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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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Van Vu and I have just uploaded to the arXiv our short survey article, “Random matrices: The Four Moment Theorem for Wigner ensembles“, submitted to the MSRI book series, as part of the proceedings on the MSRI semester program on random matrix theory from last year.  This is a highly condensed version (at 17 pages) of a much longer survey (currently at about 48 pages, though not completely finished) that we are currently working on, devoted to the recent advances in understanding the universality phenomenon for spectral statistics of Wigner matrices.  In this abridged version of the survey, we focus on a key tool in the subject, namely the Four Moment Theorem which roughly speaking asserts that the statistics of a Wigner matrix depend only on the first four moments of the entries.  We give a sketch of proof of this theorem, and two sample applications: a central limit theorem for individual eigenvalues of a Wigner matrix (extending a result of Gustavsson in the case of GUE), and the verification of a conjecture of Wigner, Dyson, and Mehta on the universality of the asymptotic k-point correlation functions even for discrete ensembles (provided that we interpret convergence in the vague topology sense).

For reasons of space, this paper is very far from an exhaustive survey even of the narrow topic of universality for Wigner matrices, but should hopefully be an accessible entry point into the subject nevertheless.

In the previous set of notes we introduced the notion of expansion in arbitrary {k}-regular graphs. For the rest of the course, we will now focus attention primarily to a special type of {k}-regular graph, namely a Cayley graph.

Definition 1 (Cayley graph) Let {G = (G,\cdot)} be a group, and let {S} be a finite subset of {G}. We assume that {S} is symmetric (thus {s^{-1} \in S} whenever {s \in S}) and does not contain the identity {1} (this is to avoid loops). Then the (right-invariant) Cayley graph {Cay(G,S)} is defined to be the graph with vertex set {G} and edge set {\{ \{sx,x\}: x \in G, s \in S \}}, thus each vertex {x \in G} is connected to the {|S|} elements {sx} for {s \in S}, and so {Cay(G,S)} is a {|S|}-regular graph.

Example 1 The graph in Exercise 3 of Notes 1 is the Cayley graph on {{\bf Z}/N{\bf Z}} with generators {S = \{-1,+1\}}.

Remark 1 We call the above Cayley graphs right-invariant because every right translation {x\mapsto xg} on {G} is a graph automorphism of {Cay(G,S)}. This group of automorphisms acts transitively on the vertex set of the Cayley graph. One can thus view a Cayley graph as a homogeneous space of {G}, as it “looks the same” from every vertex. One could of course also consider left-invariant Cayley graphs, in which {x} is connected to {xs} rather than {sx}. However, the two such graphs are isomorphic using the inverse map {x \mapsto x^{-1}}, so we may without loss of generality restrict our attention throughout to left Cayley graphs.

Remark 2 For minor technical reasons, it will be convenient later on to allow {S} to contain the identity and to come with multiplicity (i.e. it will be a multiset rather than a set). If one does so, of course, the resulting Cayley graph will now contain some loops and multiple edges.

For the purposes of building expander families, we would of course want the underlying group {G} to be finite. However, it will be convenient at various times to “lift” a finite Cayley graph up to an infinite one, and so we permit {G} to be infinite in our definition of a Cayley graph.

We will also sometimes consider a generalisation of a Cayley graph, known as a Schreier graph:

Definition 2 (Schreier graph) Let {G} be a finite group that acts (on the left) on a space {X}, thus there is a map {(g,x) \mapsto gx} from {G \times X} to {X} such that {1x = x} and {(gh)x = g(hx)} for all {g,h \in G} and {x \in X}. Let {S} be a symmetric subset of {G} which acts freely on {X} in the sense that {sx \neq x} for all {s \in S} and {x \in X}, and {sx \neq s'x} for all distinct {s,s' \in S} and {x \in X}. Then the Schreier graph {Sch(X,S)} is defined to be the graph with vertex set {X} and edge set {\{ \{sx,x\}: x \in X, s \in S \}}.

Example 2 Every Cayley graph {Cay(G,S)} is also a Schreier graph {Sch(G,S)}, using the obvious left-action of {G} on itself. The {k}-regular graphs formed from {l} permutations {\pi_1,\ldots,\pi_l \in S_n} that were studied in the previous set of notes is also a Schreier graph provided that {\pi_i(v) \neq v, \pi_i^{-1}(v), \pi_j(v)} for all distinct {1 \leq i,j \leq l}, with the underlying group being the permutation group {S_n} (which acts on the vertex set {X = \{1,\ldots,n\}} in the obvious manner), and {S := \{\pi_1,\ldots,\pi_l,\pi_1^{-1},\ldots,\pi_l^{-1}\}}.

Exercise 1 If {k} is an even integer, show that every {k}-regular graph is a Schreier graph involving a set {S} of generators of cardinality {k/2}. (Hint: first show that every {k}-regular graph can be decomposed into {k/2} unions of cycles, each of which partition the vertex set, then use the previous example.

We return now to Cayley graphs. It is easy to characterise qualitative expansion properties of Cayley graphs:

Exercise 2 (Qualitative expansion) Let {Cay(G,S)} be a finite Cayley graph.

  • (i) Show that {Cay(G,S)} is a one-sided {\epsilon}-expander for {G} for some {\epsilon>0} if and only if {S} generates {G}.
  • (ii) Show that {Cay(G,S)} is a two-sided {\epsilon}-expander for {G} for some {\epsilon>0} if and only if {S} generates {G}, and furthermore {S} intersects each index {2} subgroup of {G}.

We will however be interested in more quantitative expansion properties, in which the expansion constant {\epsilon} is independent of the size of the Cayley graph, so that one can construct non-trivial expander families {Cay(G_n,S_n)} of Cayley graphs.

One can analyse the expansion of Cayley graphs in a number of ways. For instance, by taking the edge expansion viewpoint, one can study Cayley graphs combinatorially, using the product set operation

\displaystyle  A \cdot B := \{ab: a \in A, b \in B \}

of subsets of {G}.

Exercise 3 (Combinatorial description of expansion) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs. Show that {Cay(G_n,S_n)} is a one-sided expander family if and only if there is a constant {c>0} independent of {n} such that {|E_n \cup E_n S_n| \geq (1+c) |E_n|} for all sufficiently large {n} and all subsets {E_n} of {G_n} with {|E_n| \leq |G_n|/2}.

One can also give a combinatorial description of two-sided expansion, but it is more complicated and we will not use it here.

Exercise 4 (Abelian groups do not expand) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs, with the {G_n} all abelian, and the {S_n} generating {G_n}. Show that {Cay(G_n,S_n)} are a one-sided expander family if and only if the Cayley graphs have bounded cardinality (i.e. {\sup_n |G_n| < \infty}). (Hint: assume for contradiction that {Cay(G_n,S_n)} is a one-sided expander family with {|G_n| \rightarrow \infty}, and show by two different arguments that {\sup_n |S_n^m|} grows at least exponentially in {m} and also at most polynomially in {m}, giving the desired contradiction.)

The left-invariant nature of Cayley graphs also suggests that such graphs can be profitably analysed using some sort of Fourier analysis; as the underlying symmetry group is not necessarily abelian, one should use the Fourier analysis of non-abelian groups, which is better known as (unitary) representation theory. The Fourier-analytic nature of Cayley graphs can be highlighted by recalling the operation of convolution of two functions {f, g \in \ell^2(G)}, defined by the formula

\displaystyle  f * g(x) := \sum_{y \in G} f(y) g(y^{-1} x) = \sum_{y \in G} f(x y^{-1}) g(y).

This convolution operation is bilinear and associative (at least when one imposes a suitable decay condition on the functions, such as compact support), but is not commutative unless {G} is abelian. (If one is more algebraically minded, one can also identify {\ell^2(G)} (when {G} is finite, at least) with the group algebra {{\bf C} G}, in which case convolution is simply the multiplication operation in this algebra.) The adjacency operator {A} on a Cayley graph {Cay(G,S)} can then be viewed as a convolution

\displaystyle  Af = |S| \mu * f,

where {\mu} is the probability density

\displaystyle  \mu := \frac{1}{|S|} \sum_{s \in S} \delta_s \ \ \ \ \ (1)

where {\delta_s} is the Kronecker delta function on {s}. Using the spectral definition of expansion, we thus see that {Cay(G,S)} is a one-sided expander if and only if

\displaystyle  \langle f, \mu*f \rangle \leq (1-\epsilon) \|f\|_{\ell^2(G)} \ \ \ \ \ (2)

whenever {f \in \ell^2(G)} is orthogonal to the constant function {1}, and is a two-sided expander if

\displaystyle  \| \mu*f \|_{\ell^2(G)} \leq (1-\epsilon) \|f\|_{\ell^2(G)} \ \ \ \ \ (3)

whenever {f \in \ell^2(G)} is orthogonal to the constant function {1}.

We remark that the above spectral definition of expansion can be easily extended to symmetric sets {S} which contain the identity or have multiplicity (i.e. are multisets). (We retain symmetry, though, in order to keep the operation of convolution by {\mu} self-adjoint.) In particular, one can say (with some slight abuse of notation) that a set of elements {s_1,\ldots,s_l} of {G} (possibly with repetition, and possibly with some elements equalling the identity) generates a one-sided or two-sided {\epsilon}-expander if the associated symmetric probability density

\displaystyle  \mu := \frac{1}{2l} \sum_{i=1}^l \delta_{s_i} + \delta_{s_i^{-1}}

obeys either (2) or (3).

We saw in the last set of notes that expansion can be characterised in terms of random walks. One can of course specialise this characterisation to the Cayley graph case:

Exercise 5 (Random walk description of expansion) Let {Cay(G_n,S_n)} be a family of finite {k}-regular Cayley graphs, and let {\mu_n} be the associated probability density functions. Let {A > 1/2} be a constant.

  • Show that the {Cay(G_n,S_n)} are a two-sided expander family if and only if there exists a {C>0} such that for all sufficiently large {n}, one has {\| \mu_n^{*m} - \frac{1}{|G_n|} \|_{\ell^2(G_n)} \leq \frac{1}{|G_n|^A}} for some {m \leq C \log |G_n|}, where {\mu_n^{*m} := \mu_n * \ldots * \mu_n} denotes the convolution of {m} copies of {\mu_n}.
  • Show that the {Cay(G_n,S_n)} are a one-sided expander family if and only if there exists a {C>0} such that for all sufficiently large {n}, one has {\| (\frac{1}{2} \delta_1 + \frac{1}{2} \mu_n)^{*m} - \frac{1}{|G_n|} \|_{\ell^2(G_n)} \leq \frac{1}{|G_n|^A}} for some {m \leq C \log |G_n|}.

In this set of notes, we will connect expansion of Cayley graphs to an important property of certain infinite groups, known as Kazhdan’s property (T) (or property (T) for short). In 1973, Margulis exploited this property to create the first known explicit and deterministic examples of expanding Cayley graphs. As it turns out, property (T) is somewhat overpowered for this purpose; in particular, we now know that there are many families of Cayley graphs for which the associated infinite group does not obey property (T) (or weaker variants of this property, such as property {\tau}). In later notes we will therefore turn to other methods of creating Cayley graphs that do not rely on property (T). Nevertheless, property (T) is of substantial intrinsic interest, and also has many connections to other parts of mathematics than the theory of expander graphs, so it is worth spending some time to discuss it here.

The material here is based in part on this recent text on property (T) by Bekka, de la Harpe, and Valette (available online here).

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The objective of this course is to present a number of recent constructions of expander graphs, which are a type of sparse but “pseudorandom” graph of importance in computer science, the theory of random walks, geometric group theory, and in number theory. The subject of expander graphs and their applications is an immense one, and we will not possibly be able to cover it in full in this course. In particular, we will say almost nothing about the important applications of expander graphs to computer science, for instance in constructing good pseudorandom number generators, derandomising a probabilistic algorithm, constructing error correcting codes, or in building probabilistically checkable proofs. For such topics, I recommend the survey of Hoory-Linial-Wigderson. We will also only pass very lightly over the other applications of expander graphs, though if time permits I may discuss at the end of the course the application of expander graphs in finite groups such as {SL_2(F_p)} to certain sieving problems in analytic number theory, following the work of Bourgain, Gamburd, and Sarnak.

Instead of focusing on applications, this course will concern itself much more with the task of constructing expander graphs. This is a surprisingly non-trivial problem. On one hand, an easy application of the probabilistic method shows that a randomly chosen (large, regular, bounded-degree) graph will be an expander graph with very high probability, so expander graphs are extremely abundant. On the other hand, in many applications, one wants an expander graph that is more deterministic in nature (requiring either no or very few random choices to build), and of a more specialised form. For the applications to number theory or geometric group theory, it is of particular interest to determine the expansion properties of a very symmetric type of graph, namely a Cayley graph; we will also occasionally work with the more general concept of a Schreier graph. It turns out that such questions are related to deep properties of various groups {G} of Lie type (such as {SL_2({\bf R})} or {SL_2({\bf Z})}), such as Kazhdan’s property (T), the first nontrivial eigenvalue of a Laplacian on a symmetric space {G/\Gamma} associated to {G}, the quasirandomness of {G} (as measured by the size of irreducible representations), and the product theory of subsets of {G}. These properties are of intrinsic interest to many other fields of mathematics (e.g. ergodic theory, operator algebras, additive combinatorics, representation theory, finite group theory, number theory, etc.), and it is quite remarkable that a single problem – namely the construction of expander graphs – is so deeply connected with such a rich and diverse array of mathematical topics. (Perhaps this is because so many of these fields are all grappling with aspects of a single general problem in mathematics, namely when to determine whether a given mathematical object or process of interest “behaves pseudorandomly” or not, and how this is connected with the symmetry group of that object or process.)

(There are also other important constructions of expander graphs that are not related to Cayley or Schreier graphs, such as those graphs constructed by the zigzag product construction, but we will not discuss those types of graphs in this course, again referring the reader to the survey of Hoory, Linial, and Wigderson.)

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Van Vu and I have just uploaded to the arXiv our paper A central limit theorem for the determinant of a Wigner matrix, submitted to Adv. Math.. It studies the asymptotic distribution of the determinant {\det M_n} of a random Wigner matrix (such as a matrix drawn from the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE)).

Before we get to these results, let us first discuss the simpler problem of studying the determinant {\det A_n} of a random iid matrix {A_n = (\zeta_{ij})_{1 \leq i,j \leq n}}, such as a real gaussian matrix (where all entries are independently and identically distributed using the standard real normal distribution {\zeta_{ij} \equiv N(0,1)_{\bf R}}), a complex gaussian matrix (where all entries are independently and identically distributed using the standard complex normal distribution {\zeta_{ij} \equiv N(0,1)_{\bf C}}, thus the real and imaginary parts are independent with law {N(0,1/2)_{\bf R}}), or the random sign matrix (in which all entries are independently and identically distributed according to the Bernoulli distribution {\zeta_{ij} \equiv \pm 1} (with a {1/2} chance of either sign). More generally, one can consider a matrix {A_n} in which all the entries {\zeta_{ij}} are independently and identically distributed with mean zero and variance {1}.

We can expand {\det A_n} using the Leibniz expansion

\displaystyle  \det A_n = \sum_{\sigma \in S_n} I_\sigma, \ \ \ \ \ (1)

where {\sigma: \{1,\ldots,n\} \rightarrow \{1,\ldots,n\}} ranges over the permutations of {\{1,\ldots,n\}}, and {I_\sigma} is the product

\displaystyle  I_\sigma := \hbox{sgn}(\sigma) \prod_{i=1}^n \zeta_{i\sigma(i)}.

From the iid nature of the {\zeta_{ij}}, we easily see that each {I_\sigma} has mean zero and variance one, and are pairwise uncorrelated as {\sigma} varies. We conclude that {\det A_n} has mean zero and variance {n!} (an observation first made by Turán). In particular, from Chebyshev’s inequality we see that {\det A_n} is typically of size {O(\sqrt{n!})}.

It turns out, though, that this is not quite best possible. This is easiest to explain in the real gaussian case, by performing a computation first made by Goodman. In this case, {\det A_n} is clearly symmetrical, so we can focus attention on the magnitude {|\det A_n|}. We can interpret this quantity geometrically as the volume of an {n}-dimensional parallelopiped whose generating vectors {X_1,\ldots,X_n} are independent real gaussian vectors in {{\bf R}^n} (i.e. their coefficients are iid with law {N(0,1)_{\bf R}}). Using the classical base-times-height formula, we thus have

\displaystyle  |\det A_n| = \prod_{i=1}^n \hbox{dist}(X_i, V_i) \ \ \ \ \ (2)

where {V_i} is the {i-1}-dimensional linear subspace of {{\bf R}^n} spanned by {X_1,\ldots,X_{i-1}} (note that {X_1,\ldots,X_n}, having an absolutely continuous joint distribution, are almost surely linearly independent). Taking logarithms, we conclude

\displaystyle  \log |\det A_n| = \sum_{i=1}^n \log \hbox{dist}(X_i, V_i).

Now, we take advantage of a fundamental symmetry property of the Gaussian vector distribution, namely its invariance with respect to the orthogonal group {O(n)}. Because of this, we see that if we fix {X_1,\ldots,X_{i-1}} (and thus {V_i}, the random variable {\hbox{dist}(X_i,V_i)} has the same distribution as {\hbox{dist}(X_i,{\bf R}^{i-1})}, or equivalently the {\chi} distribution

\displaystyle  \chi_{n-i+1} := (\sum_{j=1}^{n-i+1} \xi_{n-i+1,j}^2)^{1/2}

where {\xi_{n-i+1,1},\ldots,\xi_{n-i+1,n-i+1}} are iid copies of {N(0,1)_{\bf R}}. As this distribution does not depend on the {X_1,\ldots,X_{i-1}}, we conclude that the law of {\log |\det A_n|} is given by the sum of {n} independent {\chi}-variables:

\displaystyle  \log |\det A_n| \equiv \sum_{j=1}^{n} \log \chi_j.

A standard computation shows that each {\chi_j^2} has mean {j} and variance {2j}, and then a Taylor series (or Ito calculus) computation (using concentration of measure tools to control tails) shows that {\log \chi_j} has mean {\frac{1}{2} \log j - \frac{1}{2j} + O(1/j^{3/2})} and variance {\frac{1}{2j}+O(1/j^{3/2})}. As such, {\log |\det A_n|} has mean {\frac{1}{2} \log n! - \frac{1}{2} \log n + O(1)} and variance {\frac{1}{2} \log n + O(1)}. Applying a suitable version of the central limit theorem, one obtains the asymptotic law

\displaystyle  \frac{\log |\det A_n| - \frac{1}{2} \log n! + \frac{1}{2} \log n}{\sqrt{\frac{1}{2}\log n}} \rightarrow N(0,1)_{\bf R}, \ \ \ \ \ (3)

where {\rightarrow} denotes convergence in distribution. A bit more informally, we have

\displaystyle  |\det A_n| \approx n^{-1/2} \sqrt{n!} \exp( N( 0, \log n / 2 )_{\bf R} ) \ \ \ \ \ (4)

when {A_n} is a real gaussian matrix; thus, for instance, the median value of {|\det A_n|} is {n^{-1/2+o(1)} \sqrt{n!}}. At first glance, this appears to conflict with the second moment bound {\mathop{\bf E} |\det A_n|^2 = n!} of Turán mentioned earlier, but once one recalls that {\exp(N(0,t)_{\bf R})} has a second moment of {\exp(2t)}, we see that the two facts are in fact perfectly consistent; the upper tail of the normal distribution in the exponent in (4) ends up dominating the second moment.

It turns out that the central limit theorem (3) is valid for any real iid matrix with mean zero, variance one, and an exponential decay condition on the entries; this was first claimed by Girko, though the arguments in that paper appear to be incomplete. Another proof of this result, with more quantitative bounds on the convergence rate has been recently obtained by Hoi Nguyen and Van Vu. The basic idea in these arguments is to express the sum in (2) in terms of a martingale and apply the martingale central limit theorem.

If one works with complex gaussian random matrices instead of real gaussian random matrices, the above computations change slightly (one has to replace the real {\chi} distribution with the complex {\chi} distribution, in which the {\xi_{i,j}} are distributed according to the complex gaussian {N(0,1)_{\bf C}} instead of the real one). At the end of the day, one ends up with the law

\displaystyle  \frac{\log |\det A_n| - \frac{1}{2} \log n! + \frac{1}{4} \log n}{\sqrt{\frac{1}{4}\log n}} \rightarrow N(0,1)_{\bf R}, \ \ \ \ \ (5)

or more informally

\displaystyle  |\det A_n| \approx n^{-1/4} \sqrt{n!} \exp( N( 0, \log n / 4 )_{\bf R} ) \ \ \ \ \ (6)

(but note that this new asymptotic is still consistent with Turán’s second moment calculation).

We can now turn to the results of our paper. Here, we replace the iid matrices {A_n} by Wigner matrices {M_n = (\zeta_{ij})_{1 \leq i,j \leq n}}, which are defined similarly but are constrained to be Hermitian (or real symmetric), thus {\zeta_{ij} = \overline{\zeta_{ji}}} for all {i,j}. Model examples here include the Gaussian Unitary Ensemble (GUE), in which {\zeta_{ij} \equiv N(0,1)_{\bf C}} for {1 \leq i < j \leq n} and {\zeta_{ij} \equiv N(0,1)_{\bf R}} for {1 \leq i=j \leq n}, the Gaussian Orthogonal Ensemble (GOE), in which {\zeta_{ij} \equiv N(0,1)_{\bf R}} for {1 \leq i < j \leq n} and {\zeta_{ij} \equiv N(0,2)_{\bf R}} for {1 \leq i=j \leq n}, and the symmetric Bernoulli ensemble, in which {\zeta_{ij} \equiv \pm 1} for {1 \leq i \leq j \leq n} (with probability {1/2} of either sign). In all cases, the upper triangular entries of the matrix are assumed to be jointly independent. For a more precise definition of the Wigner matrix ensembles we are considering, see the introduction to our paper.

The determinants {\det M_n} of these matrices still have a Leibniz expansion. However, in the Wigner case, the mean and variance of the {I_\sigma} are slightly different, and what is worse, they are not all pairwise uncorrelated any more. For instance, the mean of {I_\sigma} is still usually zero, but equals {(-1)^{n/2}} in the exceptional case when {\sigma} is a perfect matching (i.e. the union of exactly {n/2} {2}-cycles, a possibility that can of course only happen when {n} is even). As such, the mean {\mathop{\bf E} \det M_n} still vanishes when {n} is odd, but for even {n} it is equal to

\displaystyle  (-1)^{n/2} \frac{n!}{(n/2)!2^{n/2}}

(the fraction here simply being the number of perfect matchings on {n} vertices). Using Stirling’s formula, one then computes that {|\mathop{\bf E} \det A_n|} is comparable to {n^{-1/4} \sqrt{n!}} when {n} is large and even. The second moment calculation is more complicated (and uses facts about the distribution of cycles in random permutations, mentioned in this previous post), but one can compute that {\mathop{\bf E} |\det A_n|^2} is comparable to {n^{1/2} n!} for GUE and {n^{3/2} n!} for GOE. (The discrepancy here comes from the fact that in the GOE case, {I_\sigma} and {I_\rho} can correlate when {\rho} contains reversals of {k}-cycles of {\sigma} for {k \geq 3}, but this does not happen in the GUE case.) For GUE, much more precise asymptotics for the moments of the determinant are known, starting from the work of Brezin and Hikami, though we do not need these more sophisticated computations here.

Our main results are then as follows.

Theorem 1 Let {M_n} be a Wigner matrix.

  • If {M_n} is drawn from GUE, then

    \displaystyle  \frac{\log |\det M_n| - \frac{1}{2} \log n! + \frac{1}{4} \log n}{\sqrt{\frac{1}{2}\log n}} \rightarrow N(0,1)_{\bf R}.

  • If {M_n} is drawn from GOE, then

    \displaystyle  \frac{\log |\det M_n| - \frac{1}{2} \log n! + \frac{1}{4} \log n}{\sqrt{\log n}} \rightarrow N(0,1)_{\bf R}.

  • The previous two results also hold for more general Wigner matrices, assuming that the real and imaginary parts are independent, a finite moment condition is satisfied, and the entries match moments with those of GOE or GUE to fourth order. (See the paper for a more precise formulation of the result.)

Thus, we informally have

\displaystyle  |\det M_n| \approx n^{-1/4} \sqrt{n!} \exp( N( 0, \log n / 2 )_{\bf R} )

when {M_n} is drawn from GUE, or from another Wigner ensemble matching GUE to fourth order (and obeying some additional minor technical hypotheses); and

\displaystyle  |\det M_n| \approx n^{-1/4} \sqrt{n!} \exp( N( 0, \log n )_{\bf R} )

when {M_n} is drawn from GOE, or from another Wigner ensemble matching GOE to fourth order. Again, these asymptotic limiting distributions are consistent with the asymptotic behaviour for the second moments.

The extension from the GUE or GOE case to more general Wigner ensembles is a fairly routine application of the four moment theorem for Wigner matrices, although for various technical reasons we do not quite use the existing four moment theorems in the literature, but adapt them to the log determinant. The main idea is to express the log-determinant as an integral

\displaystyle  \log|\det M_n| = \frac{1}{2} n \log n - n \hbox{Im} \int_0^\infty s(\sqrt{-1}\eta)\ d\eta \ \ \ \ \ (7)

of the Stieltjes transform

\displaystyle  s(z) := \frac{1}{n} \hbox{tr}( \frac{1}{\sqrt{n}} M_n - z )^{-1}

of {M_n}. Strictly speaking, the integral in (7) is divergent at infinity (and also can be ill-behaved near zero), but this can be addressed by standard truncation and renormalisation arguments (combined with known facts about the least singular value of Wigner matrices), which we omit here. We then use a variant of the four moment theorem for the Stieltjes transform, as used by Erdos, Yau, and Yin (based on a previous four moment theorem for individual eigenvalues introduced by Van Vu and myself). The four moment theorem is proven by the now-standard Lindeberg exchange method, combined with the usual resolvent identities to control the behaviour of the resolvent (and hence the Stieltjes transform) with respect to modifying one or two entries, together with the delocalisation of eigenvector property (which in turn arises from local semicircle laws) to control the error terms.

Somewhat surprisingly (to us, at least), it turned out that it was the first part of the theorem (namely, the verification of the limiting law for the invariant ensembles GUE and GOE) that was more difficult than the extension to the Wigner case. Even in an ensemble as highly symmetric as GUE, the rows are no longer independent, and the formula (2) is basically useless for getting any non-trivial control on the log determinant. There is an explicit formula for the joint distribution of the eigenvalues of GUE (or GOE), which does eventually give the distribution of the cumulants of the log determinant, which then gives the required central limit theorem; but this is a lengthy computation, first performed by Delannay and Le Caer.

Following a suggestion of my colleague, Rowan Killip, we give an alternate proof of this central limit theorem in the GUE and GOE cases, by using a beautiful observation of Trotter, namely that the GUE or GOE ensemble can be conjugated into a tractable tridiagonal form. Let me state it just for GUE:

Proposition 2 (Tridiagonal form of GUE) Let {M'_n} be the random tridiagonal real symmetric matrix

\displaystyle  M'_n = \begin{pmatrix} a_1 & b_1 & 0 & \ldots & 0 & 0 \\ b_1 & a_2 & b_2 & \ldots & 0 & 0 \\ 0 & b_2 & a_3 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & a_{n-1} & b_{n-1} \\ 0 & 0 & 0 & \ldots & b_{n-1} & a_n \end{pmatrix}

where the {a_1,\ldots,a_n, b_1,\ldots,b_{n-1}} are jointly independent real random variables, with {a_1,\ldots,a_n \equiv N(0,1)_{\bf R}} being standard real Gaussians, and each {b_i} having a {\chi}-distribution:

\displaystyle  b_i = (\sum_{j=1}^i |z_{i,j}|^2)^{1/2}

where {z_{i,j} \equiv N(0,1)_{\bf C}} are iid complex gaussians. Let {M_n} be drawn from GUE. Then the joint eigenvalue distribution of {M_n} is identical to the joint eigenvalue distribution of {M'_n}.

Proof: Let {M_n} be drawn from GUE. We can write

\displaystyle  M_n = \begin{pmatrix} M_{n-1} & X_n \\ X_n^* & a_n \end{pmatrix}

where {M_{n-1}} is drawn from the {n-1\times n-1} GUE, {a_n \equiv N(0,1)_{\bf R}}, and {X_n \in {\bf C}^{n-1}} is a random gaussian vector with all entries iid with distribution {N(0,1)_{\bf C}}. Furthermore, {M_{n-1}, X_n, a_n} are jointly independent.

We now apply the tridiagonal matrix algorithm. Let {b_{n-1} := |X_n|}, then {b_n} has the {\chi}-distribution indicated in the proposition. We then conjugate {M_n} by a unitary matrix {U} that preserves the final basis vector {e_n}, and maps {X} to {b_{n-1} e_{n-1}}. Then we have

\displaystyle  U M_n U^* = \begin{pmatrix} \tilde M_{n-1} & b_{n-1} e_{n-1} \\ b_{n-1} e_{n-1}^* & a_n \end{pmatrix}

where {\tilde M_{n-1}} is conjugate to {M_{n-1}}. Now we make the crucial observation: because {M_{n-1}} is distributed according to GUE (which is a unitarily invariant ensemble), and {U} is a unitary matrix independent of {M_{n-1}}, {\tilde M_{n-1}} is also distributed according to GUE, and remains independent of both {b_{n-1}} and {a_n}.

We continue this process, expanding {U M_n U^*} as

\displaystyle \begin{pmatrix} M_{n-2} & X_{n-1} & 0 \\ X_{n-1}^* & a_{n-1} & b_{n-1} \\ 0 & b_{n-1} & a_n. \end{pmatrix}

Applying a further unitary conjugation that fixes {e_{n-1}, e_n} but maps {X_{n-1}} to {b_{n-2} e_{n-2}}, we may replace {X_{n-1}} by {b_{n-2} e_{n-2}} while transforming {M_{n-2}} to another GUE matrix {\tilde M_{n-2}} independent of {a_n, b_{n-1}, a_{n-1}, b_{n-2}}. Iterating this process, we eventually obtain a coupling of {M_n} to {M'_n} by unitary conjugations, and the claim follows. \Box

The determinant of a tridiagonal matrix is not quite as simple as the determinant of a triangular matrix (in which it is simply the product of the diagonal entries), but it is pretty close: the determinant {D_n} of the above matrix is given by solving the recursion

\displaystyle  D_i = a_i D_{i-1} + b_{i-1}^2 D_{i-2}

with {D_0=1} and {D_{-1} = 0}. Thus, instead of the product of a sequence of independent scalar {\chi} distributions as in the gaussian matrix case, the determinant of GUE ends up being controlled by the product of a sequence of independent {2\times 2} matrices whose entries are given by gaussians and {\chi} distributions. In this case, one cannot immediately take logarithms and hope to get something for which the martingale central limit theorem can be applied, but some ad hoc manipulation of these {2 \times 2} matrix products eventually does make this strategy work. (Roughly speaking, one has to work with the logarithm of the Frobenius norm of the matrix first.)

Igor Rodnianski and I have just uploaded to the arXiv our paper “Effective limiting absorption principles, and applications“, submitted to Communications in Mathematical Physics. In this paper we derive limiting absorption principles (of type discussed in this recent post) for a general class of Schrödinger operators {H = -\Delta + V} on a wide class of manifolds, namely the asymptotically conic manifolds. The precise definition of such manifolds is somewhat technical, but they include as a special case the asymptotically flat manifolds, which in turn include as a further special case the smooth compact perturbations of Euclidean space {{\bf R}^n} (i.e. the smooth Riemannian manifolds that are identical to {{\bf R}^n} outside of a compact set). The potential {V} is assumed to be a short range potential, which roughly speaking means that it decays faster than {1/|x|} as {x \rightarrow \infty}; for several of the applications (particularly at very low energies) we need to in fact assume that {V} is a strongly short range potential, which roughly speaking means that it decays faster than {1/|x|^2}.

To begin with, we make no hypotheses about the topology or geodesic geometry of the manifold {M}; in particular, we allow {M} to be trapping in the sense that it contains geodesic flows that do not escape to infinity, but instead remain trapped in a bounded subset of {M}. We also allow the potential {V} to be signed, which in particular allows bound states (eigenfunctions of negative energy) to be created. For standard technical reasons we restrict attention to dimensions three and higher: {d \geq 3}.

It is well known that such Schrödinger operators {H} are essentially self-adjoint, and their spectrum consists of purely absolutely continuous spectrum on {(0,+\infty)}, together with possibly some eigenvalues at zero and negative energy (and at zero energy and in dimensions three and four, there are also the possibility of resonances which, while not strictly eigenvalues, have a somewhat analogous effect on the dynamics of the Laplacian and related objects, such as resolvents). In particular, the resolvents {R(\lambda \pm i\epsilon) := (H - \lambda \mp i\epsilon)^{-1}} make sense as bounded operators on {L^2(M)} for any {\lambda \in {\bf R}} and {\epsilon > 0}. As discussed in the previous blog post, it is of interest to obtain bounds for the behaviour of these resolvents, as this can then be used via some functional calculus manipulations to obtain control on many other operators and PDE relating to the Schrödinger operator {H}, such as the Helmholtz equation, the time-dependent Schrödinger equation, and the wave equation. In particular, it is of interest to obtain limiting absorption estimates such as

\displaystyle  \| R(\lambda \pm i\epsilon) f \|_{H^{0,-1/2-\sigma}(M)} \leq C(M,V,\lambda,\sigma) \| f \|_{H^{0,1/2+\sigma}(M)} \ \ \ \ \ (1)

for {\lambda \in {\bf R}} (and particularly in the positive energy regime {\lambda>0}), where {\sigma,\epsilon > 0} and {f} is an arbitrary test function. The constant {C(M,V,\lambda,\sigma)} needs to be independent of {\epsilon} for such estimates to be truly useful, but it is also of interest to determine the extent to which these constants depend on {M}, {V}, and {\lambda}. The dependence on {\sigma} is relatively uninteresting and henceforth we will suppress it. In particular, our paper focused to a large extent on quantitative methods that could give effective bounds on {C(M,V,\lambda)} in terms of quantities such as the magnitude {A} of the potential {V} in a suitable norm.

It turns out to be convenient to distinguish between three regimes:

  • The high-energy regime {\lambda \gg 1};
  • The medium-energy regime {\lambda \sim 1}; and
  • The low-energy regime {0 < \lambda \ll 1}.

Our methods actually apply more or less uniformly to all three regimes, but the nature of the conclusions is quite different in each of the three regimes.

The high-energy regime {\lambda \gg 1} was essentially worked out by Burq, although we give an independent treatment of Burq’s results here. In this regime it turns out that we have an unconditional estimate of the form (1) with a constant of the shape

\displaystyle  C(M,V,\lambda) = C(M,A) e^{C(M,A) \sqrt{\lambda}}

where {C(M,A)} is a constant that depends only on {M} and on a parameter {A} that controls the size of the potential {V}. This constant, while exponentially growing, is still finite, which among other things is enough to rule out the possibility that {H} contains eigenfunctions (i.e. point spectrum) embedded in the high-energy portion of the spectrum. As is well known, if {M} contains a certain type of trapped geodesic (in particular those arising from positively curved portions of the manifold, such as the equator of a sphere), then it is possible to construct pseudomodes {f} that show that this sort of exponential growth is necessary. On the other hand, if we make the non-trapping hypothesis that all geodesics in {M} escape to infinity, then we can obtain a much stronger high-energy limiting absorption estimate, namely

\displaystyle  C(M,V,\lambda,\sigma) = C(M,A) \lambda^{-1/2}.

The exponent {1/2} here is closely related to the standard fact that on non-trapping manifolds, there is a local smoothing effect for the time-dependent Schrödinger equation that gains half a derivative of regularity (cf. previous blog post). In the high-energy regime, the dynamics are well-approximated by semi-classical methods, and in particular one can use tools such as the positive commutator method and pseudo-differential calculus to obtain the desired estimates. In case of trapping one also needs the standard technique of Carleman inequalities to control the compact (and possibly trapping) core of the manifold, and in particular needing the delicate two-weight Carleman inequalities of Burq.

In the medium and low energy regimes one needs to work harder. In the medium energy regime {\lambda \sim 1}, we were able to obtain a uniform bound

\displaystyle  C(M,V,\lambda) \leq C(M,A)

for all asymptotically conic manifolds (trapping or not) and all short-range potentials. To establish this bound, we have to supplement the existing tools of the positive commutator method and Carleman inequalities with an additional ODE-type analysis of various energies of the solution {u = R(\lambda \pm i\epsilon) f} to a Helmholtz equation on large spheres, as will be discussed in more detail below the fold.

The methods also extend to the low-energy regime {0 < \lambda \ll 1}. Here, the bounds become somewhat interesting, with a subtle distinction between effective estimates that are uniform over all potentials {V} which are bounded in a suitable sense by a parameter {A} (e.g. obeying {|V(x)| \leq A \langle x \rangle^{-2-2\sigma}} for all {x}), and ineffective estimates that exploit qualitative properties of {V} (such as the absence of eigenfunctions or resonances at zero) and are thus not uniform over {V}. On the effective side, and for potentials that are strongly short range (at least at local scales {|x| = O(\lambda^{-1/2})}; one can tolerate merely short-range behaviour at more global scales, but this is a technicality that we will not discuss further here) we were able to obtain a polynomial bound of the form

\displaystyle  C(M,V,\lambda) \leq C(M,A) \lambda^{-C(M,A)}

that blew up at a large polynomial rate at the origin. Furthermore, by carefully designing a sequence of potentials {V} that induce near-eigenfunctions that resemble two different Bessel functions of the radial variable glued together, we are able to show that this type of polynomial bound is sharp in the following sense: given any constant {C > 0}, there exists a sequence {V_n} of potentials on Euclidean space {{\bf R}^d} uniformly bounded by {A}, and a sequence {\lambda_n} of energies going to zero, such that

\displaystyle  C({\bf R}^d,V_n,\lambda_n) \geq \lambda_n^{-C}.

This shows that if one wants bounds that are uniform in the potential {V}, then arbitrary polynomial blowup is necessary.

Interestingly, though, if we fix the potential {V}, and then ask for bounds that are not necessarily uniform in {V}, then one can do better, as was already observed in a classic paper of Jensen and Kato concerning power series expansions of the resolvent near the origin. In particular, if we make the spectral assumption that {V} has no eigenfunctions or resonances at zero, then an argument (based on (a variant of) the Fredholm alternative, which as discussed in this recent blog post gives ineffective bounds) gives a bound of the form

\displaystyle  C(M,V,\lambda) \leq C(M,V) \lambda^{-1/2}

in the low-energy regime (but note carefully here that the constant {C(M,V)} on the right-hand side depends on the potential {V} itself, and not merely on the parameter {A} that upper bounds it). Even if there are eigenvalues or resonances, it turns out that one can still obtain a similar bound but with an exponent of {\lambda^{-3/2}} instead of {\lambda^{-1/2}}. This limited blowup at infinity is in sharp contrast to the arbitrarily large polynomial blowup rate that can occur if one demands uniform bounds. (This particular subtlety between uniform and non-uniform estimates confused us, by the way, for several weeks; for a long time we thought that we had somehow found a contradiction between our results and the results of Jensen and Kato.)

As applications of our limiting absorption estimates, we give local smoothing and dispersive estimates for solutions (as well as the closely related RAGE type theorems) to the time-dependent Schrödinger and wave equations, and also reprove standard facts about the spectrum of Schrödinger operators in this setting.

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Perhaps the most fundamental differential operator on Euclidean space {{\bf R}^d} is the Laplacian

\displaystyle  \Delta := \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}.

The Laplacian is a linear translation-invariant operator, and as such is necessarily diagonalised by the Fourier transform

\displaystyle  \hat f(\xi) := \int_{{\bf R}^d} f(x) e^{-2\pi i x \cdot \xi}\ dx.

Indeed, we have

\displaystyle  \widehat{\Delta f}(\xi) = - 4 \pi^2 |\xi|^2 \hat f(\xi)

for any suitably nice function {f} (e.g. in the Schwartz class; alternatively, one can work in very rough classes, such as the space of tempered distributions, provided of course that one is willing to interpret all operators in a distributional or weak sense).

Because of this explicit diagonalisation, it is a straightforward manner to define spectral multipliers {m(-\Delta)} of the Laplacian for any (measurable, polynomial growth) function {m: [0,+\infty) \rightarrow {\bf C}}, by the formula

\displaystyle  \widehat{m(-\Delta) f}(\xi) := m( 4\pi^2 |\xi|^2 ) \hat f(\xi).

(The presence of the minus sign in front of the Laplacian has some minor technical advantages, as it makes {-\Delta} positive semi-definite. One can also define spectral multipliers more abstractly from general functional calculus, after establishing that the Laplacian is essentially self-adjoint.) Many of these multipliers are of importance in PDE and analysis, such as the fractional derivative operators {(-\Delta)^{s/2}}, the heat propagators {e^{t\Delta}}, the (free) Schrödinger propagators {e^{it\Delta}}, the wave propagators {e^{\pm i t \sqrt{-\Delta}}} (or {\cos(t \sqrt{-\Delta})} and {\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}}, depending on one’s conventions), the spectral projections {1_I(\sqrt{-\Delta})}, the Bochner-Riesz summation operators {(1 + \frac{\Delta}{4\pi^2 R^2})_+^\delta}, or the resolvents {R(z) := (-\Delta-z)^{-1}}.

Each of these families of multipliers are related to the others, by means of various integral transforms (and also, in some cases, by analytic continuation). For instance:

  1. Using the Laplace transform, one can express (sufficiently smooth) multipliers in terms of heat operators. For instance, using the identity

    \displaystyle  \lambda^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{-t\lambda}\ dt

    (using analytic continuation if necessary to make the right-hand side well-defined), with {\Gamma} being the Gamma function, we can write the fractional derivative operators in terms of heat kernels:

    \displaystyle  (-\Delta)^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{t\Delta}\ dt. \ \ \ \ \ (1)

  2. Using analytic continuation, one can connect heat operators {e^{t\Delta}} to Schrödinger operators {e^{it\Delta}}, a process also known as Wick rotation. Analytic continuation is a notoriously unstable process, and so it is difficult to use analytic continuation to obtain any quantitative estimates on (say) Schrödinger operators from their heat counterparts; however, this procedure can be useful for propagating identities from one family to another. For instance, one can derive the fundamental solution for the Schrödinger equation from the fundamental solution for the heat equation by this method.
  3. Using the Fourier inversion formula, one can write general multipliers as integral combinations of Schrödinger or wave propagators; for instance, if {z} lies in the upper half plane {{\bf H} := \{ z \in {\bf C}: \hbox{Im} z > 0 \}}, one has

    \displaystyle  \frac{1}{x-z} = i\int_0^\infty e^{-itx} e^{itz}\ dt

    for any real number {x}, and thus we can write resolvents in terms of Schrödinger propagators:

    \displaystyle  R(z) = i\int_0^\infty e^{it\Delta} e^{itz}\ dt. \ \ \ \ \ (2)

    In a similar vein, if {k \in {\bf H}}, then

    \displaystyle  \frac{1}{x^2-k^2} = \frac{i}{k} \int_0^\infty \cos(tx) e^{ikt}\ dt

    for any {x>0}, so one can also write resolvents in terms of wave propagators:

    \displaystyle  R(k^2) = \frac{i}{k} \int_0^\infty \cos(t\sqrt{-\Delta}) e^{ikt}\ dt. \ \ \ \ \ (3)

  4. Using the Cauchy integral formula, one can express (sufficiently holomorphic) multipliers in terms of resolvents (or limits of resolvents). For instance, if {t > 0}, then from the Cauchy integral formula (and Jordan’s lemma) one has

    \displaystyle  e^{itx} = \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} \frac{e^{ity}}{y-x+i\epsilon}\ dy

    for any {x \in {\bf R}}, and so one can (formally, at least) write Schrödinger propagators in terms of resolvents:

    \displaystyle  e^{-it\Delta} = - \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} e^{ity} R(y+i\epsilon)\ dy. \ \ \ \ \ (4)

  5. The imaginary part of {\frac{1}{\pi} \frac{1}{x-(y+i\epsilon)}} is the Poisson kernel {\frac{\epsilon}{\pi} \frac{1}{(y-x)^2+\epsilon^2}}, which is an approximation to the identity. As a consequence, for any reasonable function {m(x)}, one has (formally, at least)

    \displaystyle  m(x) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} \frac{1}{x-(y+i\epsilon)}) m(y)\ dy

    which leads (again formally) to the ability to express arbitrary multipliers in terms of imaginary (or skew-adjoint) parts of resolvents:

    \displaystyle  m(-\Delta) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} R(y+i\epsilon)) m(y)\ dy. \ \ \ \ \ (5)

    Among other things, this type of formula (with {-\Delta} replaced by a more general self-adjoint operator) is used in the resolvent-based approach to the spectral theorem (by using the limiting imaginary part of resolvents to build spectral measure). Note that one can also express {\hbox{Im} R(y+i\epsilon)} as {\frac{1}{2i} (R(y+i\epsilon) - R(y-i\epsilon))}.

Remark 1 The ability of heat operators, Schrödinger propagators, wave propagators, or resolvents to generate other spectral multipliers can be viewed as a sort of manifestation of the Stone-Weierstrass theorem (though with the caveat that the spectrum of the Laplacian is non-compact and so the Stone-Weierstrass theorem does not directly apply). Indeed, observe the *-algebra type properties

\displaystyle  e^{s\Delta} e^{t\Delta} = e^{(s+t)\Delta}; \quad (e^{s\Delta})^* = e^{s\Delta}

\displaystyle  e^{is\Delta} e^{it\Delta} = e^{i(s+t)\Delta}; \quad (e^{is\Delta})^* = e^{-is\Delta}

\displaystyle  e^{is\sqrt{-\Delta}} e^{it\sqrt{-\Delta}} = e^{i(s+t)\sqrt{-\Delta}}; \quad (e^{is\sqrt{-\Delta}})^* = e^{-is\sqrt{-\Delta}}

\displaystyle  R(z) R(w) = \frac{R(w)-R(z)}{z-w}; \quad R(z)^* = R(\overline{z}).

Because of these relationships, it is possible (in principle, at least), to leverage one’s understanding one family of spectral multipliers to gain control on another family of multipliers. For instance, the fact that the heat operators {e^{t\Delta}} have non-negative kernel (a fact which can be seen from the maximum principle, or from the Brownian motion interpretation of the heat kernels) implies (by (1)) that the fractional integral operators {(-\Delta)^{-s/2}} for {s>0} also have non-negative kernel. Or, the fact that the wave equation enjoys finite speed of propagation (and hence that the wave propagators {\cos(t\sqrt{-\Delta})} have distributional convolution kernel localised to the ball of radius {|t|} centred at the origin), can be used (by (3)) to show that the resolvents {R(k^2)} have a convolution kernel that is essentially localised to the ball of radius {O( 1 / |\hbox{Im}(k)| )} around the origin.

In this post, I would like to continue this theme by using the resolvents {R(z) = (-\Delta-z)^{-1}} to control other spectral multipliers. These resolvents are well-defined whenever {z} lies outside of the spectrum {[0,+\infty)} of the operator {-\Delta}. In the model three-dimensional case {d=3}, they can be defined explicitly by the formula

\displaystyle  R(k^2) f(x) = \int_{{\bf R}^3} \frac{e^{ik|x-y|}}{4\pi |x-y|} f(y)\ dy

whenever {k} lives in the upper half-plane {\{ k \in {\bf C}: \hbox{Im}(k) > 0 \}}, ensuring the absolute convergence of the integral for test functions {f}. (In general dimension, explicit formulas are still available, but involve Bessel functions. But asymptotically at least, and ignoring higher order terms, one simply replaces {\frac{e^{ik|x-y|}}{4\pi |x-y|}} by {\frac{e^{ik|x-y|}}{c_d |x-y|^{d-2}}} for some explicit constant {c_d}.) It is an instructive exercise to verify that this resolvent indeed inverts the operator {-\Delta-k^2}, either by using Fourier analysis or by Green’s theorem.

Henceforth we restrict attention to three dimensions {d=3} for simplicity. One consequence of the above explicit formula is that for positive real {\lambda > 0}, the resolvents {R(\lambda+i\epsilon)} and {R(\lambda-i\epsilon)} tend to different limits as {\epsilon \rightarrow 0}, reflecting the jump discontinuity in the resolvent function at the spectrum; as one can guess from formulae such as (4) or (5), such limits are of interest for understanding many other spectral multipliers. Indeed, for any test function {f}, we see that

\displaystyle  \lim_{\epsilon \rightarrow 0^+} R(\lambda+i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy

and

\displaystyle  \lim_{\epsilon \rightarrow 0^+} R(\lambda-i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{-i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy.

Both of these functions

\displaystyle  u_\pm(x) := \int_{{\bf R}^3} \frac{e^{\pm i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy

solve the Helmholtz equation

\displaystyle  (-\Delta-\lambda) u_\pm = f, \ \ \ \ \ (6)

but have different asymptotics at infinity. Indeed, if {\int_{{\bf R}^3} f(y)\ dy = A}, then we have the asymptotic

\displaystyle  u_\pm(x) = \frac{A e^{\pm i \sqrt{\lambda}|x|}}{4\pi|x|} + O( \frac{1}{|x|^2}) \ \ \ \ \ (7)

as {|x| \rightarrow \infty}, leading also to the Sommerfeld radiation condition

\displaystyle  u_\pm(x) = O(\frac{1}{|x|}); \quad (\partial_r \mp i\sqrt{\lambda}) u_\pm(x) = O( \frac{1}{|x|^2}) \ \ \ \ \ (8)

where {\partial_r := \frac{x}{|x|} \cdot \nabla_x} is the outgoing radial derivative. Indeed, one can show using an integration by parts argument that {u_\pm} is the unique solution of the Helmholtz equation (6) obeying (8) (see below). {u_+} is known as the outward radiating solution of the Helmholtz equation (6), and {u_-} is known as the inward radiating solution. Indeed, if one views the function {u_\pm(t,x) := e^{-i\lambda t} u_\pm(x)} as a solution to the inhomogeneous Schrödinger equation

\displaystyle  (i\partial_t + \Delta) u_\pm = - e^{-i\lambda t} f

and using the de Broglie law that a solution to such an equation with wave number {k \in {\bf R}^3} (i.e. resembling {A e^{i k \cdot x}} for some amplitide {A}) should propagate at (group) velocity {2k}, we see (heuristically, at least) that the outward radiating solution will indeed propagate radially away from the origin at speed {2\sqrt{\lambda}}, while inward radiating solution propagates inward at the same speed.

There is a useful quantitative version of the convergence

\displaystyle  R(\lambda \pm i\epsilon) f \rightarrow u_\pm, \ \ \ \ \ (9)

known as the limiting absorption principle:

Theorem 1 (Limiting absorption principle) Let {f} be a test function on {{\bf R}^3}, let {\lambda > 0}, and let {\sigma > 0}. Then one has

\displaystyle  \| R(\lambda \pm i\epsilon) f \|_{H^{0,-1/2-\sigma}({\bf R}^3)} \leq C_\sigma \lambda^{-1/2} \|f\|_{H^{0,1/2+\sigma}({\bf R}^3)}

for all {\epsilon > 0}, where {C_\sigma > 0} depends only on {\sigma}, and {H^{0,s}({\bf R}^3)} is the weighted norm

\displaystyle  \|f\|_{H^{0,s}({\bf R}^3)} := \| \langle x \rangle^s f \|_{L^2_x({\bf R}^3)}

and {\langle x \rangle := (1+|x|^2)^{1/2}}.

This principle allows one to extend the convergence (9) from test functions {f} to all functions in the weighted space {H^{0,1/2+\sigma}} by a density argument (though the radiation condition (8) has to be adapted suitably for this scale of spaces when doing so). The weighted space {H^{0,-1/2-\sigma}} on the left-hand side is optimal, as can be seen from the asymptotic (7); a duality argument similarly shows that the weighted space {H^{0,1/2+\sigma}} on the right-hand side is also optimal.

We prove this theorem below the fold. As observed long ago by Kato (and also reproduced below), this estimate is equivalent (via a Fourier transform in the spectral variable {\lambda}) to a useful estimate for the free Schrödinger equation known as the local smoothing estimate, which in particular implies the well-known RAGE theorem for that equation; it also has similar consequences for the free wave equation. As we shall see, it also encodes some spectral information about the Laplacian; for instance, it can be used to show that the Laplacian has no eigenvalues, resonances, or singular continuous spectrum. These spectral facts are already obvious from the Fourier transform representation of the Laplacian, but the point is that the limiting absorption principle also applies to more general operators for which the explicit diagonalisation afforded by the Fourier transform is not available. (Igor Rodnianski and I are working on a paper regarding this topic, of which I hope to say more about soon.)

In order to illustrate the main ideas and suppress technical details, I will be a little loose with some of the rigorous details of the arguments, and in particular will be manipulating limits and integrals at a somewhat formal level.

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A few days ago, I found myself needing to use the Fredholm alternative in functional analysis:

Theorem 1 (Fredholm alternative) Let {X} be a Banach space, let {T: X \rightarrow X} be a compact operator, and let {\lambda \in {\bf C}} be non-zero. Then exactly one of the following statements hold:

  • (Eigenvalue) There is a non-trivial solution {x \in X} to the equation {Tx = \lambda x}.
  • (Bounded resolvent) The operator {T-\lambda} has a bounded inverse {(T-\lambda)^{-1}} on {X}.

Among other things, the Fredholm alternative can be used to establish the spectral theorem for compact operators. A hypothesis such as compactness is necessary; the shift operator {U} on {\ell^2({\bf Z})}, for instance, has no eigenfunctions, but {U-z} is not invertible for any unit complex number {z}. The claim is also false when {\lambda=0}; consider for instance the multiplication operator {Tf(n) := \frac{1}{n} f(n)} on {\ell^2({\bf N})}, which is compact and has no eigenvalue at zero, but is not invertible.

It had been a while since I had studied the spectral theory of compact operators, and I found that I could not immediately reconstruct a proof of the Fredholm alternative from first principles. So I set myself the exercise of doing so. I thought that I had managed to establish the alternative in all cases, but as pointed out in comments, my argument is restricted to the case where the compact operator {T} is approximable, which means that it is the limit of finite rank operators in the uniform topology. Many Banach spaces (and in particular, all Hilbert spaces) have the approximation property that implies (by a result of Grothendieck) that all compact operators on that space are almost finite rank. For instance, if {X} is a Hilbert space, then any compact operator is approximable, because any compact set can be approximated by a finite-dimensional subspace, and in a Hilbert space, the orthogonal projection operator to a subspace is always a contraction. (In more general Banach spaces, finite-dimensional subspaces are still complemented, but the operator norm of the projection can be large.) Unfortunately, there are examples of Banach spaces for which the approximation property fails; the first such examples were discovered by Enflo, and a subsequent paper of by Alexander demonstrated the existence of compact operators in certain Banach spaces that are not approximable.

I also found out that this argument was essentially also discovered independently by by MacCluer-Hull and by Uuye. Nevertheless, I am recording this argument here, together with two more traditional proofs of the Fredholm alternative (based on the Riesz lemma and a continuity argument respectively).

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