soft analysis” reformulation of the elementary proof of the prime number theorem in terms of Banach algebras, and specifically in Banach algebra structures on (completions of) the space {C_c({\bf R})} of compactly supported continuous functions {f: {\bf R} \rightarrow {\bf C}} equipped with the convolution operation

\displaystyle  f*g(t) := \int_{\bf R} f(u) g(t-u)\ du.

This soft argument does not easily give any quantitative decay rate in the prime number theorem, but by the same token it avoids many of the quantitative calculations in the traditional proofs of this theorem. Ultimately, the key “soft analysis” fact used is the spectral radius formula

\displaystyle  \lim_{n \rightarrow \infty} \|f^n\|^{1/n} = \sup_{\lambda \in \hat B} |\lambda(f)| \ \ \ \ \ (6)

for any element {f} of a unital commutative Banach algebra {B}, where {\hat B} is the space of characters (i.e., continuous unital algebra homomorphisms from {B} to {{\bf C}}) of {B}. This formula is due to Gelfand and may be found in any text on Banach algebras; we will assume it here as a black box.

The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.

Theorem 1 (Construction of a Banach algebra norm) For any {G \in C_c({\bf R})}, let {\|G\|} denote the quantity

\displaystyle  \|G\| := \limsup_{x \rightarrow \infty} |\sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) - \int_{\bf R} G(t)\ dt|.

Then {\| \|} is a seminorm on {C_c({\bf R})} with the bound

\displaystyle  \|G\| \leq \|G\|_{L^1({\bf R})} := \int_{\bf R} |G(t)|\ dt \ \ \ \ \ (7)

for all {G \in C_c({\bf R})}. Furthermore, we have the Banach algebra bound

\displaystyle  \| G * H \| \leq \|G\| \|H\| \ \ \ \ \ (8)

for all {G,H \in C_c({\bf R})}.

We prove this theorem below the fold. The prime number theorem then follows from Theorem 1 and the following two assertions. The first is an application of the spectral radius formula (6) and some basic Fourier analysis:

Theorem 2 (Non-trivial Banach algebras have non-trivial spectrum) Let {\| \|} be a seminorm on {C_c({\bf R})} obeying (7), (8). Suppose that {\| \|} is not identically zero. Then there exists {\xi \in {\bf R}} such that

\displaystyle  |\int_{\bf R} G(t) e^{-it\xi}\ dt| \leq \|G\|

for all {G \in C_c}. In particular, by (7), one has

\displaystyle  \|G\| = \| G \|_{L^1({\bf R})}

whenever {G(t) e^{-it\xi}} is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that {\Lambda} is real (as well as Mertens’ theorem, in the {\xi=0} case), and is closely related to the non-vanishing of the Riemann zeta function {\zeta} on the line {\{ 1+i\xi: \xi \in {\bf R}\}}:

Theorem 3 (Breaking the parity barrier) Let {\xi \in {\bf R}} and {\varepsilon > 0}. Then there exists {G \in C_c({\bf R})} such that {G(t) e^{-it\xi}} is non-negative, and

\displaystyle  \|G\| < \|G\|_{L^1({\bf R})}.

Assuming Theorems 1, 2, 3, we may now quickly establish the prime number theorem as follows. Theorem 2 and Theorem 3 imply that the seminorm {\| \|} constructed in Theorem 1 is trivial, and thus

\displaystyle  \sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) = \int_{\bf R} G(t)\ dt + o(1)

as {x \rightarrow \infty} for any Schwartz function {G} (the decay rate in {o(1)} may depend on {G}). Specialising to functions of the form {G(t) = e^{-t} \eta( e^{-t} )} for some smooth compactly supported {\eta} on {(0,+\infty)}, we conclude that

\displaystyle  \sum_n \Lambda(n) \eta(\frac{n}{x}) = \int_{\bf R} \eta(u)\ du + o(x)

as {x \rightarrow \infty}; by the smooth Urysohn lemma this implies that

\displaystyle  \sum_{\varepsilon x \leq n \leq x} \Lambda(n) = x - \varepsilon x + o(x)

as {x \rightarrow \infty} for any fixed {\varepsilon>0}, and the prime number theorem then follows by a telescoping series argument.

The same argument also yields the prime number theorem in arithmetic progressions, or equivalently that

\displaystyle  \sum_{n \leq x} \Lambda(n) \chi(n) = o(x)

for any fixed Dirichlet character {\chi}; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity {L(1,\chi) = \sum_n \frac{\chi(n)}{n}} is non-vanishing.

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In the traditional foundations of probability theory, one selects a probability space {(\Omega, {\mathcal B}, {\mathbf P})}, and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state {\omega \in \Omega}, and stochastic (or random) mathematical objects, which do depend (but in a measurable fashion) on the sampled state {\omega \in \Omega}. For instance, a deterministic real number would just be an element {x \in {\bf R}}, whereas a stochastic real number (or real random variable) would be a measurable function {x: \Omega \rightarrow {\bf R}}, where in this post {{\bf R}} will always be endowed with the Borel {\sigma}-algebra. (For readers familiar with nonstandard analysis, the adjectives “deterministic” and “stochastic” will be used here in a manner analogous to the uses of the adjectives “standard” and “nonstandard” in nonstandard analysis. The analogy is particularly close when comparing with the “cheap nonstandard analysis” discussed in this previous blog post. We will also use “relative to {\Omega}” as a synonym for “stochastic”.)

Actually, for our purposes we will adopt the philosophy of identifying stochastic objects that agree almost surely, so if one was to be completely precise, we should define a stochastic real number to be an equivalence class {[x]} of measurable functions {x: \Omega \rightarrow {\bf R}}, up to almost sure equivalence. However, we shall often abuse notation and write {[x]} simply as {x}.

More generally, given any measurable space {X = (X, {\mathcal X})}, we can talk either about deterministic elements {x \in X}, or about stochastic elements of {X}, that is to say equivalence classes {[x]} of measurable maps {x: \Omega \rightarrow X} up to almost sure equivalence. We will use {\Gamma(X|\Omega)} to denote the set of all stochastic elements of {X}. (For readers familiar with sheaves, it may helpful for the purposes of this post to think of {\Gamma(X|\Omega)} as the space of measurable global sections of the trivial {X}-bundle over {\Omega}.) Of course every deterministic element {x} of {X} can also be viewed as a stochastic element {x|\Omega \in \Gamma(X|\Omega)} given by (the equivalence class of) the constant function {\omega \mapsto x}, thus giving an embedding of {X} into {\Gamma(X|\Omega)}. We do not attempt here to give an interpretation of {\Gamma(X|\Omega)} for sets {X} that are not equipped with a {\sigma}-algebra {{\mathcal X}}.

Remark 1 In my previous post on the foundations of probability theory, I emphasised the freedom to extend the sample space {(\Omega, {\mathcal B}, {\mathbf P})} to a larger sample space whenever one wished to inject additional sources of randomness. This is of course an important freedom to possess (and in the current formalism, is the analogue of the important operation of base change in algebraic geometry), but in this post we will focus on a single fixed sample space {(\Omega, {\mathcal B}, {\mathbf P})}, and not consider extensions of this space, so that one only has to consider two types of mathematical objects (deterministic and stochastic), as opposed to having many more such types, one for each potential choice of sample space (with the deterministic objects corresponding to the case when the sample space collapses to a point).

Any (measurable) {k}-ary operation on deterministic mathematical objects then extends to their stochastic counterparts by applying the operation pointwise. For instance, the addition operation {+: {\bf R} \times {\bf R} \rightarrow {\bf R}} on deterministic real numbers extends to an addition operation {+: \Gamma({\bf R}|\Omega) \times \Gamma({\bf R}|\Omega) \rightarrow \Gamma({\bf R}|\Omega)}, by defining the class {[x]+[y]} for {x,y: \Omega \rightarrow {\bf R}} to be the equivalence class of the function {\omega \mapsto x(\omega) + y(\omega)}; this operation is easily seen to be well-defined. More generally, any measurable {k}-ary deterministic operation {O: X_1 \times \dots \times X_k \rightarrow Y} between measurable spaces {X_1,\dots,X_k,Y} extends to an stochastic operation {O: \Gamma(X_1|\Omega) \times \dots \Gamma(X_k|\Omega) \rightarrow \Gamma(Y|\Omega)} in the obvious manner.

There is a similar story for {k}-ary relations {R: X_1 \times \dots \times X_k \rightarrow \{\hbox{true},\hbox{false}\}}, although here one has to make a distinction between a deterministic reading of the relation and a stochastic one. Namely, if we are given stochastic objects {x_i \in \Gamma(X_i|\Omega)} for {i=1,\dots,k}, the relation {R(x_1,\dots,x_k)} does not necessarily take values in the deterministic Boolean algebra {\{ \hbox{true}, \hbox{false}\}}, but only in the stochastic Boolean algebra {\Gamma(\{ \hbox{true}, \hbox{false}\}|\Omega)} – thus {R(x_1,\dots,x_k)} may be true with some positive probability and also false with some positive probability (with the event that {R(x_1,\dots,x_k)} being stochastically true being determined up to null events). Of course, the deterministic Boolean algebra embeds in the stochastic one, so we can talk about a relation {R(x_1,\dots,x_k)} being determinstically true or deterministically false, which (due to our identification of stochastic objects that agree almost surely) means that {R(x_1(\omega),\dots,x_k(\omega))} is almost surely true or almost surely false respectively. For instance given two stochastic objects {x,y}, one can view their equality relation {x=y} as having a stochastic truth value. This is distinct from the way the equality symbol {=} is used in mathematical logic, which we will now call “equality in the deterministic sense” to reduce confusion. Thus, {x=y} in the deterministic sense if and only if the stochastic truth value of {x=y} is equal to {\hbox{true}}, that is to say that {x(\omega)=y(\omega)} for almost all {\omega}.

Any universal identity for deterministic operations (or universal implication between identities) extends to their stochastic counterparts: for instance, addition is commutative, associative, and cancellative on the space of deterministic reals {{\bf R}}, and is therefore commutative, associative, and cancellative on stochastic reals {\Gamma({\bf R}|\Omega)} as well. However, one has to be more careful when working with mathematical laws that are not expressible as universal identities, or implications between identities. For instance, {{\bf R}} is an integral domain: if {x_1,x_2 \in {\bf R}} are deterministic reals such that {x_1 x_2=0}, then one must have {x_1=0} or {x_2=0}. However, if {x_1, x_2 \in \Gamma({\bf R}|\Omega)} are stochastic reals such that {x_1 x_2 = 0} (in the deterministic sense), then it is no longer necessarily the case that {x_1=0} (in the deterministic sense) or that {x_2=0} (in the deterministic sense); however, it is still true that “{x_1=0} or {x_2=0}” is true in the deterministic sense if one interprets the boolean operator “or” stochastically, thus “{x_1(\omega)=0} or {x_2(\omega)=0}” is true for almost all {\omega}. Another way to properly obtain a stochastic interpretation of the integral domain property of {{\bf R}} is to rewrite it as

\displaystyle  x_1,x_2 \in {\bf R}, x_1 x_2 = 0 \implies x_i=0 \hbox{ for some } i \in \{1,2\}

and then make all sets stochastic to obtain the true statement

\displaystyle  x_1,x_2 \in \Gamma({\bf R}|\Omega), x_1 x_2 = 0 \implies x_i=0 \hbox{ for some } i \in \Gamma(\{1,2\}|\Omega),

thus we have to allow the index {i} for which vanishing {x_i=0} occurs to also be stochastic, rather than deterministic. (A technical note: when one proves this statement, one has to select {i} in a measurable fashion; for instance, one can choose {i(\omega)} to equal {1} when {x_1(\omega)=0}, and {2} otherwise (so that in the “tie-breaking” case when {x_1(\omega)} and {x_2(\omega)} both vanish, one always selects {i(\omega)} to equal {1}).)

Similarly, the law of the excluded middle fails when interpreted deterministically, but remains true when interpreted stochastically: if {S} is a stochastic statement, then it is not necessarily the case that {S} is either deterministically true or deterministically false; however the sentence “{S} or not-{S}” is still deterministically true if the boolean operator “or” is interpreted stochastically rather than deterministically.

To avoid having to keep pointing out which operations are interpreted stochastically and which ones are interpreted deterministically, we will use the following convention: if we assert that a mathematical sentence {S} involving stochastic objects is true, then (unless otherwise specified) we mean that {S} is deterministically true, assuming that all relations used inside {S} are interpreted stochastically. For instance, if {x,y} are stochastic reals, when we assert that “Exactly one of {x < y}, {x=y}, or {x>y} is true”, then by default it is understood that the relations {<}, {=}, {>} and the boolean operator “exactly one of” are interpreted stochastically, and the assertion is that the sentence is deterministically true.

In the above discussion, the stochastic objects {x} being considered were elements of a deterministic space {X}, such as the reals {{\bf R}}. However, it can often be convenient to generalise this situation by allowing the ambient space {X} to also be stochastic. For instance, one might wish to consider a stochastic vector {v(\omega)} inside a stochastic vector space {V(\omega)}, or a stochastic edge {e} of a stochastic graph {G(\omega)}. In order to formally describe this situation within the classical framework of measure theory, one needs to place all the ambient spaces {X(\omega)} inside a measurable space. This can certainly be done in many contexts (e.g. when considering random graphs on a deterministic set of vertices, or if one is willing to work up to equivalence and place the ambient spaces inside a suitable moduli space), but is not completely natural in other contexts. For instance, if one wishes to consider stochastic vector spaces of potentially unbounded dimension (in particular, potentially larger than any given cardinal that one might specify in advance), then the class of all possible vector spaces is so large that it becomes a proper class rather than a set (even if one works up to equivalence), making it problematic to give this class the structure of a measurable space; furthermore, even once one does so, one needs to take additional care to pin down what it would mean for a random vector {\omega \mapsto v_\omega} lying in a random vector space {\omega \mapsto V_\omega} to depend “measurably” on {\omega}.

Of course, in any reasonable application one can avoid the set theoretic issues at least by various ad hoc means, for instance by restricting the dimension of all spaces involved to some fixed cardinal such as {2^{\aleph_0}}. However, the measure-theoretic issues can require some additional effort to resolve properly.

In this post I would like to describe a different way to formalise stochastic spaces, and stochastic elements of these spaces, by viewing the spaces as measure-theoretic analogue of a sheaf, but being over the probability space {\Omega} rather than over a topological space; stochastic objects are then sections of such sheaves. Actually, for minor technical reasons it is convenient to work in the slightly more general setting in which the base space {\Omega} is a finite measure space {(\Omega, {\mathcal B}, \mu)} rather than a probability space, thus {\mu(\Omega)} can take any value in {[0,+\infty)} rather than being normalised to equal {1}. This will allow us to easily localise to subevents {\Omega'} of {\Omega} without the need for normalisation, even when {\Omega'} is a null event (though we caution that the map {x \mapsto x|\Omega'} from deterministic objects {x} ceases to be injective in this latter case). We will however still continue to use probabilistic terminology. despite the lack of normalisation; thus for instance, sets {E} in {{\mathcal B}} will be referred to as events, the measure {\mu(E)} of such a set will be referred to as the probability (which is now permitted to exceed {1} in some cases), and an event whose complement is a null event shall be said to hold almost surely. It is in fact likely that almost all of the theory below extends to base spaces which are {\sigma}-finite rather than finite (for instance, by damping the measure to become finite, without introducing any further null events), although we will not pursue this further generalisation here.

The approach taken in this post is “topos-theoretic” in nature (although we will not use the language of topoi explicitly here), and is well suited to a “pointless” or “point-free” approach to probability theory, in which the role of the stochastic state {\omega \in \Omega} is suppressed as much as possible; instead, one strives to always adopt a “relative point of view”, with all objects under consideration being viewed as stochastic objects relative to the underlying base space {\Omega}. In this perspective, the stochastic version of a set is as follows.

Definition 1 (Stochastic set) Unless otherwise specified, we assume that we are given a fixed finite measure space {\Omega = (\Omega, {\mathcal B}, \mu)} (which we refer to as the base space). A stochastic set (relative to {\Omega}) is a tuple {X|\Omega = (\Gamma(X|E)_{E \in {\mathcal B}}, ((|E))_{E \subset F, E,F \in {\mathcal B}})} consisting of the following objects:

  • A set {\Gamma(X|E)} assigned to each event {E \in {\mathcal B}}; and
  • A restriction map {x \mapsto x|E} from {\Gamma(X|F)} to {\Gamma(X|E)} to each pair {E \subset F} of nested events {E,F \in {\mathcal B}}. (Strictly speaking, one should indicate the dependence on {F} in the notation for the restriction map, e.g. using {x \mapsto x|(E \leftarrow F)} instead of {x \mapsto x|E}, but we will abuse notation by omitting the {F} dependence.)

We refer to elements of {\Gamma(X|E)} as local stochastic elements of the stochastic set {X|\Omega}, localised to the event {E}, and elements of {\Gamma(X|\Omega)} as global stochastic elements (or simply elements) of the stochastic set. (In the language of sheaves, one would use “sections” instead of “elements” here, but I prefer to use the latter terminology here, for compatibility with conventional probabilistic notation, where for instance measurable maps from {\Omega} to {{\bf R}} are referred to as real random variables, rather than sections of the reals.)

Furthermore, we impose the following axioms:

  • (Category) The map {x \mapsto x|E} from {\Gamma(X|E)} to {\Gamma(X|E)} is the identity map, and if {E \subset F \subset G} are events in {{\mathcal B}}, then {((x|F)|E) = (x|E)} for all {x \in \Gamma(X|G)}.
  • (Null events trivial) If {E \in {\mathcal B}} is a null event, then the set {\Gamma(X|E)} is a singleton set. (In particular, {\Gamma(X|\emptyset)} is always a singleton set; this is analogous to the convention that {x^0=1} for any number {x}.)
  • (Countable gluing) Suppose that for each natural number {n}, one has an event {E_n \in {\mathcal B}} and an element {x_n \in \Gamma(X|E_n)} such that {x_n|(E_n \cap E_m) = x_m|(E_n \cap E_m)} for all {n,m}. Then there exists a unique {x\in \Gamma(X|\bigcup_{n=1}^\infty E_n)} such that {x_n = x|E_n} for all {n}.

If {\Omega'} is an event in {\Omega}, we define the localisation {X|\Omega'} of the stochastic set {X|\Omega} to {\Omega'} to be the stochastic set

\displaystyle X|\Omega' := (\Gamma(X|E)_{E \in {\mathcal B}; E \subset \Omega'}, ((|E))_{E \subset F \subset \Omega', E,F \in {\mathcal B}})

relative to {\Omega'}. (Note that there is no need to renormalise the measure on {\Omega'}, as we are not demanding that our base space have total measure {1}.)

The following fact is useful for actually verifying that a given object indeed has the structure of a stochastic set:

Exercise 1 Show that to verify the countable gluing axiom of a stochastic set, it suffices to do so under the additional hypothesis that the events {E_n} are disjoint. (Note that this is quite different from the situation with sheaves over a topological space, in which the analogous gluing axiom is often trivial in the disjoint case but has non-trivial content in the overlapping case. This is ultimately because a {\sigma}-algebra is closed under all Boolean operations, whereas a topology is only closed under union and intersection.)

Let us illustrate the concept of a stochastic set with some examples.

Example 1 (Discrete case) A simple case arises when {\Omega} is a discrete space which is at most countable. If we assign a set {X_\omega} to each {\omega \in \Omega}, with {X_\omega} a singleton if {\mu(\{\omega\})=0}. One then sets {\Gamma(X|E) := \prod_{\omega \in E} X_\omega}, with the obvious restriction maps, giving rise to a stochastic set {X|\Omega}. (Thus, a local element {x} of {\Gamma(X|E)} can be viewed as a map {\omega \mapsto x(\omega)} on {E} that takes values in {X_\omega} for each {\omega \in E}.) Conversely, it is not difficult to see that any stochastic set over an at most countable discrete probability space {\Omega} is of this form up to isomorphism. In this case, one can think of {X|\Omega} as a bundle of sets {X_\omega} over each point {\omega} (of positive probability) in the base space {\Omega}. One can extend this bundle interpretation of stochastic sets to reasonably nice sample spaces {\Omega} (such as standard Borel spaces) and similarly reasonable {X}; however, I would like to avoid this interpretation in the formalism below in order to be able to easily work in settings in which {\Omega} and {X} are very “large” (e.g. not separable in any reasonable sense). Note that we permit some of the {X_\omega} to be empty, thus it can be possible for {\Gamma(X|\Omega)} to be empty whilst {\Gamma(X|E)} for some strict subevents {E} of {\Omega} to be non-empty. (This is analogous to how it is possible for a sheaf to have local sections but no global sections.) As such, the space {\Gamma(X|\Omega)} of global elements does not completely determine the stochastic set {X|\Omega}; one sometimes needs to localise to an event {E} in order to see the full structure of such a set. Thus it is important to distinguish between a stochastic set {X|\Omega} and its space {\Gamma(X|\Omega)} of global elements. (As such, it is a slight abuse of the axiom of extensionality to refer to global elements of {X|\Omega} simply as “elements”, but hopefully this should not cause too much confusion.)

Example 2 (Measurable spaces as stochastic sets) Returning now to a general base space {\Omega}, any (deterministic) measurable space {X} gives rise to a stochastic set {X|\Omega}, with {\Gamma(X|E)} being defined as in previous discussion as the measurable functions from {E} to {X} modulo almost everywhere equivalence (in particular, {\Gamma(X|E)} a singleton set when {E} is null), with the usual restriction maps. The constraint of measurability on the maps {x: E \rightarrow \Omega}, together with the quotienting by almost sure equivalence, means that {\Gamma(X|E)} is now more complicated than a plain Cartesian product {\prod_{\omega \in E} X_\omega} of fibres, but this still serves as a useful first approximation to what {\Gamma(X|E)} is for the purposes of developing intuition. Indeed, the measurability constraint is so weak (as compared for instance to topological or smooth constraints in other contexts, such as sheaves of continuous or smooth sections of bundles) that the intuition of essentially independent fibres is quite an accurate one, at least if one avoids consideration of an uncountable number of objects simultaneously.

Example 3 (Extended Hilbert modules) This example is the one that motivated this post for me. Suppose that one has an extension {(\tilde \Omega, \tilde {\mathcal B}, \tilde \mu)} of the base space {(\Omega, {\mathcal B},\mu)}, thus we have a measurable factor map {\pi: \tilde \Omega \rightarrow \Omega} such that the pushforward of the measure {\tilde \mu} by {\pi} is equal to {\mu}. Then we have a conditional expectation operator {\pi_*: L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu) \rightarrow L^2(\Omega,{\mathcal B},\mu)}, defined as the adjoint of the pullback map {\pi^*: L^2(\Omega,{\mathcal B},\mu) \rightarrow L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu)}. As is well known, the conditional expectation operator also extends to a contraction {\pi_*: L^1(\tilde \Omega,\tilde {\mathcal B},\tilde \mu) \rightarrow L^1(\Omega,{\mathcal B}, \mu)}; by monotone convergence we may also extend {\pi_*} to a map from measurable functions from {\tilde \Omega} to the extended non-negative reals {[0,+\infty]}, to measurable functions from {\Omega} to {[0,+\infty]}. We then define the “extended Hilbert module” {L^2(\tilde \Omega|\Omega)} to be the space of functions {f \in L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu)} with {\pi_*(|f|^2)} finite almost everywhere. This is an extended version of the Hilbert module {L^\infty_{\Omega} L^2(\tilde \Omega|\Omega)}, which is defined similarly except that {\pi_*(|f|^2)} is required to lie in {L^\infty(\Omega,{\mathcal B},\mu)}; this is a Hilbert module over {L^\infty(\Omega, {\mathcal B}, \mu)} which is of particular importance in the Furstenberg-Zimmer structure theory of measure-preserving systems. We can then define the stochastic set {L^2_\pi(\tilde \Omega)|\Omega} by setting

\displaystyle  \Gamma(L^2_\pi(\tilde \Omega)|E) := L^2( \pi^{-1}(E) | E )

with the obvious restriction maps. In the case that {\Omega,\Omega'} are standard Borel spaces, one can disintegrate {\mu'} as an integral {\mu' = \int_\Omega \nu_\omega\ d\mu(\omega)} of probability measures {\nu_\omega} (supported in the fibre {\pi^{-1}(\{\omega\})}), in which case this stochastic set can be viewed as having fibres {L^2( \tilde \Omega, \tilde {\mathcal B}, \nu_\omega )} (though if {\Omega} is not discrete, there are still some measurability conditions in {\omega} on the local and global elements that need to be imposed). However, I am interested in the case when {\Omega,\Omega'} are not standard Borel spaces (in fact, I will take them to be algebraic probability spaces, as defined in this previous post), in which case disintegrations are not available. However, it appears that the stochastic analysis developed in this blog post can serve as a substitute for the tool of disintegration in this context.

We make the remark that if {X|\Omega} is a stochastic set and {E, F} are events that are equivalent up to null events, then one can identify {\Gamma(X|E)} with {\Gamma(X|F)} (through their common restriction to {\Gamma(X|(E \cap F))}, with the restriction maps now being bijections). As such, the notion of a stochastic set does not require the full structure of a concrete probability space {(\Omega, {\mathcal B}, {\mathbf P})}; one could also have defined the notion using only the abstract {\sigma}-algebra consisting of {{\mathcal B}} modulo null events as the base space, or equivalently one could define stochastic sets over the algebraic probability spaces defined in this previous post. However, we will stick with the classical formalism of concrete probability spaces here so as to keep the notation reasonably familiar.

As a corollary of the above observation, we see that if the base space {\Omega} has total measure {0}, then all stochastic sets are trivial (they are just points).

Exercise 2 If {X|\Omega} is a stochastic set, show that there exists an event {\Omega'} with the property that for any event {E}, {\Gamma(X|E)} is non-empty if and only if {E} is contained in {\Omega'} modulo null events. (In particular, {\Omega'} is unique up to null events.) Hint: consider the numbers {\mu( E )} for {E} ranging over all events with {\Gamma(X|E)} non-empty, and form a maximising sequence for these numbers. Then use all three axioms of a stochastic set.

One can now start take many of the fundamental objects, operations, and results in set theory (and, hence, in most other categories of mathematics) and establish analogues relative to a finite measure space. Implicitly, what we will be doing in the next few paragraphs is endowing the category of stochastic sets with the structure of an elementary topos. However, to keep things reasonably concrete, we will not explicitly emphasise the topos-theoretic formalism here, although it is certainly lurking in the background.

Firstly, we define a stochastic function {f: X|\Omega \rightarrow Y|\Omega} between two stochastic sets {X|\Omega, Y|\Omega} to be a collection of maps {f: \Gamma(X|E) \rightarrow \Gamma(Y|E)} for each {E \in {\mathcal B}} which form a natural transformation in the sense that {f(x|E) = f(x)|E} for all {x \in \Gamma(X|F)} and nested events {E \subset F}. In the case when {\Omega} is discrete and at most countable (and after deleting all null points), a stochastic function is nothing more than a collection of functions {f_\omega: X_\omega \rightarrow Y_\omega} for each {\omega \in \Omega}, with the function {f: \Gamma(X|E) \rightarrow \Gamma(Y|E)} then being a direct sum of the factor functions {f_\omega}:

\displaystyle  f( (x_\omega)_{\omega \in E} ) = ( f_\omega(x_\omega) )_{\omega \in E}.

Thus (in the discrete, at most countable setting, at least) stochastic functions do not mix together information from different states {\omega} in a sample space; the value of {f(x)} at {\omega} depends only on the value of {x} at {\omega}. The situation is a bit more subtle for continuous probability spaces, due to the identification of stochastic objects that agree almost surely, nevertheness it is still good intuition to think of stochastic functions as essentially being “pointwise” or “local” in nature.

One can now form the stochastic set {\hbox{Hom}(X \rightarrow Y)|\Omega} of functions from {X|\Omega} to {Y|\Omega}, by setting {\Gamma(\hbox{Hom}(X \rightarrow Y)|E)} for any event {E} to be the set of local stochastic functions {f: X|E \rightarrow Y|E} of the localisations of {X|\Omega, Y|\Omega} to {E}; this is a stochastic set if we use the obvious restriction maps. In the case when {\Omega} is discrete and at most countable, the fibre {\hbox{Hom}(X \rightarrow Y)_\omega} at a point {\omega} of positive measure is simply the set {Y_\omega^{X_\omega}} of functions from {X_\omega} to {Y_\omega}.

In a similar spirit, we say that one stochastic set {Y|\Omega} is a (stochastic) subset of another {X|\Omega}, and write {Y|\Omega \subset X|\Omega}, if we have a stochastic inclusion map, thus {\Gamma(Y|E) \subset \Gamma(X|E)} for all events {E}, with the restriction maps being compatible. We can then define the power set {2^X|\Omega} of a stochastic set {X|\Omega} by setting {\Gamma(2^X|E)} for any event {E} to be the set of all stochastic subsets {Y|E} of {X|E} relative to {E}; it is easy to see that {2^X|\Omega} is a stochastic set with the obvious restriction maps (one can also identify {2^X|\Omega} with {\hbox{Hom}(X, \{\hbox{true},\hbox{false}\})|\Omega} in the obvious fashion). Again, when {\Omega} is discrete and at most countable, the fibre of {2^X|\Omega} at a point {\omega} of positive measure is simply the deterministic power set {2^{X_\omega}}.

Note that if {f: X|\Omega \rightarrow Y|\Omega} is a stochastic function and {Y'|\Omega} is a stochastic subset of {Y|\Omega}, then the inverse image {f^{-1}(Y')|\Omega}, defined by setting {\Gamma(f^{-1}(Y')|E)} for any event {E} to be the set of those {x \in \Gamma(X|E)} with {f(x) \in \Gamma(Y'|E)}, is a stochastic subset of {X|\Omega}. In particular, given a {k}-ary relation {R: X_1 \times \dots \times X_k|\Omega \rightarrow \{\hbox{true}, \hbox{false}\}|\Omega}, the inverse image {R^{-1}( \{ \hbox{true} \}|\Omega )} is a stochastic subset of {X_1 \times \dots \times X_k|\Omega}, which by abuse of notation we denote as

\displaystyle  \{ (x_1,\dots,x_k) \in X_1 \times \dots \times X_k: R(x_1,\dots,x_k) \hbox{ is true} \}|\Omega.

In a similar spirit, if {X'|\Omega} is a stochastic subset of {X|\Omega} and {f: X|\Omega \rightarrow Y|\Omega} is a stochastic function, we can define the image {f(X')|\Omega} by setting {\Gamma(f(X')|E)} to be the set of those {f(x)} with {x \in \Gamma(X'|E)}; one easily verifies that this is a stochastic subset of {Y|\Omega}.

Remark 2 One should caution that in the definition of the subset relation {Y|\Omega \subset X|\Omega}, it is important that {\Gamma(Y|E) \subset \Gamma(X|E)} for all events {E}, not just the global event {\Omega}; in particular, just because a stochastic set {X|\Omega} has no global sections, does not mean that it is contained in the stochastic empty set {\emptyset|\Omega}.

Now we discuss Boolean operations on stochastic subsets of a given stochastic set {X|\Omega}. Given two stochastic subsets {X_1|\Omega, X_2|\Omega} of {X|\Omega}, the stochastic intersection {(X_1 \cap X_2)|\Omega} is defined by setting {\Gamma((X_1 \cap X_2)|E)} to be the set of {x \in \Gamma(X|E)} that lie in both {\Gamma(X_1|E)} and {\Gamma(X_2|E)}:

\displaystyle  \Gamma(X_1 \cap X_2)|E) := \Gamma(X_1|E) \cap \Gamma(X_2|E).

This is easily verified to again be a stochastic subset of {X|\Omega}. More generally one may define stochastic countable intersections {(\bigcap_{n=1}^\infty X_n)|\Omega} for any sequence {X_n|\Omega} of stochastic subsets of {X|\Omega}. One could extend this definition to uncountable families if one wished, but I would advise against it, because some of the usual laws of Boolean algebra (e.g. the de Morgan laws) may break down in this setting.

Stochastic unions are a bit more subtle. The set {\Gamma((X_1 \cup X_2)|E)} should not be defined to simply be the union of {\Gamma(X_1|E)} and {\Gamma(X_2|E)}, as this would not respect the gluing axiom. Instead, we define {\Gamma((X_1 \cup X_2)|E)} to be the set of all {x \in \Gamma(X|E)} such that one can cover {E} by measurable subevents {E_1,E_2} such that {x_i|E_i \in \Gamma(X_i|E_i)} for {i=1,2}; then {(X_1 \cup X_2)|\Omega} may be verified to be a stochastic subset of {X|\Omega}. Thus for instance {\{0,1\}|\Omega} is the stochastic union of {\{0\}|\Omega} and {\{1\}|\Omega}. Similarly for countable unions {(\bigcup_{n=1}^\infty X_n)|\Omega} of stochastic subsets {X_n|\Omega} of {X|\Omega}, although for uncountable unions are extremely problematic (they are disliked by both the measure theory and the countable gluing axiom) and will not be defined here. Finally, the stochastic difference set {\Gamma((X_1 \backslash X_2)|E)} is defined as the set of all {x|E} in {\Gamma(X_1|E)} such that {x|F \not \in \Gamma(X_2|F)} for any subevent {F} of {E} of positive probability. One may verify that in the case when {\Omega} is discrete and at most countable, these Boolean operations correspond to the classical Boolean operations applied separately to each fibre {X_{i,\omega}} of the relevant sets {X_i}. We also leave as an exercise to the reader to verify the usual laws of Boolean arithmetic, e.g. the de Morgan laws, provided that one works with at most countable unions and intersections.

One can also consider a stochastic finite union {(\bigcup_{n=1}^N X_n)|\Omega} in which the number {N} of sets in the union is itself stochastic. More precisely, let {X|\Omega} be a stochastic set, let {N \in {\bf N}|\Omega} be a stochastic natural number, and let {n \mapsto X_n|\Omega} be a stochastic function from the stochastic set {\{ n \in {\bf N}: n \leq N\}|\Omega} (defined by setting {\Gamma(\{n \in {\bf N}: n\leq N\}|E) := \{ n \in {\bf N}|E: n \leq N|E\}})) to the stochastic power set {2^X|\Omega}. Here we are considering {0} to be a natural number, to allow for unions that are possibly empty, with {{\bf N}_+ := {\bf N} \backslash \{0\}} used for the positive natural numbers. We also write {(X_n)_{n=1}^N|\Omega} for the stochastic function {n \mapsto X_n|\Omega}. Then we can define the stochastic union {\bigcup_{n=1}^N X_n|\Omega} by setting {\Gamma(\bigcup_{n=1}^N X_n|E)} for an event {E} to be the set of local elements {x \in \Gamma(X|E)} with the property that there exists a covering of {E} by measurable subevents {E_{n_0}} for {n_0 \in {\bf N}_+}, such that one has {n_0 \leq N|E_{n_0}} and {x|E_{n_0} \in \Gamma(X_{n_0}|E_{n_0})}. One can verify that {\bigcup_{n=1}^N X_n|\Omega} is a stochastic set (with the obvious restriction maps). Again, in the model case when {\Omega} is discrete and at most countable, the fibre {(\bigcup_{n=1}^N X_n)_\omega} is what one would expect it to be, namely {\bigcup_{n=1}^{N(\omega)} (X_n)_\omega}.

The Cartesian product {(X \times Y)|\Omega} of two stochastic sets may be defined by setting {\Gamma((X \times Y)|E) := \Gamma(X|E) \times \Gamma(Y|E)} for all events {E}, with the obvious restriction maps; this is easily seen to be another stochastic set. This lets one define the concept of a {k}-ary operation {f: (X_1 \times \dots \times X_k)|\Omega \rightarrow Y|\Omega} from {k} stochastic sets {X_1,\dots,X_k} to another stochastic set {Y}, or a {k}-ary relation {R: (X_1 \times \dots \times X_k)|\Omega \rightarrow \{\hbox{true}, \hbox{false}\}|\Omega}. In particular, given {x_i \in X_i|\Omega} for {i=1,\dots,k}, the relation {R(x_1,\dots,x_k)} may be deterministically true, deterministically false, or have some other stochastic truth value.

Remark 3 In the degenerate case when {\Omega} is null, stochastic logic becomes a bit weird: all stochastic statements are deterministically true, as are their stochastic negations, since every event in {\Omega} (even the empty set) now holds with full probability. Among other pathologies, the empty set now has a global element over {\Omega} (this is analogous to the notorious convention {0^0=1}), and any two deterministic objects {x,y} become equal over {\Omega}: {x|\Omega=y|\Omega}.

The following simple observation is crucial to subsequent discussion. If {(x_n)_{n \in {\bf N}_+}} is a sequence taking values in the global elements {\Gamma(X|\Omega)} of a stochastic space {X|\Omega}, then we may also define global elements {x_n \in \Gamma(X|\Omega)} for stochastic indices {n \in {\bf N}_+|\Omega} as well, by appealing to the countable gluing axiom to glue together {x_{n_0}} restricted to the set {\{ \omega \in \Omega: n(\omega) = n_0\}} for each deterministic natural number {n_0} to form {x_n}. With this definition, the map {n \mapsto x_n} is a stochastic function from {{\bf N}_+|\Omega} to {X|\Omega}; indeed, this creates a one-to-one correspondence between external sequences (maps {n \mapsto x_n} from {{\bf N}_+} to {\Gamma(X|\Omega)}) and stochastic sequences (stochastic functions {n \mapsto x_n} from {{\bf N}_+|\Omega} to {X|\Omega}). Similarly with {{\bf N}_+} replaced by any other at most countable set. This observation will be important in allowing many deterministic arguments involving sequences will be able to be carried over to the stochastic setting.

We now specialise from the extremely broad discipline of set theory to the more focused discipline of real analysis. There are two fundamental axioms that underlie real analysis (and in particular distinguishes it from real algebra). The first is the Archimedean property, which we phrase in the “no infinitesimal” formulation as follows:

Proposition 2 (Archimedean property) Let {x \in {\bf R}} be such that {x \leq 1/n} for all positive natural numbers {n}. Then {x \leq 0}.

The other is the least upper bound axiom:

Proposition 3 (Least upper bound axiom) Let {S} be a non-empty subset of {{\bf R}} which has an upper bound {M \in {\bf R}}, thus {x \leq M} for all {x \in S}. Then there exists a unique real number {\sup S \in {\bf R}} with the following properties:

  • {x \leq \sup S} for all {x \in S}.
  • For any real {L < \sup S}, there exists {x \in S} such that {L < x \leq \sup S}.
  • {\sup S \leq M}.

Furthermore, {\sup S} does not depend on the choice of {M}.

The Archimedean property extends easily to the stochastic setting:

Proposition 4 (Stochastic Archimedean property) Let {x \in \Gamma({\bf R}|\Omega)} be such that {x \leq 1/n} for all deterministic natural numbers {n}. Then {x \leq 0}.

Remark 4 Here, incidentally, is one place in which this stochastic formalism deviates from the nonstandard analysis formalism, as the latter certainly permits the existence of infinitesimal elements. On the other hand, we caution that stochastic real numbers are permitted to be unbounded, so that formulation of Archimedean property is not valid in the stochastic setting.

The proof is easy and is left to the reader. The least upper bound axiom also extends nicely to the stochastic setting, but the proof requires more work (in particular, our argument uses the monotone convergence theorem):

Theorem 5 (Stochastic least upper bound axiom) Let {S|\Omega} be a stochastic subset of {{\bf R}|\Omega} which has a global upper bound {M \in {\bf R}|\Omega}, thus {x \leq M} for all {x \in \Gamma(S|\Omega)}, and is globally non-empty in the sense that there is at least one global element {x \in \Gamma(S|\Omega)}. Then there exists a unique stochastic real number {\sup S \in \Gamma({\bf R}|\Omega)} with the following properties:

  • {x \leq \sup S} for all {x \in \Gamma(S|\Omega)}.
  • For any stochastic real {L < \sup S}, there exists {x \in \Gamma(S|\Omega)} such that {L < x \leq \sup S}.
  • {\sup S \leq M}.

Furthermore, {\sup S} does not depend on the choice of {M}.

For future reference, we note that the same result holds with {{\bf R}} replaced by {{\bf N} \cup \{+\infty\}} throughout, since the latter may be embedded in the former, for instance by mapping {n} to {1 - \frac{1}{n+1}} and {+\infty} to {1}. In applications, the above theorem serves as a reasonable substitute for the countable axiom of choice, which does not appear to hold in unrestricted generality relative to a measure space; in particular, it can be used to generate various extremising sequences for stochastic functionals on various stochastic function spaces.

Proof: Uniqueness is clear (using the Archimedean property), as well as the independence on {M}, so we turn to existence. By using an order-preserving map from {{\bf R}} to {(-1,1)} (e.g. {x \mapsto \frac{2}{\pi} \hbox{arctan}(x)}) we may assume that {S|\Omega} is a subset of {(-1,1)|\Omega}, and that {M < 1}.

We observe that {\Gamma(S|\Omega)} is a lattice: if {x, y \in \Gamma(S|\Omega)}, then {\max(x,y)} and {\min(x,y)} also lie in {\Gamma(S|\Omega)}. Indeed, {\max(x,y)} may be formed by appealing to the countable gluing axiom to glue {y} (restricted the set {\{ \omega \in \Omega: x(\omega) < y(\omega) \}}) with {x} (restricted to the set {\{ \omega \in \Omega: x(\omega) \geq y(\omega) \}}), and similarly for {\min(x,y)}. (Here we use the fact that relations such as {<} are Borel measurable on {{\bf R}}.)

Let {A \in {\bf R}} denote the deterministic quantity

\displaystyle  A := \sup \{ \int_\Omega x(\omega)\ d\mu(\omega): x \in \Gamma(S|\Omega) \}

then (by Proposition 3!) {A} is well-defined; here we use the hypothesis that {\mu(\Omega)} is finite. Thus we may find a sequence {(x_n)_{n \in {\bf N}}} of elements {x_n} of {\Gamma(S|\Omega)} such that

\displaystyle  \int_\Omega x_n(\omega)\ d\mu(\omega) \rightarrow A \hbox{ as } n \rightarrow \infty. \ \ \ \ \ (1)

Using the lattice property, we may assume that the {x_n} are non-decreasing: {x_n \leq x_m} whenever {n \leq m}. If we then define {\sup S(\omega) := \sup_n x_n(\omega)} (after choosing measurable representatives of each equivalence class {x_n}), then {\sup S} is a stochastic real with {\sup S \leq M}.

If {x \in \Gamma(S|\Omega)}, then {\max(x,x_n) \in \Gamma(S|\Omega)}, and so

\displaystyle  \int_\Omega \max(x,x_n)\ d\mu(\omega) \leq A.

From this and (1) we conclude that

\displaystyle  \int_\Omega \max(x-x_n,0) \rightarrow 0 \hbox{ as } n \rightarrow \infty.

From monotone convergence, we conclude that

\displaystyle  \int_\Omega \max(x-\sup S,0) = 0

and so {x \leq \sup S}, as required.

Now let {L < \sup S} be a stochastic real. After choosing measurable representatives of each relevant equivalence class, we see that for almost every {\omega \in \Omega}, we can find a natural number {n(\omega)} with {x_{n(\omega)} > L}. If we choose {n(\omega)} to be the first such positive natural number when it exists, and (say) {1} otherwise, then {n} is a stochastic positive natural number and {L < x_n}. The claim follows. \Box

Remark 5 One can abstract away the role of the measure {\mu} here, leaving only the ideal of null sets. The property that the measure is finite is then replaced by the more general property that given any non-empty family of measurable sets, there is an at most countable union of sets in that family that is an upper bound modulo null sets for all elements in that faily.

Using Proposition 4 and Theorem 5, one can then revisit many of the other foundational results of deterministic real analysis, and develop stochastic analogues; we give some examples of this below the fold (focusing on the Heine-Borel theorem and a case of the spectral theorem). As an application of this formalism, we revisit some of the Furstenberg-Zimmer structural theory of measure-preserving systems, particularly that of relatively compact and relatively weakly mixing systems, and interpret them in this framework, basically as stochastic versions of compact and weakly mixing systems (though with the caveat that the shift map is allowed to act non-trivially on the underlying probability space). As this formalism is “point-free”, in that it avoids explicit use of fibres and disintegrations, it will be well suited for generalising this structure theory to settings in which the underlying probability spaces are not standard Borel, and the underlying groups are uncountable; I hope to discuss such generalisations in future blog posts.

Remark 6 Roughly speaking, stochastic real analysis can be viewed as a restricted subset of classical real analysis in which all operations have to be “measurable” with respect to the base space. In particular, indiscriminate application of the axiom of choice is not permitted, and one should largely restrict oneself to performing countable unions and intersections rather than arbitrary unions or intersections. Presumably one can formalise this intuition with a suitable “countable transfer principle”, but I was not able to formulate a clean and general principle of this sort, instead verifying various assertions about stochastic objects by hand rather than by direct transfer from the deterministic setting. However, it would be desirable to have such a principle, since otherwise one is faced with the tedious task of redoing all the foundations of real analysis (or whatever other base theory of mathematics one is going to be working in) in the stochastic setting by carefully repeating all the arguments.

More generally, topos theory is a good formalism for capturing precisely the informal idea of performing mathematics with certain operations, such as the axiom of choice, the law of the excluded middle, or arbitrary unions and intersections, being somehow “prohibited” or otherwise “restricted”.

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One of the basic tools in modern combinatorics is the probabilistic method, introduced by Erdos, in which a deterministic solution to a given problem is shown to exist by constructing a random candidate for a solution, and showing that this candidate solves all the requirements of the problem with positive probability. When the problem requires a real-valued statistic {X} to be suitably large or suitably small, the following trivial observation is often employed:

Proposition 1 (Comparison with mean) Let {X} be a random real-valued variable, whose mean (or first moment) {\mathop{\bf E} X} is finite. Then

\displaystyle  X \leq \mathop{\bf E} X

with positive probability, and

\displaystyle  X \geq \mathop{\bf E} X

with positive probability.

This proposition is usually applied in conjunction with a computation of the first moment {\mathop{\bf E} X}, in which case this version of the probabilistic method becomes an instance of the first moment method. (For comparison with other moment methods, such as the second moment method, exponential moment method, and zeroth moment method, see Chapter 1 of my book with Van Vu. For a general discussion of the probabilistic method, see the book by Alon and Spencer of the same name.)

As a typical example in random matrix theory, if one wanted to understand how small or how large the operator norm {\|A\|_{op}} of a random matrix {A} could be, one might first try to compute the expected operator norm {\mathop{\bf E} \|A\|_{op}} and then apply Proposition 1; see this previous blog post for examples of this strategy (and related strategies, based on comparing {\|A\|_{op}} with more tractable expressions such as the moments {\hbox{tr} A^k}). (In this blog post, all matrices are complex-valued.)

Recently, in their proof of the Kadison-Singer conjecture (and also in their earlier paper on Ramanujan graphs), Marcus, Spielman, and Srivastava introduced an striking new variant of the first moment method, suited in particular for controlling the operator norm {\|A\|_{op}} of a Hermitian positive semi-definite matrix {A}. Such matrices have non-negative real eigenvalues, and so {\|A\|_{op}} in this case is just the largest eigenvalue {\lambda_1(A)} of {A}. Traditionally, one tries to control the eigenvalues through averaged statistics such as moments {\hbox{tr} A^k = \sum_i \lambda_i(A)^k} or Stieltjes transforms {\hbox{tr} (A-z)^{-1} = \sum_i (\lambda_i(A)-z)^{-1}}; again, see this previous blog post. Here we use {z} as short-hand for {zI_d}, where {I_d} is the {d \times d} identity matrix. Marcus, Spielman, and Srivastava instead rely on the interpretation of the eigenvalues {\lambda_i(A)} of {A} as the roots of the characteristic polynomial {p_A(z) := \hbox{det}(z-A)} of {A}, thus

\displaystyle  \|A\|_{op} = \hbox{maxroot}( p_A ) \ \ \ \ \ (1)

where {\hbox{maxroot}(p)} is the largest real root of a non-zero polynomial {p}. (In our applications, we will only ever apply {\hbox{maxroot}} to polynomials that have at least one real root, but for sake of completeness let us set {\hbox{maxroot}(p)=-\infty} if {p} has no real roots.)

Prior to the work of Marcus, Spielman, and Srivastava, I think it is safe to say that the conventional wisdom in random matrix theory was that the representation (1) of the operator norm {\|A\|_{op}} was not particularly useful, due to the highly non-linear nature of both the characteristic polynomial map {A \mapsto p_A} and the maximum root map {p \mapsto \hbox{maxroot}(p)}. (Although, as pointed out to me by Adam Marcus, some related ideas have occurred in graph theory rather than random matrix theory, for instance in the theory of the matching polynomial of a graph.) For instance, a fact as basic as the triangle inequality {\|A+B\|_{op} \leq \|A\|_{op} + \|B\|_{op}} is extremely difficult to establish through (1). Nevertheless, it turns out that for certain special types of random matrices {A} (particularly those in which a typical instance {A} of this ensemble has a simple relationship to “adjacent” matrices in this ensemble), the polynomials {p_A} enjoy an extremely rich structure (in particular, they lie in families of real stable polynomials, and hence enjoy good combinatorial interlacing properties) that can be surprisingly useful. In particular, Marcus, Spielman, and Srivastava established the following nonlinear variant of Proposition 1:

Proposition 2 (Comparison with mean) Let {m,d \geq 1}. Let {A} be a random matrix, which is the sum {A = \sum_{i=1}^m A_i} of independent Hermitian rank one {d \times d} matrices {A_i}, each taking a finite number of values. Then

\displaystyle  \hbox{maxroot}(p_A) \leq \hbox{maxroot}( \mathop{\bf E} p_A )

with positive probability, and

\displaystyle  \hbox{maxroot}(p_A) \geq \hbox{maxroot}( \mathop{\bf E} p_A )

with positive probability.

We prove this proposition below the fold. The hypothesis that each {A_i} only takes finitely many values is technical and can likely be relaxed substantially, but we will not need to do so here. Despite the superficial similarity with Proposition 1, the proof of Proposition 2 is quite nonlinear; in particular, one needs the interlacing properties of real stable polynomials to proceed. Another key ingredient in the proof is the observation that while the determinant {\hbox{det}(A)} of a matrix {A} generally behaves in a nonlinar fashion on the underlying matrix {A}, it becomes (affine-)linear when one considers rank one perturbations, and so {p_A} depends in an affine-multilinear fashion on the {A_1,\ldots,A_m}. More precisely, we have the following deterministic formula, also proven below the fold:

Proposition 3 (Deterministic multilinearisation formula) Let {A} be the sum of deterministic rank one {d \times d} matrices {A_1,\ldots,A_m}. Then we have

\displaystyle  p_A(z) = \mu[A_1,\ldots,A_m](z) \ \ \ \ \ (2)

for all {z \in C}, where the mixed characteristic polynomial {\mu[A_1,\ldots,A_m](z)} of any {d \times d} matrices {A_1,\ldots,A_m} (not necessarily rank one) is given by the formula

\displaystyle  \mu[A_1,\ldots,A_m](z) \ \ \ \ \ (3)

\displaystyle  = (\prod_{i=1}^m (1 - \frac{\partial}{\partial z_i})) \hbox{det}( z + \sum_{i=1}^m z_i A_i ) |_{z_1=\ldots=z_m=0}.

Among other things, this formula gives a useful representation of the mean characteristic polynomial {\mathop{\bf E} p_A}:

Corollary 4 (Random multilinearisation formula) Let {A} be the sum of jointly independent rank one {d \times d} matrices {A_1,\ldots,A_m}. Then we have

\displaystyle  \mathop{\bf E} p_A(z) = \mu[ \mathop{\bf E} A_1, \ldots, \mathop{\bf E} A_m ](z) \ \ \ \ \ (4)

for all {z \in {\bf C}}.

Proof: For fixed {z}, the expression {\hbox{det}( z + \sum_{i=1}^m z_i A_i )} is a polynomial combination of the {z_i A_i}, while the differential operator {(\prod_{i=1}^m (1 - \frac{\partial}{\partial z_i}))} is a linear combination of differential operators {\frac{\partial^j}{\partial z_{i_1} \ldots \partial z_{i_j}}} for {1 \leq i_1 < \ldots < i_j \leq d}. As a consequence, we may expand (3) as a linear combination of terms, each of which is a multilinear combination of {A_{i_1},\ldots,A_{i_j}} for some {1 \leq i_1 < \ldots < i_j \leq d}. Taking expectations of both sides of (2) and using the joint independence of the {A_i}, we obtain the claim. \Box

In view of Proposition 2, we can now hope to control the operator norm {\|A\|_{op}} of certain special types of random matrices {A} (and specifically, the sum of independent Hermitian positive semi-definite rank one matrices) by first controlling the mean {\mathop{\bf E} p_A} of the random characteristic polynomial {p_A}. Pursuing this philosophy, Marcus, Spielman, and Srivastava establish the following result, which they then use to prove the Kadison-Singer conjecture:

Theorem 5 (Marcus-Spielman-Srivastava theorem) Let {m,d \geq 1}. Let {v_1,\ldots,v_m \in {\bf C}^d} be jointly independent random vectors in {{\bf C}^d}, with each {v_i} taking a finite number of values. Suppose that we have the normalisation

\displaystyle  \mathop{\bf E} \sum_{i=1}^m v_i v_i^* = 1

where we are using the convention that {1} is the {d \times d} identity matrix {I_d} whenever necessary. Suppose also that we have the smallness condition

\displaystyle  \mathop{\bf E} \|v_i\|^2 \leq \varepsilon

for some {\varepsilon>0} and all {i=1,\ldots,m}. Then one has

\displaystyle  \| \sum_{i=1}^m v_i v_i^* \|_{op} \leq (1+\sqrt{\varepsilon})^2 \ \ \ \ \ (5)

with positive probability.

Note that the upper bound in (5) must be at least {1} (by taking {v_i} to be deterministic) and also must be at least {\varepsilon} (by taking the {v_i} to always have magnitude at least {\sqrt{\varepsilon}}). Thus the bound in (5) is asymptotically tight both in the regime {\varepsilon\rightarrow 0} and in the regime {\varepsilon \rightarrow \infty}; the latter regime will be particularly useful for applications to Kadison-Singer. It should also be noted that if one uses more traditional random matrix theory methods (based on tools such as Proposition 1, as well as more sophisticated variants of these tools, such as the concentration of measure results of Rudelson and Ahlswede-Winter), one obtains a bound of {\| \sum_{i=1}^m v_i v_i^* \|_{op} \ll_\varepsilon \log d} with high probability, which is insufficient for the application to the Kadison-Singer problem; see this article of Tropp. Thus, Theorem 5 obtains a sharper bound, at the cost of trading in “high probability” for “positive probability”.

In the paper of Marcus, Spielman and Srivastava, Theorem 5 is used to deduce a conjecture {KS_2} of Weaver, which was already known to imply the Kadison-Singer conjecture; actually, a slight modification of their argument gives the paving conjecture of Kadison and Singer, from which the original Kadison-Singer conjecture may be readily deduced. We give these implications below the fold. (See also this survey article for some background on the Kadison-Singer problem.)

Let us now summarise how Theorem 5 is proven. In the spirit of semi-definite programming, we rephrase the above theorem in terms of the rank one Hermitian positive semi-definite matrices {A_i := v_iv_i^*}:

Theorem 6 (Marcus-Spielman-Srivastava theorem again) Let {A_1,\ldots,A_m} be jointly independent random rank one Hermitian positive semi-definite {d \times d} matrices such that the sum {A :=\sum_{i=1}^m A_i} has mean

\displaystyle  \mathop{\bf E} A = I_d

and such that

\displaystyle  \mathop{\bf E} \hbox{tr} A_i \leq \varepsilon

for some {\varepsilon>0} and all {i=1,\ldots,m}. Then one has

\displaystyle  \| A \|_{op} \leq (1+\sqrt{\varepsilon})^2

with positive probability.

In view of (1) and Proposition 2, this theorem follows from the following control on the mean characteristic polynomial:

Theorem 7 (Control of mean characteristic polynomial) Let {A_1,\ldots,A_m} be jointly independent random rank one Hermitian positive semi-definite {d \times d} matrices such that the sum {A :=\sum_{i=1}^m A_i} has mean

\displaystyle  \mathop{\bf E} A = 1

and such that

\displaystyle  \mathop{\bf E} \hbox{tr} A_i \leq \varepsilon

for some {\varepsilon>0} and all {i=1,\ldots,m}. Then one has

\displaystyle  \hbox{maxroot}(\mathop{\bf E} p_A) \leq (1 +\sqrt{\varepsilon})^2.

This result is proven using the multilinearisation formula (Corollary 4) and some convexity properties of real stable polynomials; we give the proof below the fold.

Thanks to Adam Marcus, Assaf Naor and Sorin Popa for many useful explanations on various aspects of the Kadison-Singer problem.

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Let {F} be a field. A definable set over {F} is a set of the form

\displaystyle  \{ x \in F^n | \phi(x) \hbox{ is true} \} \ \ \ \ \ (1)

where {n} is a natural number, and {\phi(x)} is a predicate involving the ring operations {+,\times} of {F}, the equality symbol {=}, an arbitrary number of constants and free variables in {F}, the quantifiers {\forall, \exists}, boolean operators such as {\vee,\wedge,\neg}, and parentheses and colons, where the quantifiers are always understood to be over the field {F}. Thus, for instance, the set of quadratic residues

\displaystyle  \{ x \in F | \exists y: x = y \times y \}

is definable over {F}, and any algebraic variety over {F} is also a definable set over {F}. Henceforth we will abbreviate “definable over {F}” simply as “definable”.

If {F} is a finite field, then every subset of {F^n} is definable, since finite sets are automatically definable. However, we can obtain a more interesting notion in this case by restricting the complexity of a definable set. We say that {E \subset F^n} is a definable set of complexity at most {M} if {n \leq M}, and {E} can be written in the form (1) for some predicate {\phi} of length at most {M} (where all operators, quantifiers, relations, variables, constants, and punctuation symbols are considered to have unit length). Thus, for instance, a hypersurface in {n} dimensions of degree {d} would be a definable set of complexity {O_{n,d}(1)}. We will then be interested in the regime where the complexity remains bounded, but the field size (or field characteristic) becomes large.

In a recent paper, I established (in the large characteristic case) the following regularity lemma for dense definable graphs, which significantly strengthens the Szemerédi regularity lemma in this context, by eliminating “bad” pairs, giving a polynomially strong regularity, and also giving definability of the cells:

Lemma 1 (Algebraic regularity lemma) Let {F} be a finite field, let {V,W} be definable non-empty sets of complexity at most {M}, and let {E \subset V \times W} also be definable with complexity at most {M}. Assume that the characteristic of {F} is sufficiently large depending on {M}. Then we may partition {V = V_1 \cup \ldots \cup V_m} and {W = W_1 \cup \ldots \cup W_n} with {m,n = O_M(1)}, with the following properties:

  • (Definability) Each of the {V_1,\ldots,V_m,W_1,\ldots,W_n} are definable of complexity {O_M(1)}.
  • (Size) We have {|V_i| \gg_M |V|} and {|W_j| \gg_M |W|} for all {i=1,\ldots,m} and {j=1,\ldots,n}.
  • (Regularity) We have

    \displaystyle  |E \cap (A \times B)| = d_{ij} |A| |B| + O_M( |F|^{-1/4} |V| |W| ) \ \ \ \ \ (2)

    for all {i=1,\ldots,m}, {j=1,\ldots,n}, {A \subset V_i}, and {B\subset W_j}, where {d_{ij}} is a rational number in {[0,1]} with numerator and denominator {O_M(1)}.

My original proof of this lemma was quite complicated, based on an explicit calculation of the “square”

\displaystyle  \mu(w,w') := \{ v \in V: (v,w), (v,w') \in E \}

of {E} using the Lang-Weil bound and some facts about the étale fundamental group. It was the reliance on the latter which was the main reason why the result was restricted to the large characteristic setting. (I then applied this lemma to classify expanding polynomials over finite fields of large characteristic, but I will not discuss these applications here; see this previous blog post for more discussion.)

Recently, Anand Pillay and Sergei Starchenko (and independently, Udi Hrushovski) have observed that the theory of the étale fundamental group is not necessary in the argument, and the lemma can in fact be deduced from quite general model theoretic techniques, in particular using (a local version of) the concept of stability. One of the consequences of this new proof of the lemma is that the hypothesis of large characteristic can be omitted; the lemma is now known to be valid for arbitrary finite fields {F} (although its content is trivial if the field is not sufficiently large depending on the complexity at most {M}).

Inspired by this, I decided to see if I could find yet another proof of the algebraic regularity lemma, again avoiding the theory of the étale fundamental group. It turns out that the spectral proof of the Szemerédi regularity lemma (discussed in this previous blog post) adapts very nicely to this setting. The key fact needed about definable sets over finite fields is that their cardinality takes on an essentially discrete set of values. More precisely, we have the following fundamental result of Chatzidakis, van den Dries, and Macintyre:

Proposition 2 Let {F} be a finite field, and let {M > 0}.

  • (Discretised cardinality) If {E} is a non-empty definable set of complexity at most {M}, then one has

    \displaystyle  |E| = c |F|^d + O_M( |F|^{d-1/2} ) \ \ \ \ \ (3)

    where {d = O_M(1)} is a natural number, and {c} is a positive rational number with numerator and denominator {O_M(1)}. In particular, we have {|F|^d \ll_M |E| \ll_M |F|^d}.

  • (Definable cardinality) Assume {|F|} is sufficiently large depending on {M}. If {V, W}, and {E \subset V \times W} are definable sets of complexity at most {M}, so that {E_w := \{ v \in V: (v,w) \in W \}} can be viewed as a definable subset of {V} that is definably parameterised by {w \in W}, then for each natural number {d = O_M(1)} and each positive rational {c} with numerator and denominator {O_M(1)}, the set

    \displaystyle  \{ w \in W: |E_w| = c |F|^d + O_M( |F|^{d-1/2} ) \} \ \ \ \ \ (4)

    is definable with complexity {O_M(1)}, where the implied constants in the asymptotic notation used to define (4) are the same as those that appearing in (3). (Informally: the “dimension” {d} and “measure” {c} of {E_w} depends definably on {w}.)

We will take this proposition as a black box; a proof can be obtained by combining the description of definable sets over pseudofinite fields (discussed in this previous post) with the Lang-Weil bound (discussed in this previous post). (The former fact is phrased using nonstandard analysis, but one can use standard compactness-and-contradiction arguments to convert such statements to statements in standard analysis, as discussed in this post.)

The above proposition places severe restrictions on the cardinality of definable sets; for instance, it shows that one cannot have a definable set of complexity at most {M} and cardinality {|F|^{1/2}}, if {|F|} is sufficiently large depending on {M}. If {E \subset V} are definable sets of complexity at most {M}, it shows that {|E| = (c+ O_M(|F|^{-1/2})) |V|} for some rational {0\leq c \leq 1} with numerator and denominator {O_M(1)}; furthermore, if {c=0}, we may improve this bound to {|E| = O_M( |F|^{-1} |V|)}. In particular, we obtain the following “self-improving” properties:

  • If {E \subset V} are definable of complexity at most {M} and {|E| \leq \epsilon |V|} for some {\epsilon>0}, then (if {\epsilon} is sufficiently small depending on {M} and {F} is sufficiently large depending on {M}) this forces {|E| = O_M( |F|^{-1} |V| )}.
  • If {E \subset V} are definable of complexity at most {M} and {||E| - c |V|| \leq \epsilon |V|} for some {\epsilon>0} and positive rational {c}, then (if {\epsilon} is sufficiently small depending on {M,c} and {F} is sufficiently large depending on {M,c}) this forces {|E| = c |V| + O_M( |F|^{-1/2} |V| )}.

It turns out that these self-improving properties can be applied to the coefficients of various matrices (basically powers of the adjacency matrix associated to {E}) that arise in the spectral proof of the regularity lemma to significantly improve the bounds in that lemma; we describe how this is done below the fold. We also make some connections to the stability-based proofs of Pillay-Starchenko and Hrushovski.

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Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma:

Lemma 1 (Szemerédi regularity lemma) Let {G = (V,E)} be a graph on {n} vertices, and let {\epsilon > 0}. Then there exists a partition {V = V_1 \cup \ldots \cup V_M} for some {M \leq M(\epsilon)} with the property that for all but at most {\epsilon M^2} of the pairs {1 \leq i \leq j \leq M}, the pair {V_i, V_j} is {\epsilon}-regular in the sense that

\displaystyle  | d( A, B ) - d( V_i, V_j ) | \leq \epsilon

whenever {A \subset V_i, B \subset V_j} are such that {|A| \geq \epsilon |V_i|} and {|B| \geq \epsilon |V_j|}, and {d(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|/|A| |B|} is the edge density between {A} and {B}. Furthermore, the partition is equitable in the sense that {||V_i| - |V_j|| \leq 1} for all {1 \leq i \leq j \leq M}.

There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of {G}, which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.

For reasons of exposition, it is convenient to first establish a slightly weaker form of the lemma, in which one drops the hypothesis of equitability (but then has to weight the cells {V_i} by their magnitude when counting bad pairs):

Lemma 2 (Szemerédi regularity lemma, weakened variant) . Let {G = (V,E)} be a graph on {n} vertices, and let {\epsilon > 0}. Then there exists a partition {V = V_1 \cup \ldots \cup V_M} for some {M \leq M(\epsilon)} with the property that for all pairs {(i,j) \in \{1,\ldots,M\}^2} outside of an exceptional set {\Sigma}, one has

\displaystyle  | E(A,B) - d_{ij} |A| |B| | \ll \epsilon |V_i| |V_j| \ \ \ \ \ (1)

whenever {A \subset V_i, B \subset V_j}, for some real number {d_{ij}}, where {E(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|} is the number of edges between {A} and {B}. Furthermore, we have

\displaystyle  \sum_{(i,j) \in \Sigma} |V_i| |V_j| \ll \epsilon |V|^2. \ \ \ \ \ (2)

Let us now prove Lemma 2. We enumerate {V} (after relabeling) as {V = \{1,\ldots,n\}}. The adjacency matrix {T} of the graph {G} is then a self-adjoint {n \times n} matrix, and thus admits an eigenvalue decomposition

\displaystyle  T = \sum_{i=1}^n \lambda_i u_i^* u_i

for some orthonormal basis {u_1,\ldots,u_n} of {{\bf C}^n} and some eigenvalues {\lambda_1,\ldots,\lambda_n \in {\bf R}}, which we arrange in decreasing order of magnitude:

\displaystyle  |\lambda_1| \geq \ldots \geq |\lambda_n|.

We can compute the trace of {T^2} as

\displaystyle  \hbox{tr}(T^2) = \sum_{i=1}^n |\lambda_i|^2.

But we also have {\hbox{tr}(T^2) = 2|E| \leq n^2}, so

\displaystyle  \sum_{i=1}^n |\lambda_i|^2 \leq n^2. \ \ \ \ \ (3)

Among other things, this implies that

\displaystyle  |\lambda_i| \leq \frac{n}{\sqrt{i}} \ \ \ \ \ (4)

for all {i \geq 1}.

Let {F: {\bf N} \rightarrow {\bf N}} be a function (depending on {\epsilon}) to be chosen later, with {F(i) \geq i} for all {i}. Applying (3) and the pigeonhole principle (or the finite convergence principle, see this blog post), we can find {J \leq C(F,\epsilon)} such that

\displaystyle  \sum_{J \leq i < F(J)} |\lambda_i|^2 \leq \epsilon^3 n^2.

(Indeed, the bound on {J} is basically {F} iterated {1/\epsilon^3} times.) We can now split

\displaystyle  T = T_1 + T_2 + T_3, \ \ \ \ \ (5)

where {T_1} is the “structured” component

\displaystyle  T_1 := \sum_{i < J} \lambda_i u_i^* u_i, \ \ \ \ \ (6)

{T_2} is the “small” component

\displaystyle  T_2 := \sum_{J \leq i < F(J)} \lambda_i u_i^* u_i, \ \ \ \ \ (7)

and {T_3} is the “pseudorandom” component

\displaystyle  T_3 := \sum_{i > F(J)} \lambda_i u_i^* u_i. \ \ \ \ \ (8)

We now design a vertex partition to make {T_1} approximately constant on most cells. For each {i < J}, we partition {V} into {O_{J,\epsilon}(1)} cells on which {u_i} (viewed as a function from {V} to {{\bf C}}) only fluctuates by {O(\epsilon n^{-1/2} /J)}, plus an exceptional cell of size {O( \frac{\epsilon}{J} |V|)} coming from the values where {|u_i|} is excessively large (larger than {\sqrt{\frac{J}{\epsilon}} n^{-1/2}}). Combining all these partitions together, we can write {V = V_1 \cup \ldots \cup V_{M-1} \cup V_M} for some {M = O_{J,\epsilon}(1)}, where {V_M} has cardinality at most {\epsilon |V|}, and for all {1 \leq i \leq M-1}, the eigenfunctions {u_1,\ldots,u_{J-1}} all fluctuate by at most {O(\epsilon/J)}. In particular, if {1 \leq i,j \leq M-1}, then (by (4) and (6)) the entries of {T_1} fluctuate by at most {O(\epsilon)} on each block {V_i \times V_j}. If we let {d_{ij}} be the mean value of these entries on {V_i \times V_j}, we thus have

\displaystyle  1_B^* T_1 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (9)

for any {1 \leq i,j \leq M-1} and {A \subset V_i, B \subset V_j}, where we view the indicator functions {1_A, 1_B} as column vectors of dimension {n}.

Next, we observe from (3) and (7) that {\hbox{tr} T_2^2 \leq \epsilon^3 n^2}. If we let {x_{ab}} be the coefficients of {T_2}, we thus have

\displaystyle  \sum_{a,b \in V} |x_{ab}|^2 \leq \epsilon^3 n^2

and hence by Markov’s inequality we have

\displaystyle  \sum_{a \in V_i} \sum_{b \in V_j} |x_{ab}|^2 \leq \epsilon^2 |V_i| |V_j| \ \ \ \ \ (10)

for all pairs {(i,j) \in \{1,\ldots,M-1\}^2} outside of an exceptional set {\Sigma_1} with

\displaystyle  \sum_{(i,j) \in \Sigma_1} |V_i| |V_j| \leq \epsilon |V|^2.

If {(i,j) \in \{1,\ldots,M-1\}^2} avoids {\Sigma_1}, we thus have

\displaystyle  1_B^* T_2 1_A = O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (11)

for any {A \subset V_i, B \subset V_j}, by (10) and the Cauchy-Schwarz inequality.

Finally, to control {T_3} we see from (4) and (8) that {T_3} has an operator norm of at most {n/\sqrt{F(J)}}. In particular, we have from the Cauchy-Schwarz inequality that

\displaystyle  1_B^* T_3 1_A = O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (12)

for any {A, B \subset V}.

Let {\Sigma} be the set of all pairs {(i,j) \in \{1,\ldots,M\}^2} where either {(i,j) \in \Sigma_1}, {i = M}, {j=M}, or

\displaystyle  \min(|V_i|, |V_j|) \leq \frac{\epsilon}{M} n.

One easily verifies that (2) holds. If {(i,j) \in \{1,\ldots,M\}^2} is not in {\Sigma}, then by summing (9), (11), (12) and using (5), we see that

\displaystyle  1_B^* T 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) + O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (13)

for all {A \subset V_i, B \subset V_j}. The left-hand side is just {E(A,B)}. As {(i,j) \not \in \Sigma}, we have

\displaystyle  |V_i|, |V_j| > \frac{\epsilon}{M} n

and so (since {M = O_{J,\epsilon}(1)})

\displaystyle  n^2 / \sqrt{F(J)} \ll_{J,\epsilon} |V_i| |V_j| / \sqrt{F(J)}.

If we let {F} be a sufficiently rapidly growing function of {J} that depends on {\epsilon}, the second error term in (13) can be absorbed in the first, and (1) follows. This concludes the proof of Lemma 2.

To prove Lemma 1, one argues similarly (after modifying {\epsilon} as necessary), except that the initial partition {V_1,\ldots,V_M} of {V} constructed above needs to be subdivided further into equitable components (of size {\epsilon |V|/M+O(1)}), plus some remainder sets which can be aggregated into an exceptional component of size {O( \epsilon |V| )} (and which can then be redistributed amongst the other components to arrive at a truly equitable partition). We omit the details.

Remark 1 It is easy to verify that {F} needs to be growing exponentially in {J} in order for the above argument to work, which leads to tower-exponential bounds in the number of cells {M} in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying {F}, one basically obtains the strong regularity lemma first established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting {F(J) := J} essentially gives the weak regularity lemma of Frieze and Kannan.

Remark 2 If we specialise to a Cayley graph, in which {V = (V,+)} is a finite abelian group and {E = \{ (a,b): a-b \in A \}} for some (symmetric) subset {A} of {V}, then the eigenvectors are characters, and one essentially recovers the arithmetic regularity lemma of Green, in which the vertex partition classes {V_i} are given by Bohr sets (and one can then place additional regularity properties on these Bohr sets with some additional arguments). The components {T_1, T_2, T_3} of {T}, representing high, medium, and low eigenvalues of {T}, then become a decomposition associated to high, medium, and low Fourier coefficients of {A}.

Remark 3 The use of spectral theory here is parallel to the use of Fourier analysis to establish results such as Roth’s theorem on arithmetic progressions of length three. In analogy with this, one could view hypergraph regularity as being a sort of “higher order spectral theory”, although this spectral perspective is not as convenient as it is in the graph case.

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of local spectral statistics of non-Hermitian matrices“. The main result of this paper is a “Four Moment Theorem” that establishes universality for local spectral statistics of non-Hermitian matrices with independent entries, under the additional hypotheses that the entries of the matrix decay exponentially, and match moments with either the real or complex gaussian ensemble to fourth order. This is the non-Hermitian analogue of a long string of recent results establishing universality of local statistics in the Hermitian case (as discussed for instance in this recent survey of Van and myself, and also in several other places).

The complex case is somewhat easier to describe. Given a (non-Hermitian) random matrix ensemble {M_n} of {n \times n} matrices, one can arbitrarily enumerate the (geometric) eigenvalues as {\lambda_1(M_n),\ldots,\lambda_n(M_n) \in {\bf C}}, and one can then define the {k}-point correlation functions {\rho^{(k)}_n: {\bf C}^k \rightarrow {\bf R}^+} to be the symmetric functions such that

\displaystyle  \int_{{\bf C}^k} F(z_1,\ldots,z_k) \rho^{(k)}_n(z_1,\ldots,z_k)\ dz_1 \ldots dz_k

\displaystyle  = {\bf E} \sum_{1 \leq i_1 < \ldots < i_k \leq n} F(\lambda_1(M_n),\ldots,\lambda_k(M_n)).

In the case when {M_n} is drawn from the complex gaussian ensemble, so that all the entries are independent complex gaussians of mean zero and variance one, it is a classical result of Ginibre that the asymptotics of {\rho^{(k)}_n} near some point {z \sqrt{n}} as {n \rightarrow \infty} and {z \in {\bf C}} is fixed are given by the determinantal rule

\displaystyle  \rho^{(k)}_n(z\sqrt{n} + w_1,\ldots,z\sqrt{n}+w_k) \rightarrow \hbox{det}( K(w_i,w_j) )_{1 \leq i,j \leq k} \ \ \ \ \ (1)

for {|z| < 1} and

\displaystyle  \rho^{(k)}_n(z\sqrt{n} + w_1,\ldots,z\sqrt{n}+w_k) \rightarrow 0

for {|z| > 1}, where {K} is the reproducing kernel

\displaystyle  K(z,w) := \frac{1}{\pi} e^{-|z|^2/2 - |w|^2/2 + z \overline{w}}.

(There is also an asymptotic for the boundary case {|z|=1}, but it is more complicated to state.) In particular, we see that {\rho^{(k)}_n(z \sqrt{n}) \rightarrow \frac{1}{\pi} 1_{|z| \leq 1}} for almost every {z}, which is a manifestation of the well-known circular law for these matrices; but the circular law only captures the macroscopic structure of the spectrum, whereas the asymptotic (1) describes the microscopic structure.

Our first main result is that the asymptotic (1) for {|z|<1} also holds (in the sense of vague convergence) when {M_n} is a matrix whose entries are independent with mean zero, variance one, exponentially decaying tails, and which all match moments with the complex gaussian to fourth order. (Actually we prove a stronger result than this which is valid for all bounded {z} and has more uniform bounds, but is a bit more technical to state.) An analogous result is also established for real gaussians (but now one has to separate the correlation function into components depending on how many eigenvalues are real and how many are strictly complex; also, the limiting distribution is more complicated, being described by Pfaffians rather than determinants). Among other things, this allows us to partially extend some known results on complex or real gaussian ensembles to more general ensembles. For instance, there is a central limit theorem of Rider which establishes a central limit theorem for the number of eigenvalues of a complex gaussian matrix in a mesoscopic disk; from our results, we can extend this central limit theorem to matrices that match the complex gaussian ensemble to fourth order, provided that the disk is small enough (for technical reasons, our error bounds are not strong enough to handle large disks). Similarly, extending some results of Edelman-Kostlan-Shub and of Forrester-Nagao, we can show that for a matrix matching the real gaussian ensemble to fourth order, the number of real eigenvalues is {\sqrt{\frac{2n}{\pi}} + O(n^{1/2-c})} with probability {1-O(n^{-c})} for some absolute constant {c>0}.

There are several steps involved in the proof. The first step is to apply the Girko Hermitisation trick to replace the problem of understanding the spectrum of a non-Hermitian matrix, with that of understanding the spectrum of various Hermitian matrices. The two identities that realise this trick are, firstly, Jensen’s formula

\displaystyle  \log |\det(M_n-z_0)| = - \sum_{1 \leq i \leq n: \lambda_i(M_n) \in B(z_0,r)} \log \frac{r}{|\lambda_i(M_n)-z_0|}

\displaystyle + \frac{1}{2\pi} \int_0^{2\pi} \log |\det(M_n-z_0-re^{i\theta})|\ d\theta

that relates the local distribution of eigenvalues to the log-determinants {\log |\det(M_n-z_0)|}, and secondly the elementary identity

\displaystyle  \log |\det(M_n - z)| = \frac{1}{2} \log|\det W_{n,z}| + \frac{1}{2} n \log n

that relates the log-determinants of {M_n-z} to the log-determinants of the Hermitian matrices

\displaystyle  W_{n,z} := \frac{1}{\sqrt{n}} \begin{pmatrix} 0 & M_n -z \\ (M_n-z)^* & 0 \end{pmatrix}.

The main difficulty is then to obtain concentration and universality results for the Hermitian log-determinants {\log|\det W_{n,z}|}. This turns out to be a task that is analogous to the task of obtaining concentration for Wigner matrices (as we did in this recent paper), as well as central limit theorems for log-determinants of Wigner matrices (as we did in this other recent paper). In both of these papers, the main idea was to use the Four Moment Theorem for Wigner matrices (which can now be proven relatively easily by a combination of the local semi-circular law and resolvent swapping methods), combined with (in the latter paper) a central limit theorem for the gaussian unitary ensemble (GUE). This latter task was achieved by using the convenient Trotter normal form to tridiagonalise a GUE matrix, which has the effect of revealing the determinant of that matrix as the solution to a certain linear stochastic difference equation, and one can analyse the distribution of that solution via such tools as the martingale central limit theorem.

The matrices {W_{n,z}} are somewhat more complicated than Wigner matrices (for instance, the semi-circular law must be replaced by a distorted Marchenko-Pastur law), but the same general strategy works to obtain concentration and universality for their log-determinants. The main new difficulty that arises is that the analogue of the Trotter norm for gaussian random matrices is not tridiagonal, but rather Hessenberg (i.e. upper-triangular except for the lower diagonal). This ultimately has the effect of expressing the relevant determinant as the solution to a nonlinear stochastic difference equation, which is a bit trickier to solve for. Fortunately, it turns out that one only needs good lower bounds on the solution, as one can use the second moment method to upper bound the determinant and hence the log-determinant (following a classical computation of Turan). This simplifies the analysis on the equation somewhat.

While this result is the first local universality result in the category of random matrices with independent entries, there are still two limitations to the result which one would like to remove. The first is the moment matching hypotheses on the matrix. Very recently, one of the ingredients of our paper, namely the local circular law, was proved without moment matching hypotheses by Bourgade, Yau, and Yin (provided one stays away from the edge of the spectrum); however, as of this time of writing the other main ingredient – the universality of the log-determinant – still requires moment matching. (The standard tool for obtaining universality without moment matching hypotheses is the heat flow method (and more specifically, the local relaxation flow method), but the analogue of Dyson Brownian motion in the non-Hermitian setting appears to be somewhat intractible, being a coupled flow on both the eigenvalues and eigenvectors rather than just on the eigenvalues alone.)

I’ve just uploaded to the arXiv my paper The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, submitted to Probability Theory and Related Fields. This paper (like several of my previous papers) is concerned with the asymptotic distribution of the eigenvalues {\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)} of a random Wigner matrix {M_n} in the limit {n \rightarrow \infty}, with a particular focus on matrices drawn from the Gaussian Unitary Ensemble (GUE). This paper is focused on the bulk of the spectrum, i.e. to eigenvalues {\lambda_i(M_n)} with {\delta n \leq i \leq (1-\delta) i n} for some fixed {\delta>0}.

The location of an individual eigenvalue {\lambda_i(M_n)} is by now quite well understood. If we normalise the entries of the matrix {M_n} to have mean zero and variance {1}, then in the asymptotic limit {n \rightarrow \infty}, the Wigner semicircle law tells us that with probability {1-o(1)} one has

\displaystyle  \lambda_i(M_n) =\sqrt{n} u + o(\sqrt{n})

where the classical location {u = u_{i/n} \in [-2,2]} of the eigenvalue is given by the formula

\displaystyle  \int_{-2}^{u} \rho_{sc}(x)\ dx = \frac{i}{n}

and the semicircular distribution {\rho_{sc}(x)\ dx} is given by the formula

\displaystyle  \rho_{sc}(x) := \frac{1}{2\pi} (4-x^2)_+^{1/2}.

Actually, one can improve the error term here from {o(\sqrt{n})} to {O( \log^{1/2+\epsilon} n)} for any {\epsilon>0} (see this previous recent paper of Van and myself for more discussion of these sorts of estimates, sometimes known as eigenvalue rigidity estimates).

From the semicircle law (and the fundamental theorem of calculus), one expects the {i^{th}} eigenvalue spacing {\lambda_{i+1}(M_n)-\lambda_i(M_n)} to have an average size of {\frac{1}{\sqrt{n} \rho_{sc}(u)}}. It is thus natural to introduce the normalised eigenvalue spacing

\displaystyle  X_i := \frac{\lambda_{i+1}(M_n) - \lambda_i(M_n)}{1/\sqrt{n} \rho_{sc}(u)}

and ask what the distribution of {X_i} is.

As mentioned previously, we will focus on the bulk case {\delta n \leq i\leq (1-\delta)n}, and begin with the model case when {M_n} is drawn from GUE. (In the edge case when {i} is close to {1} or to {n}, the distribution is given by the famous Tracy-Widom law.) Here, the distribution was almost (but as we shall see, not quite) worked out by Gaudin and Mehta. By using the theory of determinantal processes, they were able to compute a quantity closely related to {X_i}, namely the probability

\displaystyle  {\bf P}( N_{[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]} = 0) \ \ \ \ \ (1)

that an interval {[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]} near {\sqrt{n} u} of length comparable to the expected eigenvalue spacing {1/\sqrt{n} \rho_{sc}(u)} is devoid of eigenvalues. For {u} in the bulk and fixed {x,y}, they showed that this probability is equal to

\displaystyle  \det( 1 - 1_{[x,y]} P 1_{[x,y]} ) + o(1),

where {P} is the Dyson projection

\displaystyle  P f(x) = \int_{\bf R} \frac{\sin(\pi(x-y))}{\pi(x-y)} f(y)\ dy

to Fourier modes in {[-1/2,1/2]}, and {\det} is the Fredholm determinant. As shown by Jimbo, Miwa, Tetsuji, Mori, and Sato, this determinant can also be expressed in terms of a solution to a Painleve V ODE, though we will not need this fact here. In view of this asymptotic and some standard integration by parts manipulations, it becomes plausible to propose that {X_i} will be asymptotically distributed according to the Gaudin-Mehta distribution {p(x)\ dx}, where

\displaystyle  p(x) := \frac{d^2}{dx^2} \det( 1 - 1_{[0,x]} P 1_{[0,x]} ).

A reasonably accurate approximation for {p} is given by the Wigner surmise {p(x) \approx \frac{1}{2} \pi x e^{-\pi x^2/4}}, which was presciently proposed by Wigner as early as 1957; it is exact for {n=2} but not in the asymptotic limit {n \rightarrow \infty}.

Unfortunately, when one tries to make this argument rigorous, one finds that the asymptotic for (1) does not control a single gap {X_i}, but rather an ensemble of gaps {X_i}, where {i} is drawn from an interval {[i_0 - L, i_0 + L]} of some moderate size {L} (e.g. {L = \log n}); see for instance this paper of Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou for a more precise formalisation of this statement (which is phrased slightly differently, in which one samples all gaps inside a fixed window of spectrum, rather than inside a fixed range of eigenvalue indices {i}). (This result is stated for GUE, but can be extended to other Wigner ensembles by the Four Moment Theorem, at least if one assumes a moment matching condition; see this previous paper with Van Vu for details. The moment condition can in fact be removed, as was done in this subsequent paper with Erdos, Ramirez, Schlein, Vu, and Yau.)

The problem is that when one specifies a given window of spectrum such as {[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]}, one cannot quite pin down in advance which eigenvalues {\lambda_i(M_n)} are going to lie to the left or right of this window; even with the strongest eigenvalue rigidity results available, there is a natural uncertainty of {\sqrt{\log n}} or so in the {i} index (as can be quantified quite precisely by this central limit theorem of Gustavsson).

The main difficulty here is that there could potentially be some strange coupling between the event (1) of an interval being devoid of eigenvalues, and the number {N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)} of eigenvalues to the left of that interval. For instance, one could conceive of a possible scenario in which the interval in (1) tends to have many eigenvalues when {N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)} is even, but very few when {N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)} is odd. In this sort of situation, the gaps {X_i} may have different behaviour for even {i} than for odd {i}, and such anomalies would not be picked up in the averaged statistics in which {i} is allowed to range over some moderately large interval.

The main result of the current paper is that these anomalies do not actually occur, and that all of the eigenvalue gaps {X_i} in the bulk are asymptotically governed by the Gaudin-Mehta law without the need for averaging in the {i} parameter. Again, this is shown first for GUE, and then extended to other Wigner matrices obeying a matching moment condition using the Four Moment Theorem. (It is likely that the moment matching condition can be removed here, but I was unable to achieve this, despite all the recent advances in establishing universality of local spectral statistics for Wigner matrices, mainly because the universality results in the literature are more focused on specific energy levels {u} than on specific eigenvalue indices {i}. To make matters worse, in some cases universality is currently known only after an additional averaging in the energy parameter.)

The main task in the proof is to show that the random variable {N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)} is largely decoupled from the event in (1) when {M_n} is drawn from GUE. To do this we use some of the theory of determinantal processes, and in particular the nice fact that when one conditions a determinantal process to the event that a certain spatial region (such as an interval) contains no points of the process, then one obtains a new determinantal process (with a kernel that is closely related to the original kernel). The main task is then to obtain a sufficiently good control on the distance between the new determinantal kernel and the old one, which we do by some functional-analytic considerations involving the manipulation of norms of operators (and specifically, the operator norm, Hilbert-Schmidt norm, and nuclear norm). Amusingly, the Fredholm alternative makes a key appearance, as I end up having to invert a compact perturbation of the identity at one point (specifically, I need to invert {1 - 1_{[x,y]}P1_{[x,y]}}, where {P} is the Dyson projection and {[x,y]} is an interval). As such, the bounds in my paper become ineffective, though I am sure that with more work one can invert this particular perturbation of the identity by hand, without the need to invoke the Fredholm alternative.

Van Vu and I have just uploaded to the arXiv our paper Random matrices: Sharp concentration of eigenvalues, submitted to the Electronic Journal of Probability. As with many of our previous papers, this paper is concerned with the distribution of the eigenvalues {\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)} of a random Wigner matrix {M_n} (such as a matrix drawn from the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE)). To simplify the discussion we shall mostly restrict attention to the bulk of the spectrum, i.e. to eigenvalues {\lambda_i(M_n)} with {\delta n \leq i \leq (1-\delta) i n} for some fixed {\delta>0}, although analogues of most of the results below have also been obtained at the edge of the spectrum.

If we normalise the entries of the matrix {M_n} to have mean zero and variance {1/n}, then in the asymptotic limit {n \rightarrow \infty}, we have the Wigner semicircle law, which asserts that the eigenvalues are asymptotically distributed according to the semicircular distribution {\rho_{sc}(x)\ dx}, where

\displaystyle  \rho_{sc}(x) := \frac{1}{2\pi} (4-x^2)_+^{1/2}.

An essentially equivalent way of saying this is that for large {n}, we expect the {i^{th}} eigenvalue {\lambda_i(M_n)} of {M_n} to stay close to the classical location {\gamma_i \in [-2,2]}, defined by the formula

\displaystyle  \int_{-2}^{\gamma_i} \rho_{sc}(x)\ dx = \frac{i}{n}.

In particular, from the Wigner semicircle law it can be shown that asymptotically almost surely, one has

\displaystyle  \lambda_i(M_n) = \gamma_i + o(1) \ \ \ \ \ (1)

for all {1 \leq i \leq n}.

In the modern study of the spectrum of Wigner matrices (and in particular as a key tool in establishing universality results), it has become of interest to improve the error term in (1) as much as possible. A typical early result in this direction was by Bai, who used the Stieltjes transform method to obtain polynomial convergence rates of the shape {O(n^{-c})} for some absolute constant {c>0}; see also the subsequent papers of Alon-Krivelevich-Vu and of of Meckes, who were able to obtain such convergence rates (with exponentially high probability) by using concentration of measure tools, such as Talagrand’s inequality. On the other hand, in the case of the GUE ensemble it is known (by this paper of Gustavsson) that {\lambda_i(M_n)} has variance comparable to {\frac{\log n}{n^2}} in the bulk, so that the optimal error term in (1) should be about {O(\log^{1/2} n/n)}. (One may think that if one wanted bounds on (1) that were uniform in {i}, one would need to enlarge the error term further, but this does not appear to be the case, due to strong correlations between the {\lambda_i}; note for instance this recent result of Ben Arous and Bourgarde that the largest gap between eigenvalues in the bulk is typically of order {O(\log^{1/2} n/n)}.)

A significant advance in this direction was achieved by Erdos, Schlein, and Yau in a series of papers where they used a combination of Stieltjes transform and concentration of measure methods to obtain local semicircle laws which showed, among other things, that one had asymptotics of the form

\displaystyle  N(I) = (1+o(1)) \int_I \rho_{sc}(x)\ dx

with exponentially high probability for intervals {I} in the bulk that were as short as {n^{-1+\epsilon}} for some {\epsilon>0}, where {N(I)} is the number of eigenvalues. These asymptotics are consistent with a good error term in (1), and are already sufficient for many applications, but do not quite imply a strong concentration result for individual eigenvalues {\lambda_i} (basically because they do not preclude long-range or “secular” shifts in the spectrum that involve large blocks of eigenvalues at mesoscopic scales). Nevertheless, this was rectified in a subsequent paper of Erdos, Yau, and Yin, which roughly speaking obtained a bound of the form

\displaystyle  \lambda_i(M_n) = \gamma_i + O( \frac{\log^{O(\log\log n)} n}{n} )

in the bulk with exponentially high probability, for Wigner matrices obeying some exponential decay conditions on the entries. This was achieved by a rather delicate high moment calculation, in which the contribution of the diagonal entries of the resolvent (whose average forms the Stieltjes transform) was shown to mostly cancel each other out.

As the GUE computations show, this concentration result is sharp up to the quasilogarithmic factor {\log^{O(\log\log n)} n}. The main result of this paper is to improve the concentration result to one more in line with the GUE case, namely

\displaystyle  \lambda_i(M_n) = \gamma_i + O( \frac{\log^{O(1)} n}{n} )

with exponentially high probability (see the paper for a more precise statement of results). The one catch is that an additional hypothesis is required, namely that the entries of the Wigner matrix have vanishing third moment. We also obtain similar results for the edge of the spectrum (but with a different scaling).

Our arguments are rather different from those of Erdos, Yau, and Yin, and thus provide an alternate approach to establishing eigenvalue concentration. The main tool is the Lindeberg exchange strategy, which is also used to prove the Four Moment Theorem (although we do not directly invoke the Four Moment Theorem in our analysis). The main novelty is that this exchange strategy is now used to establish large deviation estimates (i.e. exponentially small tail probabilities) rather than universality of the limiting distribution. Roughly speaking, the basic point is as follows. The Lindeberg exchange strategy seeks to compare a function {F(X_1,\ldots,X_n)} of many independent random variables {X_1,\ldots,X_n} with the same function {F(Y_1,\ldots,Y_n)} of a different set of random variables (which match moments with the original set of variables to some order, such as to second or fourth order) by exchanging the random variables one at a time. Typically, one tries to upper bound expressions such as

\displaystyle  {\bf E} \phi(F(X_1,\ldots,X_n)) - \phi(F(X_1,\ldots,X_{n-1},Y_n))

for various smooth test functions {\phi}, by performing a Taylor expansion in the variable being swapped and taking advantage of the matching moment hypotheses. In previous implementations of this strategy, {\phi} was a bounded test function, which allowed one to get control of the bulk of the distribution of {F(X_1,\ldots,X_n)}, and in particular in controlling probabilities such as

\displaystyle  {\bf P}( a \leq F(X_1,\ldots,X_n) \leq b )

for various thresholds {a} and {b}, but did not give good control on the tail as the error terms tended to be polynomially decaying in {n} rather than exponentially decaying. However, it turns out that one can modify the exchange strategy to deal with moments such as

\displaystyle  {\bf E} (1 + F(X_1,\ldots,X_n)^2)^k

for various moderately large {k} (e.g. of size comparable to {\log n}), obtaining results such as

\displaystyle  {\bf E} (1 + F(Y_1,\ldots,Y_n)^2)^k = (1+o(1)) {\bf E} (1 + F(X_1,\ldots,X_n)^2)^k

after performing all the relevant exchanges. As such, one can then use large deviation estimates on {F(X_1,\ldots,X_n)} to deduce large deviation estimates on {F(Y_1,\ldots,Y_n)}.

In this paper we also take advantage of a simplification, first noted by Erdos, Yau, and Yin, that Four Moment Theorems become somewhat easier to prove if one works with resolvents {(M_n-z)^{-1}} (and the closely related Stieltjes transform {s(z) := \frac{1}{n} \hbox{tr}( (M_n-z)^{-1} )}) rather than with individual eigenvalues, as the Taylor expansion of resolvents are very simple (essentially being a Neumann series). The relationship between the Stieltjes transform and the location of individual eigenvalues can be seen by taking advantage of the identity

\displaystyle  \frac{\pi}{2} - \frac{\pi}{n} N((-\infty,E)) = \int_0^\infty \hbox{Re} s(E + i \eta)\ d\eta

for any energy level {E \in {\bf R}}, which can be verified from elementary calculus. (In practice, we would truncate {\eta} near zero and near infinity to avoid some divergences, but this is a minor technicality.) As such, a concentration result for the Stieltjes transform can be used to establish an analogous concentration result for the eigenvalue counting functions {N((-\infty,E))}, which in turn can be used to deduce concentration results for individual eigenvalues {\lambda_i(M_n)} by some basic combinatorial manipulations.

Let {L: H \rightarrow H} be a self-adjoint operator on a finite-dimensional Hilbert space {H}. The behaviour of this operator can be completely described by the spectral theorem for finite-dimensional self-adjoint operators (i.e. Hermitian matrices, when viewed in coordinates), which provides a sequence {\lambda_1,\ldots,\lambda_n \in {\bf R}} of eigenvalues and an orthonormal basis {e_1,\ldots,e_n} of eigenfunctions such that {L e_i = \lambda_i e_i} for all {i=1,\ldots,n}. In particular, given any function {m: \sigma(L) \rightarrow {\bf C}} on the spectrum {\sigma(L) := \{ \lambda_1,\ldots,\lambda_n\}} of {L}, one can then define the linear operator {m(L): H \rightarrow H} by the formula

\displaystyle  m(L) e_i := m(\lambda_i) e_i,

which then gives a functional calculus, in the sense that the map {m \mapsto m(L)} is a {C^*}-algebra isometric homomorphism from the algebra {BC(\sigma(L) \rightarrow {\bf C})} of bounded continuous functions from {\sigma(L)} to {{\bf C}}, to the algebra {B(H \rightarrow H)} of bounded linear operators on {H}. Thus, for instance, one can define heat operators {e^{-tL}} for {t>0}, Schrödinger operators {e^{itL}} for {t \in {\bf R}}, resolvents {\frac{1}{L-z}} for {z \not \in \sigma(L)}, and (if {L} is positive) wave operators {e^{it\sqrt{L}}} for {t \in {\bf R}}. These will be bounded operators (and, in the case of the Schrödinger and wave operators, unitary operators, and in the case of the heat operators with {L} positive, they will be contractions). Among other things, this functional calculus can then be used to solve differential equations such as the heat equation

\displaystyle  u_t + Lu = 0; \quad u(0) = f \ \ \ \ \ (1)

the Schrödinger equation

\displaystyle  u_t + iLu = 0; \quad u(0) = f \ \ \ \ \ (2)

the wave equation

\displaystyle  u_{tt} + Lu = 0; \quad u(0) = f; \quad u_t(0) = g \ \ \ \ \ (3)

or the Helmholtz equation

\displaystyle  (L-z) u = f. \ \ \ \ \ (4)

The functional calculus can also be associated to a spectral measure. Indeed, for any vectors {f, g \in H}, there is a complex measure {\mu_{f,g}} on {\sigma(L)} with the property that

\displaystyle  \langle m(L) f, g \rangle_H = \int_{\sigma(L)} m(x) d\mu_{f,g}(x);

indeed, one can set {\mu_{f,g}} to be the discrete measure on {\sigma(L)} defined by the formula

\displaystyle  \mu_{f,g}(E) := \sum_{i: \lambda_i \in E} \langle f, e_i \rangle_H \langle e_i, g \rangle_H.

One can also view this complex measure as a coefficient

\displaystyle  \mu_{f,g} = \langle \mu f, g \rangle_H

of a projection-valued measure {\mu} on {\sigma(L)}, defined by setting

\displaystyle  \mu(E) f := \sum_{i: \lambda_i \in E} \langle f, e_i \rangle_H e_i.

Finally, one can view {L} as unitarily equivalent to a multiplication operator {M: f \mapsto g f} on {\ell^2(\{1,\ldots,n\})}, where {g} is the real-valued function {g(i) := \lambda_i}, and the intertwining map {U: \ell^2(\{1,\ldots,n\}) \rightarrow H} is given by

\displaystyle  U ( (c_i)_{i=1}^n ) := \sum_{i=1}^n c_i e_i,

so that {L = U M U^{-1}}.

It is an important fact in analysis that many of these above assertions extend to operators on an infinite-dimensional Hilbert space {H}, so long as one one is careful about what “self-adjoint operator” means; these facts are collectively referred to as the spectral theorem. For instance, it turns out that most of the above claims have analogues for bounded self-adjoint operators {L: H \rightarrow H}. However, in the theory of partial differential equations, one often needs to apply the spectral theorem to unbounded, densely defined linear operators {L: D \rightarrow H}, which (initially, at least), are only defined on a dense subspace {D} of the Hilbert space {H}. A very typical situation arises when {H = L^2(\Omega)} is the square-integrable functions on some domain or manifold {\Omega} (which may have a boundary or be otherwise “incomplete”), and {D = C^\infty_c(\Omega)} are the smooth compactly supported functions on {\Omega}, and {L} is some linear differential operator. It is then of interest to obtain the spectral theorem for such operators, so that one build operators such as {e^{-tL}, e^{itL}, \frac{1}{L-z}, e^{it\sqrt{L}}} or to solve equations such as (1), (2), (3), (4).

In order to do this, some necessary conditions on the densely defined operator {L: D \rightarrow H} must be imposed. The most obvious is that of symmetry, which asserts that

\displaystyle  \langle Lf, g \rangle_H = \langle f, Lg \rangle_H \ \ \ \ \ (5)

for all {f, g \in D}. In some applications, one also wants to impose positive definiteness, which asserts that

\displaystyle  \langle Lf, f \rangle_H \geq 0 \ \ \ \ \ (6)

for all {f \in D}. These hypotheses are sufficient in the case when {L} is bounded, and in particular when {H} is finite dimensional. However, as it turns out, for unbounded operators these conditions are not, by themselves, enough to obtain a good spectral theory. For instance, one consequence of the spectral theorem should be that the resolvents {(L-z)^{-1}} are well-defined for any strictly complex {z}, which by duality implies that the image of {L-z} should be dense in {H}. However, this can fail if one just assumes symmetry, or symmetry and positive definiteness. A well-known example occurs when {H} is the Hilbert space {H := L^2((0,1))}, {D := C^\infty_c((0,1))} is the space of test functions, and {L} is the one-dimensional Laplacian {L := -\frac{d^2}{dx^2}}. Then {L} is symmetric and positive, but the operator {L-k^2} does not have dense image for any complex {k}, since

\displaystyle  \langle (L-\overline{k}^2) f, e^{\overline{k}x} \rangle_H = 0

for all test functions {f \in C^\infty_c((0,1))}, as can be seen from a routine integration by parts. As such, the resolvent map is not everywhere uniquely defined. There is also a lack of uniqueness for the wave, heat, and Schrödinger equations for this operator (note that there are no spatial boundary conditions specified in these equations).

Another example occurs when {H := L^2((0,+\infty))}, {D := C^\infty_c((0,+\infty))}, {L} is the momentum operator {L := i \frac{d}{dx}}. Then the resolvent {(L-z)^{-1}} can be uniquely defined for {z} in the upper half-plane, but not in the lower half-plane, due to the obstruction

\displaystyle  \langle (L-z) f, e^{i \bar{z} x} \rangle_H = 0

for all test functions {f} (note that the function {e^{i\bar{z} x}} lies in {L^2((0,+\infty))} when {z} is in the lower half-plane). For related reasons, the translation operators {e^{itL}} have a problem with either uniqueness or existence (depending on whether {t} is positive or negative), due to the unspecified boundary behaviour at the origin.

The key property that lets one avoid this bad behaviour is that of essential self-adjointness. Once {L} is essentially self-adjoint, then spectral theorem becomes applicable again, leading to all the expected behaviour (e.g. existence and uniqueness for the various PDE given above).

Unfortunately, the concept of essential self-adjointness is defined rather abstractly, and is difficult to verify directly; unlike the symmetry condition (5) or the positive condition (6), it is not a “local” condition that can be easily verified just by testing {L} on various inputs, but is instead a more “global” condition. In practice, to verify this property, one needs to invoke one of a number of a partial converses to the spectral theorem, which roughly speaking asserts that if at least one of the expected consequences of the spectral theorem is true for some symmetric densely defined operator {L}, then {L} is self-adjoint. Examples of “expected consequences” include:

  • Existence of resolvents {(L-z)^{-1}} (or equivalently, dense image for {L-z});
  • Existence of a contractive heat propagator semigroup {e^{tL}} (in the positive case);
  • Existence of a unitary Schrödinger propagator group {e^{itL}};
  • Existence of a unitary wave propagator group {e^{it\sqrt{L}}} (in the positive case);
  • Existence of a “reasonable” functional calculus.
  • Unitary equivalence with a multiplication operator.

Thus, to actually verify essential self-adjointness of a differential operator, one typically has to first solve a PDE (such as the wave, Schrödinger, heat, or Helmholtz equation) by some non-spectral method (e.g. by a contraction mapping argument, or a perturbation argument based on an operator already known to be essentially self-adjoint). Once one can solve one of the PDEs, then one can apply one of the known converse spectral theorems to obtain essential self-adjointness, and then by the forward spectral theorem one can then solve all the other PDEs as well. But there is no getting out of that first step, which requires some input (typically of an ODE, PDE, or geometric nature) that is external to what abstract spectral theory can provide. For instance, if one wants to establish essential self-adjointness of the Laplace-Beltrami operator {L = -\Delta_g} on a smooth Riemannian manifold {(M,g)} (using {C^\infty_c(M)} as the domain space), it turns out (under reasonable regularity hypotheses) that essential self-adjointness is equivalent to geodesic completeness of the manifold, which is a global ODE condition rather than a local one: one needs geodesics to continue indefinitely in order to be able to (unitarily) solve PDEs such as the wave equation, which in turn leads to essential self-adjointness. (Note that the domains {(0,1)} and {(0,+\infty)} in the previous examples were not geodesically complete.) For this reason, essential self-adjointness of a differential operator is sometimes referred to as quantum completeness (with the completeness of the associated Hamilton-Jacobi flow then being the analogous classical completeness).

In these notes, I wanted to record (mostly for my own benefit) the forward and converse spectral theorems, and to verify essential self-adjointness of the Laplace-Beltrami operator on geodesically complete manifolds. This is extremely standard analysis (covered, for instance, in the texts of Reed and Simon), but I wanted to write it down myself to make sure that I really understood this foundational material properly.

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In the previous set of notes we saw how a representation-theoretic property of groups, namely Kazhdan’s property (T), could be used to demonstrate expansion in Cayley graphs. In this set of notes we discuss a different representation-theoretic property of groups, namely quasirandomness, which is also useful for demonstrating expansion in Cayley graphs, though in a somewhat different way to property (T). For instance, whereas property (T), being qualitative in nature, is only interesting for infinite groups such as {SL_d({\bf Z})} or {SL_d({\bf R})}, and only creates Cayley graphs after passing to a finite quotient, quasirandomness is a quantitative property which is directly applicable to finite groups, and is able to deduce expansion in a Cayley graph, provided that random walks in that graph are known to become sufficiently “flat” in a certain sense.

The definition of quasirandomness is easy enough to state:

Definition 1 (Quasirandom groups) Let {G} be a finite group, and let {D \geq 1}. We say that {G} is {D}-quasirandom if all non-trivial unitary representations {\rho: G \rightarrow U(H)} of {G} have dimension at least {D}. (Recall a representation is trivial if {\rho(g)} is the identity for all {g \in G}.)

Exercise 1 Let {G} be a finite group, and let {D \geq 1}. A unitary representation {\rho: G \rightarrow U(H)} is said to be irreducible if {H} has no {G}-invariant subspaces other than {\{0\}} and {H}. Show that {G} is {D}-quasirandom if and only if every non-trivial irreducible representation of {G} has dimension at least {D}.

Remark 1 The terminology “quasirandom group” was introduced explicitly (though with slightly different notational conventions) by Gowers in 2008 in his detailed study of the concept; the name arises because dense Cayley graphs in quasirandom groups are quasirandom graphs in the sense of Chung, Graham, and Wilson, as we shall see below. This property had already been used implicitly to construct expander graphs by Sarnak and Xue in 1991, and more recently by Gamburd in 2002 and by Bourgain and Gamburd in 2008. One can of course define quasirandomness for more general locally compact groups than the finite ones, but we will only need this concept in the finite case. (A paper of Kunze and Stein from 1960, for instance, exploits the quasirandomness properties of the locally compact group {SL_2({\bf R})} to obtain mixing estimates in that group.)

Quasirandomness behaves fairly well with respect to quotients and short exact sequences:

Exercise 2 Let {0 \rightarrow H \rightarrow G \rightarrow K \rightarrow 0} be a short exact sequence of finite groups {H,G,K}.

  • (i) If {G} is {D}-quasirandom, show that {K} is {D}-quasirandom also. (Equivalently: any quotient of a {D}-quasirandom finite group is again a {D}-quasirandom finite group.)
  • (ii) Conversely, if {H} and {K} are both {D}-quasirandom, show that {G} is {D}-quasirandom also. (In particular, the direct or semidirect product of two {D}-quasirandom finite groups is again a {D}-quasirandom finite group.)

Informally, we will call {G} quasirandom if it is {D}-quasirandom for some “large” {D}, though the precise meaning of “large” will depend on context. For applications to expansion in Cayley graphs, “large” will mean “{D \geq |G|^c} for some constant {c>0} independent of the size of {G}“, but other regimes of {D} are certainly of interest.

The way we have set things up, the trivial group {G = \{1\}} is infinitely quasirandom (i.e. it is {D}-quasirandom for every {D}). This is however a degenerate case and will not be discussed further here. In the non-trivial case, a finite group can only be quasirandom if it is large and has no large subgroups:

Exercise 3 Let {D \geq 1}, and let {G} be a finite {D}-quasirandom group.

  • (i) Show that if {G} is non-trivial, then {|G| \geq D+1}. (Hint: use the mean zero component {\tau\downharpoonright_{\ell^2(G)_0}} of the regular representation {\tau: G \rightarrow U(\ell^2(G))}.) In particular, non-trivial finite groups cannot be infinitely quasirandom.
  • (ii) Show that any proper subgroup {H} of {G} has index {[G:H] \geq D+1}. (Hint: use the mean zero component of the quasiregular representation.)

The following exercise shows that quasirandom groups have to be quite non-abelian, and in particular perfect:

Exercise 4 (Quasirandomness, abelianness, and perfection) Let {G} be a finite group.

  • (i) If {G} is abelian and non-trivial, show that {G} is not {2}-quasirandom. (Hint: use Fourier analysis or the classification of finite abelian groups.)
  • (ii) Show that {G} is {2}-quasirandom if and only if it is perfect, i.e. the commutator group {[G,G]} is equal to {G}. (Equivalently, {G} is {2}-quasirandom if and only if it has no non-trivial abelian quotients.)

Later on we shall see that there is a converse to the above two exercises; any non-trivial perfect finite group with no large subgroups will be quasirandom.

Exercise 5 Let {G} be a finite {D}-quasirandom group. Show that for any subgroup {G'} of {G}, {G'} is {D/[G:G']}-quasirandom, where {[G:G'] := |G|/|G'|} is the index of {G'} in {G}. (Hint: use induced representations.)

Now we give an example of a more quasirandom group.

Lemma 2 (Frobenius lemma) If {F_p} is a field of some prime order {p}, then {SL_2(F_p)} is {\frac{p-1}{2}}-quasirandom.

This should be compared with the cardinality {|SL_2(F_p)|} of the special linear group, which is easily computed to be {(p^2-1) \times p = p^3 - p}.

Proof: We may of course take {p} to be odd. Suppose for contradiction that we have a non-trivial representation {\rho: SL_2(F_p) \rightarrow U_d({\bf C})} on a unitary group of some dimension {d} with {d < \frac{p-1}{2}}. Set {a} to be the group element

\displaystyle  a := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},

and suppose first that {\rho(a)} is non-trivial. Since {a^p=1}, we have {\rho(a)^p=1}; thus all the eigenvalues of {\rho(a)} are {p^{th}} roots of unity. On the other hand, by conjugating {a} by diagonal matrices in {SL_2(F_p)}, we see that {a} is conjugate to {a^m} (and hence {\rho(a)} conjugate to {\rho(a)^m}) whenever {m} is a quadratic residue mod {p}. As such, the eigenvalues of {\rho(a)} must be permuted by the operation {x \mapsto x^m} for any quadratic residue mod {p}. Since {\rho(a)} has at least one non-trivial eigenvalue, and there are {\frac{p-1}{2}} distinct quadratic residues, we conclude that {\rho(a)} has at least {\frac{p-1}{2}} distinct eigenvalues. But {\rho(a)} is a {d \times d} matrix with {d < \frac{p-1}{2}}, a contradiction. Thus {a} lies in the kernel of {\rho}. By conjugation, we then see that this kernel contains all unipotent matrices. But these matrices generate {SL_2(F_p)} (see exercise below), and so {\rho} is trivial, a contradiction. \Box

Exercise 6 Show that for any prime {p}, the unipotent matrices

\displaystyle  \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}

for {t} ranging over {F_p} generate {SL_2(F_p)} as a group.

Exercise 7 Let {G} be a finite group, and let {D \geq 1}. If {G} is generated by a collection {G_1,\ldots,G_k} of {D}-quasirandom subgroups, show that {G} is itself {D}-quasirandom.

Exercise 8 Show that {SL_d(F_p)} is {\frac{p-1}{2}}-quasirandom for any {d \geq 2} and any prime {p}. (This is not sharp; the optimal bound here is {\gg_d p^{d-1}}, which follows from the results of Landazuri and Seitz.)

As a corollary of the above results and Exercise 2, we see that the projective special linear group {PSL_d(F_p)} is also {\frac{p-1}{2}}-quasirandom.

Remark 2 One can ask whether the bound {\frac{p-1}{2}} in Lemma 2 is sharp, assuming of course that {p} is odd. Noting that {SL_2(F_p)} acts linearly on the plane {F_p^2}, we see that it also acts projectively on the projective line {PF_p^1 := (F_p^2 \backslash \{0\}) / F_p^\times}, which has {p+1} elements. Thus {SL_2(F_p)} acts via the quasiregular representation on the {p+1}-dimensional space {\ell^2(PF_p^1)}, and also on the {p}-dimensional subspace {\ell^2(PF_p^1)_0}; this latter representation (known as the Steinberg representation) is irreducible. This shows that the {\frac{p-1}{2}} bound cannot be improved beyond {p}. More generally, given any character {\chi: F_p^\times \rightarrow S^1}, {SL_2(F_p)} acts on the {p+1}-dimensional space {V_\chi} of functions {f \in \ell^2( F_p^2 \backslash \{0\} )} that obey the twisted dilation invariance {f(tx) = \chi(t) f(x)} for all {t \in F_p^\times} and {x \in F_p^2 \backslash \{0\}}; these are known as the principal series representations. When {\chi} is the trivial character, this is the quasiregular representation discussed earlier. For most other characters, this is an irreducible representation, but it turns out that when {\chi} is the quadratic representation (thus taking values in {\{-1,+1\}} while being non-trivial), the principal series representation splits into the direct sum of two {\frac{p+1}{2}}-dimensional representations, which comes very close to matching the bound in Lemma 2. There is a parallel series of representations to the principal series (known as the discrete series) which is more complicated to describe (roughly speaking, one has to embed {F_p} in a quadratic extension {F_{p^2}} and then use a rotated version of the above construction, to change a split torus into a non-split torus), but can generate irreducible representations of dimension {\frac{p-1}{2}}, showing that the bound in Lemma 2 is in fact exactly sharp. These constructions can be generalised to arbitrary finite groups of Lie type using Deligne-Luzstig theory, but this is beyond the scope of this course (and of my own knowledge in the subject).

Exercise 9 Let {p} be an odd prime. Show that for any {n \geq p+2}, the alternating group {A_n} is {p-1}-quasirandom. (Hint: show that all cycles of order {p} in {A_n} are conjugate to each other in {A_n} (and not just in {S_n}); in particular, a cycle is conjugate to its {j^{th}} power for all {j=1,\ldots,p-1}. Also, as {n \geq 5}, {A_n} is simple, and so the cycles of order {p} generate the entire group.)

Remark 3 By using more precise information on the representations of the alternating group (using the theory of Specht modules and Young tableaux), one can show the slightly sharper statement that {A_n} is {n-1}-quasirandom for {n \geq 6} (but is only {3}-quasirandom for {n=5} due to icosahedral symmetry, and {1}-quasirandom for {n \leq 4} due to lack of perfectness). Using Exercise 3 with the index {n} subgroup {A_{n-1}}, we see that the bound {n-1} cannot be improved. Thus, {A_n} (for large {n}) is not as quasirandom as the special linear groups {SL_d(F_p)} (for {p} large and {d} bounded), because in the latter case the quasirandomness is as strong as a power of the size of the group, whereas in the former case it is only logarithmic in size.

If one replaces the alternating group {A_n} with the slightly larger symmetric group {S_n}, then quasirandomness is destroyed (since {S_n}, having the abelian quotient {S_n/A_n}, is not perfect); indeed, {S_n} is {1}-quasirandom and no better.

Remark 4 Thanks to the monumental achievement of the classification of finite simple groups, we know that apart from a finite number (26, to be precise) of sporadic exceptions, all finite simple groups (up to isomorphism) are either a cyclic group {{\bf Z}/p{\bf Z}}, an alternating group {A_n}, or is a finite simple group of Lie type such as {PSL_d(F_p)}. (We will define the concept of a finite simple group of Lie type more precisely in later notes, but suffice to say for now that such groups are constructed from reductive algebraic groups, for instance {PSL_d(F_p)} is constructed from {SL_d} in characteristic {p}.) In the case of finite simple groups {G} of Lie type with bounded rank {r=O(1)}, it is known from the work of Landazuri and Seitz that such groups are {\gg |G|^c}-quasirandom for some {c>0} depending only on the rank. On the other hand, by the previous remark, the large alternating groups do not have this property, and one can show that the finite simple groups of Lie type with large rank also do not have this property. Thus, we see using the classification that if a finite simple group {G} is {|G|^c}-quasirandom for some {c>0} and {|G|} is sufficiently large depending on {c}, then {G} is a finite simple group of Lie type with rank {O_c(1)}. It would be of interest to see if there was an alternate way to establish this fact that did not rely on the classification, as it may lead to an alternate approach to proving the classification (or perhaps a weakened version thereof).

A key reason why quasirandomness is desirable for the purposes of demonstrating expansion is that quasirandom groups happen to be rapidly mixing at large scales, as we shall see below the fold. As such, quasirandomness is an important tool for demonstrating expansion in Cayley graphs, though because expansion is a phenomenon that must hold at all scales, one needs to supplement quasirandomness with some additional input that creates mixing at small or medium scales also before one can deduce expansion. As an example of this technique of combining quasirandomness with mixing at small and medium scales, we present a proof (due to Sarnak-Xue, and simplified by Gamburd) of a weak version of the famous “3/16 theorem” of Selberg on the least non-trivial eigenvalue of the Laplacian on a modular curve, which among other things can be used to construct a family of expander Cayley graphs in {SL_2({\bf Z}/N{\bf Z})} (compare this with the property (T)-based methods in the previous notes, which could construct expander Cayley graphs in {SL_d({\bf Z}/N{\bf Z})} for any fixed {d \geq 3}).

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