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Our study of random matrices, to date, has focused on somewhat general ensembles, such as iid random matrices or Wigner random matrices, in which the distribution of the individual entries of the matrices was essentially arbitrary (as long as certain moments, such as the mean and variance, were normalised). In these notes, we now focus on two much more special, and much more symmetric, ensembles:

  • The Gaussian Unitary Ensemble (GUE), which is an ensemble of random {n \times n} Hermitian matrices {M_n} in which the upper-triangular entries are iid with distribution {N(0,1)_{\bf C}}, and the diagonal entries are iid with distribution {N(0,1)_{\bf R}}, and independent of the upper-triangular ones; and
  • The Gaussian random matrix ensemble, which is an ensemble of random {n \times n} (non-Hermitian) matrices {M_n} whose entries are iid with distribution {N(0,1)_{\bf C}}.

The symmetric nature of these ensembles will allow us to compute the spectral distribution by exact algebraic means, revealing a surprising connection with orthogonal polynomials and with determinantal processes. This will, for instance, recover the semi-circular law for GUE, but will also reveal fine spacing information, such as the distribution of the gap between adjacent eigenvalues, which is largely out of reach of tools such as the Stieltjes transform method and the moment method (although the moment method, with some effort, is able to control the extreme edges of the spectrum).

Similarly, we will see for the first time the circular law for eigenvalues of non-Hermitian matrices.

There are a number of other highly symmetric ensembles which can also be treated by the same methods, most notably the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). However, for simplicity we shall focus just on the above two ensembles. For a systematic treatment of these ensembles, see the text by Deift.

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When solving the initial value problem to an ordinary differential equation, such as

\displaystyle  \partial_t u = F(u); \quad u(0) = u_0, \ \ \ \ \ (1)

where {u: {\bf R} \rightarrow V} is the unknown solution (taking values in some finite-dimensional vector space {V}), {u_0 \in V} is the initial datum, and {F: V \rightarrow V} is some nonlinear function (which we will take to be smooth for sake of argument), then one can construct a solution locally in time via the Picard iteration method. There are two basic ideas. The first is to use the fundamental theorem of calculus to rewrite the initial value problem (1) as the problem of solving an integral equation,

\displaystyle  u(t) = u_0 + \int_0^t F(u(s))\ ds. \ \ \ \ \ (2)

The second idea is to solve this integral equation by the contraction mapping theorem, showing that the integral operator {{\mathcal N}} defined by

\displaystyle  {\mathcal N}(u) (t) := u_0 + \int_0^t F(u(s))\ ds

is a contraction on a suitable complete metric space (e.g. a closed ball in the function space {C^0([0,T]; V)}), and thus has a unique fixed point in this space. This method works as long as one only seeks to construct local solutions (for time {t} in {[0,T]} for sufficiently small {T>0}), but the solutions constructed have a number of very good properties, including

  • Existence: A solution {u} exists in the space {C^0([0,T];V)} (and even in {C^\infty([0,T];V)}) for {T} sufficiently small.
  • Uniqueness: There is at most one solution {u} to the initial value problem in the space {C^0([0,T];V)} (or in smoother spaces, such as {C^\infty([0,T];V)}). (For solutions in the weaker space {C^0([0,T];V)} we use the integral formulation (2) to define the solution concept.)
  • Lipschitz continuous dependence on the data: If {u_0^{(n)}} is a sequence of initial data converging to {u_0}, then the associated solutions {u^{(n)}} converge uniformly to {u} on {[0,T]} (possibly after shrinking {T} slightly). In fact we have the Lipschitz bound {\| u^{(n)}(t) - u(t) \|_V \leq C \| u^{(n)}_0 - u_0 \|_V} for {n} large enough and {t \in [0,T]}, where {C} is an absolute constant.

This package of properties is referred to as (Lipschitz) wellposedness.

This method extends to certain partial differential equations, particularly those of a semilinear nature (linear except for lower order nonlinear terms). For instance, if trying to solve an initial value problem of the form

\displaystyle  \partial_t u + Lu = F(u); \quad u(0,x) = u_0(x),

where now {u: {\bf R} \rightarrow V} takes values in a function space {V} (e.g. a Sobolev space {H^k({\bf R}^d)}), {u_0 \in V} is an initial datum, {L} is some (differential) operator (independent of {u}) that is (densely) defined on {V}, and {F} is a nonlinearity which is also (densely) defined on {V}, then (formally, at least) one can solve this problem by using Duhamel’s formula to convert the problem to that of solving an integral equation

\displaystyle  u(t) = e^{-tL} u_0 + \int_0^t e^{-(t-s)L} F(u(s))\ ds

and one can then hope to show that the associated nonlinear integral operator

\displaystyle  u \mapsto e^{-tL} u_0 + \int_0^t e^{-(t-s)L} F(u(s))\ ds

is a contraction in a subset of a suitably chosen function space.

This method turns out to work surprisingly well for many semilinear partial differential equations, and in particular for semilinear parabolic, semilinear dispersive, and semilinear wave equations. As in the ODE case, when the method works, it usually gives the entire package of Lipschitz well-posedness: existence, uniqueness, and Lipschitz continuous dependence on the initial data, for short times at least.

However, when one moves from semilinear initial value problems to quasilinear initial value problems such as

\displaystyle  \partial_t u + L_u u = F(u); \quad u(0,x) = u_0(x)

in which the top order operator {L_u} now depends on the solution {u} itself, then the nature of well-posedness changes; one can still hope to obtain (local) existence and uniqueness, and even continuous dependence on the data, but one usually is forced to give up Lipschitz continuous dependence at the highest available regularity (though one can often recover it at lower regularities). As a consequence, the Picard iteration method is not directly suitable for constructing solutions to such equations.

One can already see this phenomenon with a very simple equation, namely the one-dimensional constant-velocity transport equation

\displaystyle  \partial_t u + c \partial_x u = 0; \quad u(0,x) = u_0(x) \ \ \ \ \ (3)

where we consider {c = c_0} as part of the initial data. (If one wishes, one could view this equation as a rather trivial example of a system.

\displaystyle  \partial_t u + c \partial_x u = 0; \quad \partial_t c = 0

\displaystyle  u(0,x) = u_0(x); \quad c(0) = c_0,

to emphasis this viewpoint, but this would be somewhat idiosyncratic.) One can solve this equation explicitly of course to get the solution

\displaystyle  u(t,x) = u_0(x-ct).

In particular, if we look at the solution just at time {t=1} for simplicity, we have

\displaystyle  u(1,x) = u_0(x-c).

Now let us see how this solution {u(1,x)} depends on the parameter {c}. One can ask whether this dependence is Lipschitz in {c}, in some function space {V}:

\displaystyle  \| u_0(\cdot - c) - u_0(\cdot - c') \|_V \leq A |c-c'|

for some finite {A}. But using the Newton approximation

\displaystyle  u_0(\cdot - c) - u_0(\cdot - c') \approx (c-c') \partial_x u_0(\cdot - c)

we see that we should only expect such a bound when {\partial_x u_0} (and its translates) lie in {V}. Thus, we see a loss of derivatives phenomenon with regard to Lipschitz well-posedness; if the initial data {u_0} is in some regularity space, say {C^3}, then one only obtains Lipschitz dependence on {c} in a lower regularity space such as {C^2}.

We have just seen that if all one knows about the initial data {u_0} is that it is bounded in a function space {V}, then one usually cannot hope to make the dependence of {u} on the velocity parameter {c} Lipschitz continuous. Indeed, one cannot even make it continuous uniformly in {V}. Given two values of {c} that are close together, e.g. {c = 0} and {c=\epsilon}, and a reasonable function space {V} (e.g. a Sobolev space {H^k}, or a classical regularity space {C^k}) one can easily cook up a function {u_0} that is bounded in {V} but whose two solutions {u_0(\cdot)} and {u_0(\cdot-\epsilon)} separate in the {V} norm at time {1}, simply by choosing {u_0} to be supported on an interval of width {\epsilon}.

(Part of the problem here is that using a subtractive method {\|u-v\|_V} to determine the distance between two solutions {u, v} is not a physically natural operation when transport mechanisms are present that could cause the key features of {u, v} (such as singularities) to be situated in slightly different locations. In such cases, the correct notion of distance may need to take transport into account, e.g. by using metrics of Wasserstein type.)

On the other hand, one still has non-uniform continuous dependence on the initial parameters: if {u_0} lies in some reasonable function space {V}, then the map {c \mapsto u_0(\cdot-c)} is continuous in the {V} topology, even if it is not uniformly continuous with respect to {v_0}. (More succinctly: translation is a continuous but not uniformly continuous operation in most function spaces.) The reason for this is that we already have established this continuity in the case when {u_0} is so smooth that an additional derivative of {u_0} lies in {V}; and such smooth functions tend to be dense in the original space {V}, so the general case can then be established by a limiting argument, approximating a general function in {V} by a smoother function. We then see that the non-uniformity ultimately comes from the fact that a given function in {V} may be arbitrarily rough (or concentrated at an arbitrarily fine scale), and so the ability to approximate such a function by a smooth one can be arbitrarily poor.

In many quasilinear PDE, one often encounters qualitatively similar phenomena. Namely, one often has local well-posedness in sufficiently smooth function spaces {V} (so that if the initial data lies in {V}, then for short times one has existence, uniqueness, and continuous dependence on the data in the {V} topology), but Lipschitz or uniform continuity in the {V} topology is usually false. However, if the data (and solution) is known to be in a high-regularity function space {V}, one can often recover Lipschitz or uniform continuity in a lower-regularity topology.

Because the continuous dependence on the data in quasilinear equations is necessarily non-uniform, the arguments needed to establish this dependence can be remarkably delicate. As with the simple example of the transport equation, the key is to approximate a rough solution by a smooth solution first, by smoothing out the data (this is the non-uniform step, as it depends on the physical scale (or wavelength) that the data features are located). But for quasilinear equations, keeping the rough and smooth solution together can require a little juggling of function space norms, in particular playing the low-frequency nature of the smooth solution against the high-frequency nature of the residual between the rough and smooth solutions.

Below the fold I will illustrate this phenomenon with one of the simplest quasilinear equations, namely the initial value problem for the inviscid Burgers’ equation

\displaystyle  \partial_t u + u u_x = 0; \quad u(0,x) = u_0(x) \ \ \ \ \ (4)

which is a modification of the transport equation (3) in which the velocity {c} is no longer a parameter, but now depends (and is, in this case, actually equal to) the solution. To avoid technicalities we will work only with the classical function spaces {C^k} of {k} times continuously differentiable functions, though one can certainly work with other spaces (such as Sobolev spaces) by exploiting the Sobolev embedding theorem. To avoid having to distinguish continuity from uniform continuity, we shall work in a compact domain by assuming periodicity in space, thus for instance restricting {x} to the unit circle {{\bf R}/{\bf Z}}.

This discussion is inspired by this survey article of Nikolay Tzvetkov, which further explores the distinction between well-posedness and ill-posedness in both semilinear and quasilinear settings.

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A celebrated theorem of Gromov reads:

Theorem 1 Every finitely generated group of polynomial growth is virtually nilpotent.

The original proof of Gromov’s theorem was quite non-elementary, using an infinitary limit and exploiting the work surrounding the solution to Hilbert’s fifth problem. More recently, Kleiner provided a proof which was more elementary (based in large part on an earlier paper of Colding and Minicozzi), though still not entirely so, relying in part on (a weak form of the) Tits alternative and also on an ultrafilter argument of Korevaar-Schoen and Mok. I discuss Kleiner’s argument more in this previous blog post.

Recently, Yehuda Shalom and I established a quantitative version of Gromov’s theorem by making every component of Kleiner’s argument finitary. Technically, this provides a fully elementary proof of Gromov’s theorem (we do use one infinitary limit to simplify the argument a little bit, but this is not truly necessary); however, because we were trying to quantify as much of the result as possible, the argument became quite lengthy.

In this note I want to record a short version of the argument of Yehuda and myself which is not quantitative, but gives a self-contained and largely elementary proof of Gromov’s theorem. The argument is not too far from the Kleiner argument, but has a number of simplifications at various places. In a number of places, there was a choice to take between a short argument that was “inefficient” in the sense that it did not lead to a good quantitative bound, and a lengthier argument which led to better quantitative bounds. I have opted for the former in all such cases.

Yehuda and I plan to write a short paper containing this argument as well as some additional material, but due to some interest in this particular proof, we are detailing it here on this blog in advance of our paper.

Note: this post will assume familiarity with the basic terminology of group theory, and will move somewhat quickly through the technical details.

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Ben Green, and I have just uploaded to the arXiv a short (six-page) paper “Yet another proof of Szemeredi’s theorem“, submitted to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we put in print a folklore observation, namely that the inverse conjecture for the Gowers norm, together with the density increment argument, easily implies Szemerédi’s famous theorem on arithmetic progressions. This is unsurprising, given that Gowers’ proof of Szemerédi’s theorem proceeds through a weaker version of the inverse conjecture and a density increment argument, and also given that it is possible to derive Szemerédi’s theorem from knowledge of the characteristic factor for multiple recurrence (the ergodic theory analogue of the inverse conjecture, first established by Host and Kra), as was done by Bergelson, Leibman, and Lesigne (and also implicitly in the earlier paper of Bergelson, Host, and Kra); but to our knowledge the exact derivation of Szemerédi’s theorem from the inverse conjecture was not in the literature. Ordinarily this type of folklore might be considered too trifling (and too well known among experts in the field) to publish; but we felt that the venue of the Szemerédi birthday conference provided a natural venue for this particular observation.

The key point is that one can show (by an elementary argument relying primarily an induction on dimension argument and the Weyl recurrence theorem, i.e. that given any real {\alpha} and any integer {s \geq 1}, that the expression {\alpha n^s} gets arbitrarily close to an integer) that given a (polynomial) nilsequence {n \mapsto F(g(n)\Gamma)}, one can subdivide any long arithmetic progression (such as {[N] = \{1,\ldots,N\}}) into a number of medium-sized progressions, where the nilsequence is nearly constant on each progression. As a consequence of this and the inverse conjecture for the Gowers norm, if a set has no arithmetic progressions, then it must have an elevated density on a subprogression; iterating this observation as per the usual density-increment argument as introduced long ago by Roth, one obtains the claim. (This is very close to the scheme of Gowers’ proof.)

Technically, one might call this the shortest proof of Szemerédi’s theorem in the literature (and would be something like the sixteenth such genuinely distinct proof, by our count), but that would be cheating quite a bit, primarily due to the fact that it assumes the inverse conjecture for the Gowers norm, our current proof of which is checking in at about 100 pages…

Ben Green, and I have just uploaded to the arXiv our paper “An arithmetic regularity lemma, an associated counting lemma, and applications“, submitted (a little behind schedule) to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we describe the general-degree version of the arithmetic regularity lemma, which can be viewed as the counterpart of the Szemerédi regularity lemma, in which the object being regularised is a function {f: [N] \rightarrow [0,1]} on a discrete interval {[N] = \{1,\ldots,N\}} rather than a graph, and the type of patterns one wishes to count are additive patterns (such as arithmetic progressions {n,n+d,\ldots,n+(k-1)d}) rather than subgraphs. Very roughly speaking, this regularity lemma asserts that all such functions can be decomposed as a degree {\leq s} nilsequence (or more precisely, a variant of a nilsequence that we call an virtual irrational nilsequence), plus a small error, plus a third error which is extremely tiny in the Gowers uniformity norm {U^{s+1}[N]}. In principle, at least, the latter two errors can be readily discarded in applications, so that the regularity lemma reduces many questions in additive combinatorics to questions concerning (virtual irrational) nilsequences. To work with these nilsequences, we also establish a arithmetic counting lemma that gives an integral formula for counting additive patterns weighted by such nilsequences.

The regularity lemma is a manifestation of the “dichotomy between structure and randomness”, as discussed for instance in my ICM article or FOCS article. In the degree {1} case {s=1}, this result is essentially due to Green. It is powered by the inverse conjecture for the Gowers norms, which we and Tamar Ziegler have recently established (paper to be forthcoming shortly; the {k=4} case of our argument is discussed here). The counting lemma is established through the quantitative equidistribution theory of nilmanifolds, which Ben and I set out in this paper.

The regularity and counting lemmas are designed to be used together, and in the paper we give three applications of this combination. Firstly, we give a new proof of Szemerédi’s theorem, which proceeds via an energy increment argument rather than a density increment one. Secondly, we establish a conjecture of Bergelson, Host, and Kra, namely that if {A \subset [N]} has density {\alpha}, and {\epsilon > 0}, then there exist {\gg_{\alpha,\epsilon} N} shifts {h} for which {A} contains at least {(\alpha^4 - \epsilon)N} arithmetic progressions of length {k=4} of spacing {h}. (The {k=3} case of this conjecture was established earlier by Green; the {k=5} case is false, as was shown by Ruzsa in an appendix to the Bergelson-Host-Kra paper.) Thirdly, we establish a variant of a recent result of Gowers-Wolf, showing that the true complexity of a system of linear forms over {[N]} indeed matches the conjectured value predicted in their first paper.

In all three applications, the scheme of proof can be described as follows:

  • Apply the arithmetic regularity lemma, and decompose a relevant function {f} into three pieces, {f_{nil}, f_{sml}, f_{unf}}.
  • The uniform part {f_{unf}} is so tiny in the Gowers uniformity norm that its contribution can be easily dealt with by an appropriate “generalised von Neumann theorem”.
  • The contribution of the (virtual, irrational) nilsequence {f_{nil}} can be controlled using the arithmetic counting lemma.
  • Finally, one needs to check that the contribution of the small error {f_{sml}} does not overwhelm the main term {f_{nil}}. This is the trickiest bit; one often needs to use the counting lemma again to show that one can find a set of arithmetic patterns for {f_{nil}} that is so sufficiently “equidistributed” that it is not impacted by the small error.

To illustrate the last point, let us give the following example. Suppose we have a set {A \subset [N]} of some positive density (say {|A| = 10^{-1} N}) and we have managed to prove that {A} contains a reasonable number of arithmetic progressions of length {5} (say), e.g. it contains at least {10^{-10} N^2} such progressions. Now we perturb {A} by deleting a small number, say {10^{-2} N}, elements from {A} to create a new set {A'}. Can we still conclude that the new set {A'} contains any arithmetic progressions of length {5}?

Unfortunately, the answer could be no; conceivably, all of the {10^{-10} N^2} arithmetic progressions in {A} could be wiped out by the {10^{-2} N} elements removed from {A}, since each such element of {A} could be associated with up to {|A|} (or even {5|A|}) arithmetic progressions in {A}.

But suppose we knew that the {10^{-10} N^2} arithmetic progressions in {A} were equidistributed, in the sense that each element in {A} belonged to the same number of such arithmetic progressions, namely {5 \times 10^{-9} N}. Then each element deleted from {A} only removes at most {5 \times 10^{-9} N} progressions, and so one can safely remove {10^{-2} N} elements from {A} and still retain some arithmetic progressions. The same argument works if the arithmetic progressions are only approximately equidistributed, in the sense that the number of progressions that a given element {a \in A} belongs to concentrates sharply around its mean (for instance, by having a small variance), provided that the equidistribution is sufficiently strong. Fortunately, the arithmetic regularity and counting lemmas are designed to give precisely such a strong equidistribution result.

A succinct (but slightly inaccurate) summation of the regularity+counting lemma strategy would be that in order to solve a problem in additive combinatorics, it “suffices to check it for nilsequences”. But this should come with a caveat, due to the issue of the small error above; in addition to checking it for nilsequences, the answer in the nilsequence case must be sufficiently “dispersed” in a suitable sense, so that it can survive the addition of a small (but not completely negligible) perturbation.

One last “production note”. Like our previous paper with Emmanuel Breuillard, we used Subversion to write this paper, which turned out to be a significant efficiency boost as we could work on different parts of the paper simultaneously (this was particularly important this time round as the paper was somewhat lengthy and complicated, and there was a submission deadline). When doing so, we found it convenient to split the paper into a dozen or so pieces (one for each section of the paper, basically) in order to avoid conflicts, and to help coordinate the writing process. I’m also looking into git (a more advanced version control system), and am planning to use it for another of my joint projects; I hope to be able to comment on the relative strengths of these systems (and with plain old email) in the future.

As an experiment, I’ve recently started using Google Buzz as an outlet for various things I wanted to say or share, but which were too insubstantial to merit a mention on this blog. (In turn, one of the reasons of starting this blog was to share various bits of mathematics which were too insubstantial for a published paper.  Presumably the process becomes degenerate if iterated any further…)    I don’t know how frequently I will be updating, though.

In the foundations of modern probability, as laid out by Kolmogorov, the basic objects of study are constructed in the following order:

  1. Firstly, one selects a sample space {\Omega}, whose elements {\omega} represent all the possible states that one’s stochastic system could be in.
  2. Then, one selects a {\sigma}-algebra {{\mathcal B}} of events {E} (modeled by subsets of {\Omega}), and assigns each of these events a probability {{\bf P}(E) \in [0,1]} in a countably additive manner, so that the entire sample space has probability {1}.
  3. Finally, one builds (commutative) algebras of random variables {X} (such as complex-valued random variables, modeled by measurable functions from {\Omega} to {{\bf C}}), and (assuming suitable integrability or moment conditions) one can assign expectations {\mathop{\bf E} X} to each such random variable.

In measure theory, the underlying measure space {\Omega} plays a prominent foundational role, with the measurable sets and measurable functions (the analogues of the events and the random variables) always being viewed as somehow being attached to that space. In probability theory, in contrast, it is the events and their probabilities that are viewed as being fundamental, with the sample space {\Omega} being abstracted away as much as possible, and with the random variables and expectations being viewed as derived concepts. See Notes 0 for further discussion of this philosophy.

However, it is possible to take the abstraction process one step further, and view the algebra of random variables and their expectations as being the foundational concept, and ignoring both the presence of the original sample space, the algebra of events, or the probability measure.

There are two reasons for wanting to shed (or abstract away) these previously foundational structures. Firstly, it allows one to more easily take certain types of limits, such as the large {n} limit {n \rightarrow \infty} when considering {n \times n} random matrices, because quantities built from the algebra of random variables and their expectations, such as the normalised moments of random matrices tend to be quite stable in the large {n} limit (as we have seen in previous notes), even as the sample space and event space varies with {n}. (This theme of using abstraction to facilitate the taking of the large {n} limit also shows up in the application of ergodic theory to combinatorics via the correspondence principle; see this previous blog post for further discussion.)

Secondly, this abstract formalism allows one to generalise the classical, commutative theory of probability to the more general theory of non-commutative probability theory, which does not have a classical underlying sample space or event space, but is instead built upon a (possibly) non-commutative algebra of random variables (or “observables”) and their expectations (or “traces”). This more general formalism not only encompasses classical probability, but also spectral theory (with matrices or operators taking the role of random variables, and the trace taking the role of expectation), random matrix theory (which can be viewed as a natural blend of classical probability and spectral theory), and quantum mechanics (with physical observables taking the role of random variables, and their expected value on a given quantum state being the expectation). It is also part of a more general “non-commutative way of thinking” (of which non-commutative geometry is the most prominent example), in which a space is understood primarily in terms of the ring or algebra of functions (or function-like objects, such as sections of bundles) placed on top of that space, and then the space itself is largely abstracted away in order to allow the algebraic structures to become less commutative. In short, the idea is to make algebra the foundation of the theory, as opposed to other possible choices of foundations such as sets, measures, categories, etc..

[Note that this foundational preference is to some extent a metamathematical one rather than a mathematical one; in many cases it is possible to rewrite the theory in a mathematically equivalent form so that some other mathematical structure becomes designated as the foundational one, much as probability theory can be equivalently formulated as the measure theory of probability measures. However, this does not negate the fact that a different choice of foundations can lead to a different way of thinking about the subject, and thus to ask a different set of questions and to discover a different set of proofs and solutions. Thus it is often of value to understand multiple foundational perspectives at once, to get a truly stereoscopic view of the subject.]

It turns out that non-commutative probability can be modeled using operator algebras such as {C^*}-algebras, von Neumann algebras, or algebras of bounded operators on a Hilbert space, with the latter being accomplished via the Gelfand-Naimark-Segal construction. We will discuss some of these models here, but just as probability theory seeks to abstract away its measure-theoretic models, the philosophy of non-commutative probability is also to downplay these operator algebraic models once some foundational issues are settled.

When one generalises the set of structures in one’s theory, for instance from the commutative setting to the non-commutative setting, the notion of what it means for a structure to be “universal”, “free”, or “independent” can change. The most familiar example of this comes from group theory. If one restricts attention to the category of abelian groups, then the “freest” object one can generate from two generators {e,f} is the free abelian group of commutative words {e^n f^m} with {n,m \in {\bf Z}}, which is isomorphic to the group {{\bf Z}^2}. If however one generalises to the non-commutative setting of arbitrary groups, then the “freest” object that can now be generated from two generators {e,f} is the free group {{\Bbb F}_2} of non-commutative words {e^{n_1} f^{m_1} \ldots e^{n_k} f^{m_k}} with {n_1,m_1,\ldots,n_k,m_k \in {\bf Z}}, which is a significantly larger extension of the free abelian group {{\bf Z}^2}.

Similarly, when generalising classical probability theory to non-commutative probability theory, the notion of what it means for two or more random variables to be independent changes. In the classical (commutative) setting, two (bounded, real-valued) random variables {X, Y} are independent if one has

\displaystyle  \mathop{\bf E} f(X) g(Y) = 0

whenever {f, g: {\bf R} \rightarrow {\bf R}} are well-behaved functions (such as polynomials) such that all of {\mathop{\bf E} f(X)}, {\mathop{\bf E} g(Y)} vanishes. In the non-commutative setting, one can generalise the above definition to two commuting bounded self-adjoint variables; this concept is useful for instance in quantum probability, which is an abstraction of the theory of observables in quantum mechanics. But for two (bounded, self-adjoint) non-commutative random variables {X, Y}, the notion of classical independence no longer applies. As a substitute, one can instead consider the notion of being freely independent (or free for short), which means that

\displaystyle  \mathop{\bf E} f_1(X) g_1(Y) \ldots f_k(X) g_k(Y) = 0

whenever {f_1,g_1,\ldots,f_k,g_k: {\bf R} \rightarrow {\bf R}} are well-behaved functions such that all of {\mathop{\bf E} f_1(X), \mathop{\bf E} g_1(Y), \ldots, \mathop{\bf E} f_k(X), \mathop{\bf E} g_k(Y)} vanish.

The concept of free independence was introduced by Voiculescu, and its study is now known as the subject of free probability. We will not attempt a systematic survey of this subject here; for this, we refer the reader to the surveys of Speicher and of Biane. Instead, we shall just discuss a small number of topics in this area to give the flavour of the subject only.

The significance of free probability to random matrix theory lies in the fundamental observation that random matrices which are independent in the classical sense, also tend to be independent in the free probability sense, in the large {n} limit {n \rightarrow \infty}. (This is only possible because of the highly non-commutative nature of these matrices; as we shall see, it is not possible for non-trivial commuting independent random variables to be freely independent.) Because of this, many tedious computations in random matrix theory, particularly those of an algebraic or enumerative combinatorial nature, can be done more quickly and systematically by using the framework of free probability, which by design is optimised for algebraic tasks rather than analytical ones.

Much as free groups are in some sense “maximally non-commutative”, freely independent random variables are about as far from being commuting as possible. For instance, if {X, Y} are freely independent and of expectation zero, then {\mathop{\bf E} XYXY} vanishes, but {\mathop{\bf E} XXYY} instead factors as {(\mathop{\bf E} X^2) (\mathop{\bf E} Y^2)}. As a consequence, the behaviour of freely independent random variables can be quite different from the behaviour of their classically independent commuting counterparts. Nevertheless there is a remarkably strong analogy between the two types of independence, in that results which are true in the classically independent case often have an interesting analogue in the freely independent setting. For instance, the central limit theorem (Notes 2) for averages of classically independent random variables, which roughly speaking asserts that such averages become gaussian in the large {n} limit, has an analogue for averages of freely independent variables, the free central limit theorem, which roughly speaking asserts that such averages become semicircular in the large {n} limit. One can then use this theorem to provide yet another proof of Wigner’s semicircle law (Notes 4).

Another important (and closely related) analogy is that while the distribution of sums of independent commutative random variables can be quickly computed via the characteristic function (i.e. the Fourier transform of the distribution), the distribution of sums of freely independent non-commutative random variables can be quickly computed using the Stieltjes transform instead (or with closely related objects, such as the {R}-transform of Voiculescu). This is strongly reminiscent of the appearance of the Stieltjes transform in random matrix theory, and indeed we will see many parallels between the use of the Stieltjes transform here and in Notes 4.

As mentioned earlier, free probability is an excellent tool for computing various expressions of interest in random matrix theory, such as asymptotic values of normalised moments in the large {n} limit {n \rightarrow \infty}. Nevertheless, as it only covers the asymptotic regime in which {n} is sent to infinity while holding all other parameters fixed, there are some aspects of random matrix theory to which the tools of free probability are not sufficient by themselves to resolve (although it can be possible to combine free probability theory with other tools to then answer these questions). For instance, questions regarding the rate of convergence of normalised moments as {n \rightarrow \infty} are not directly answered by free probability, though if free probability is combined with tools such as concentration of measure (Notes 1) then such rate information can often be recovered. For similar reasons, free probability lets one understand the behaviour of {k^{th}} moments as {n \rightarrow \infty} for fixed {k}, but has more difficulty dealing with the situation in which {k} is allowed to grow slowly in {n} (e.g. {k = O(\log n)}). Because of this, free probability methods are effective at controlling the bulk of the spectrum of a random matrix, but have more difficulty with the edges of that spectrum (as well as with related concepts such as the operator norm, Notes 3) as well as with fine-scale structure of the spectrum. Finally, free probability methods are most effective when dealing with matrices that are Hermitian with bounded operator norm, largely because the spectral theory of bounded self-adjoint operators in the infinite-dimensional setting of the large {n} limit is non-pathological. (This is ultimately due to the stable nature of eigenvalues in the self-adjoint setting; see this previous blog post for discussion.) For non-self-adjoint operators, free probability needs to be augmented with additional tools, most notably by bounds on least singular values, in order to recover the required stability for the various spectral data of random matrices to behave continuously with respect to the large {n} limit. We will discuss this latter point in a later set of notes.

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I have just finished the first draft of my blog book for 2009, under the title of “An epsilon of room: pages from year three of a mathematical blog“.  It largely follows the format of my previous two blog books, “Structure and Randomness” and “Poincaré’s legacies“.

There is still some amount of work to be done on the texts; for instance, I need to create an index (which I had neglected to do in the previous two books in the series), and will probably end up splitting the book into two volumes (as was done for “Poincaré’s legacies”).

As always, any feedback or comments are very welcome.

I’ve just uploaded the D.H.J. Polymath article “Density Hales-Jewett and Moser numbers” to the arXiv, submitted to the Szemeredi birthday conference proceedings.

This article investigates the Density Hales-Jewett numbers c_{n,3}, defined as the largest subset of the n-dimensional cube [3]^n containing no combinatorial line, as well as the related Moser numbers c'_{n,3}, defined as the largest subset of the n-dimensional cube containing no geometric line.  We compute the first six numbers in each sequence in this paper.  For the DHJ numbers, they are 1,2,6,18,52,150,450 and for the Moser numbers they are 1,2,6,16,43,124,353.  The last two elements of both sequences are new; the computation c'_{6,3}=353 was the hardest and required a non-trivial amount of computer assistance.

We also establish the asymptotic lower bounds

c_{n,3} \geq 3^n \exp( - O(\sqrt{\log n}) )

and

c'_{n,3} \geq (2 \sqrt{\frac{9}{4\pi}} + o(1)) \frac{3^n}{\sqrt{n}}.

In contrast, the best known upper bound to these quantities is O(3^n / \log^{1/3}_* n), obtained by the sister project to this Polymath project.

We also show a counterexample to a certain “hyper-optimistic conjecture” which would have generalised the Lubell-Yamamoto–Meshalkin (LYM) inequality to this setting.

Thanks to all the participants for this interesting experiment in collaborative mathematics.  This is certainly a different type of project from the ones I normally am involved in, but I found it to be an enjoyable and educational experience.

There is still some time to make further (minor) corrections; I think the deadline for the final submission should be in April.

We can now turn attention to one of the centerpiece universality results in random matrix theory, namely the Wigner semi-circle law for Wigner matrices. Recall from previous notes that a Wigner Hermitian matrix ensemble is a random matrix ensemble {M_n = (\xi_{ij})_{1 \leq i,j \leq n}} of Hermitian matrices (thus {\xi_{ij} = \overline{\xi_{ji}}}; this includes real symmetric matrices as an important special case), in which the upper-triangular entries {\xi_{ij}}, {i>j} are iid complex random variables with mean zero and unit variance, and the diagonal entries {\xi_{ii}} are iid real variables, independent of the upper-triangular entries, with bounded mean and variance. Particular special cases of interest include the Gaussian Orthogonal Ensemble (GOE), the symmetric random sign matrices (aka symmetric Bernoulli ensemble), and the Gaussian Unitary Ensemble (GUE).

In previous notes we saw that the operator norm of {M_n} was typically of size {O(\sqrt{n})}, so it is natural to work with the normalised matrix {\frac{1}{\sqrt{n}} M_n}. Accordingly, given any {n \times n} Hermitian matrix {M_n}, we can form the (normalised) empirical spectral distribution (or ESD for short)

\displaystyle  \mu_{\frac{1}{\sqrt{n}} M_n} := \frac{1}{n} \sum_{j=1}^n \delta_{\lambda_j(M_n) / \sqrt{n}},

of {M_n}, where {\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)} are the (necessarily real) eigenvalues of {M_n}, counting multiplicity. The ESD is a probability measure, which can be viewed as a distribution of the normalised eigenvalues of {M_n}.

When {M_n} is a random matrix ensemble, then the ESD {\mu_{\frac{1}{\sqrt{n}} M_n}} is now a random measure – i.e. a random variable taking values in the space {\hbox{Pr}({\mathbb R})} of probability measures on the real line. (Thus, the distribution of {\mu_{\frac{1}{\sqrt{n}} M_n}} is a probability measure on probability measures!)

Now we consider the behaviour of the ESD of a sequence of Hermitian matrix ensembles {M_n} as {n \rightarrow \infty}. Recall from Notes 0 that for any sequence of random variables in a {\sigma}-compact metrisable space, one can define notions of convergence in probability and convergence almost surely. Specialising these definitions to the case of random probability measures on {{\mathbb R}}, and to deterministic limits, we see that a sequence of random ESDs {\mu_{\frac{1}{\sqrt{n}} M_n}} converge in probability (resp. converge almost surely) to a deterministic limit {\mu \in \hbox{Pr}({\mathbb R})} (which, confusingly enough, is a deterministic probability measure!) if, for every test function {\varphi \in C_c({\mathbb R})}, the quantities {\int_{\mathbb R} \varphi\ d\mu_{\frac{1}{\sqrt{n}} M_n}} converge in probability (resp. converge almost surely) to {\int_{\mathbb R} \varphi\ d\mu}.

Remark 1 As usual, convergence almost surely implies convergence in probability, but not vice versa. In the special case of random probability measures, there is an even weaker notion of convergence, namely convergence in expectation, defined as follows. Given a random ESD {\mu_{\frac{1}{\sqrt{n}} M_n}}, one can form its expectation {{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n} \in \hbox{Pr}({\mathbb R})}, defined via duality (the Riesz representation theorem) as

\displaystyle  \int_{\mathbb R} \varphi\ d{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n} := {\bf E} \int_{\mathbb R} \varphi\ d	 \mu_{\frac{1}{\sqrt{n}} M_n};

this probability measure can be viewed as the law of a random eigenvalue {\frac{1}{\sqrt{n}}\lambda_i(M_n)} drawn from a random matrix {M_n} from the ensemble. We then say that the ESDs converge in expectation to a limit {\mu \in \hbox{Pr}({\mathbb R})} if {{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n}} converges the vague topology to {\mu}, thus

\displaystyle  {\bf E} \int_{\mathbb R} \varphi\ d	 \mu_{\frac{1}{\sqrt{n}} M_n} \rightarrow \int_{\mathbb R} \varphi\ d\mu

for all {\phi \in C_c({\mathbb R})}.

In general, these notions of convergence are distinct from each other; but in practice, one often finds in random matrix theory that these notions are effectively equivalent to each other, thanks to the concentration of measure phenomenon.

Exercise 1 Let {M_n} be a sequence of {n \times n} Hermitian matrix ensembles, and let {\mu} be a continuous probability measure on {{\mathbb R}}.

  • Show that {\mu_{\frac{1}{\sqrt{n}} M_n}} converges almost surely to {\mu} if and only if {\mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)} converges almost surely to {\mu(-\infty,\lambda)} for all {\lambda \in {\mathbb R}}.
  • Show that {\mu_{\frac{1}{\sqrt{n}} M_n}} converges in probability to {\mu} if and only if {\mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)} converges in probability to {\mu(-\infty,\lambda)} for all {\lambda \in {\mathbb R}}.
  • Show that {\mu_{\frac{1}{\sqrt{n}} M_n}} converges in expectation to {\mu} if and only if {\mathop{\mathbb E} \mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)} converges to {\mu(-\infty,\lambda)} for all {\lambda \in {\mathbb R}}.

We can now state the Wigner semi-circular law.

Theorem 1 (Semicircular law) Let {M_n} be the top left {n \times n} minors of an infinite Wigner matrix {(\xi_{ij})_{i,j \geq 1}}. Then the ESDs {\mu_{\frac{1}{\sqrt{n}} M_n}} converge almost surely (and hence also in probability and in expectation) to the Wigner semi-circular distribution

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

A numerical example of this theorem in action can be seen at the MathWorld entry for this law.

The semi-circular law nicely complements the upper Bai-Yin theorem from Notes 3, which asserts that (in the case when the entries have finite fourth moment, at least), the matrices {\frac{1}{\sqrt{n}} M_n} almost surely has operator norm at most {2+o(1)}. Note that the operator norm is the same thing as the largest magnitude of the eigenvalues. Because the semi-circular distribution (1) is supported on the interval {[-2,2]} with positive density on the interior of this interval, Theorem 1 easily supplies the lower Bai-Yin theorem, that the operator norm of {\frac{1}{\sqrt{n}} M_n} is almost surely at least {2-o(1)}, and thus (in the finite fourth moment case) the norm is in fact equal to {2+o(1)}. Indeed, we have just shown that the circular law provides an alternate proof of the lower Bai-Yin bound (Proposition 11 of Notes 3).

As will hopefully become clearer in the next set of notes, the semi-circular law is the noncommutative (or free probability) analogue of the central limit theorem, with the semi-circular distribution (1) taking on the role of the normal distribution. Of course, there is a striking difference between the two distributions, in that the former is compactly supported while the latter is merely subgaussian. One reason for this is that the concentration of measure phenomenon is more powerful in the case of ESDs of Wigner matrices than it is for averages of iid variables; compare the concentration of measure results in Notes 3 with those in Notes 1.

There are several ways to prove (or at least to heuristically justify) the circular law. In this set of notes we shall focus on the two most popular methods, the moment method and the Stieltjes transform method, together with a third (heuristic) method based on Dyson Brownian motion (Notes 3b). In the next set of notes we shall also study the free probability method, and in the set of notes after that we use the determinantal processes method (although this method is initially only restricted to highly symmetric ensembles, such as GUE).

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