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Sumfree sets in groups

11 March, 2016 in math.CO, paper | Tags: , , | by | 11 comments

Van Vu and I just posted to the arXiv our paper “sum-free sets in groups” (submitted to Discrete Analysis), as well as a companion survey article (submitted to J. Comb.). Given a subset {A} of an additive group {G = (G,+)}, define the quantity {\phi(A)} to be the cardinality of the largest subset {B} of {A} which is sum-free in {A} in the sense that all the sums {b_1+b_2} with {b_1,b_2} distinct elements of {B} lie outside of {A}. For instance, if {A} is itself a group, then {\phi(A)=1}, since no two elements of {A} can sum to something outside of {A}. More generally, if {A} is the union of {k} groups, then {\phi(A)} is at most {k}, thanks to the pigeonhole principle.

If {G} is the integers, then there are no non-trivial subgroups, and one can thus expect {\phi(A)} to start growing with {A}. For instance, one has the following easy result:

Proposition 1 Let {A} be a set of {2^k} natural numbers. Then {\phi(A) > k}.

Proof: We use an argument of Ruzsa, which is based in turn on an older argument of Choi. Let {x_1} be the largest element of {A}, and then recursively, once {x_1,\dots,x_i} has been selected, let {x_{i+1}} be the largest element of {A} not equal to any of the {x_1,\dots,x_i}, such that {x_{i+1}+x_j \not \in A} for all {j=1,\dots,i}, terminating this construction when no such {x_{i+1}} can be located. This gives a sequence {x_1 > x_2 > \dots > x_m} of elements in {A} which are sum-free in {A}, and with the property that for any {y \in A}, either {y} is equal to one of the {x_i}, or else {y + x_i \in A} for some {i} with {x_i > y}. Iterating this, we see that any {y \in A} is of the form {x_{i_1} - x_{i_2} - \dots - x_{i_j}} for some {j \geq 1} and {1 \leq i_1 < i_2 < \dots \leq i_j \leq m}. The number of such expressions {x_{i_1} - x_{i_2} - \dots - x_{i_j}} is at most {2^{m}-1}, thus {2^k \leq 2^m-1} which implies {m \geq k+1}. Since {\phi(A) \geq m}, the claim follows. \Box

In particular, we have {\phi(A) \gg \log |A|} for subsets {A} of the integers. It has been possible to improve upon this easy bound, but only with remarkable effort. The best lower bound currently is

\displaystyle \phi(A) \geq \log |A| (\log\log|A|)^{1/2 - o(1)},

a result of Shao (building upon earlier work of Sudakov, Szemeredi, and Vu and of Dousse). In the opposite direction, a construction of Ruzsa gives examples of large sets {A} with {\phi(A) \leq \exp( O( \sqrt{\log |A|} ) )}.

Using the standard tool of Freiman homomorphisms, the above results for the integers extend to other torsion-free abelian groups {G}. In our paper we study the opposite case where {G} is finite (but still abelian). In this paper of Erdös (in which the quantity {\phi(A)} was first introduced), the following question was posed: if {A} is sufficiently large depending on {\phi(A)}, does this imply the existence of two elements {x,y \in A} with {x+y=0}? As it turns out, we were able to find some simple counterexamples to this statement. For instance, if {H} is any finite additive group, then the set {A := \{ 1 \hbox{ mod } 7, 2 \hbox{ mod } 7, 4 \hbox{ mod } 7\} \times H \subset {\bf Z}/7{\bf Z} \times H} has {\phi(A)=3} but with no {x,y \in A} summing to zero; this type of example in fact works with {7} replaced by any larger Mersenne prime, and we also have a counterexample in {{\bf Z}/2^n{\bf Z}} for {n} arbitrarily large. However, in the positive direction, we can show that the answer to Erdös’s question is positive if {|G|} is assumed to have no small prime factors. That is to say,

Theorem 2 For every {k \geq 1} there exists {C \geq 1} such that if {G} is a finite abelian group whose order is not divisible by any prime less than or equal to {C}, and {A} is a subset of {G} with order at least {C} and {\phi(A) \leq k}, then there exist {x,y \in A} with {x+y=0}.

There are two main tools used to prove this result. One is an “arithmetic removal lemma” proven by Král, Serra, and Vena. Note that the condition {\phi(A) \leq k} means that for any distinct {x_1,\dots,x_{k+1} \in A}, at least one of the {x_i+x_j}, {1 \leq i < j \leq k+1}, must also lie in {A}. Roughly speaking, the arithmetic removal lemma allows one to “almost” remove the requirement that {x_1,\dots,x_{k+1}} be distinct, which basically now means that {x \in A \implies 2x \in A} for almost all {x \in A}. This near-dilation symmetry, when combined with the hypothesis that {|G|} has no small prime factors, gives a lot of “dispersion” in the Fourier coefficients of {1_A} which can now be exploited to prove the theorem.

The second tool is the following structure theorem, which is the main result of our paper, and goes a fair ways towards classifying sets {A} for which {\phi(A)} is small:

Theorem 3 Let {A} be a finite subset of an arbitrary additive group {G}, with {\phi(A) \leq k}. Then one can find finite subgroups {H_1,\dots,H_m} with {m \leq k} such that {|A \cap H_i| \gg_k |H_i|} and {|A \backslash (H_1 \cup \dots \cup H_m)| \ll_k 1}. Furthermore, if {m=k}, then the exceptional set {A \backslash (H_1 \cup \dots \cup H_m)} is empty.

Roughly speaking, this theorem shows that the example of the union of {k} subgroups mentioned earlier is more or less the “only” example of sets {A} with {\phi(A) \leq k}, modulo the addition of some small exceptional sets and some refinement of the subgroups to dense subsets.

This theorem has the flavour of other inverse theorems in additive combinatorics, such as Freiman’s theorem, and indeed one can use Freiman’s theorem (and related tools, such as the Balog-Szemeredi theorem) to easily get a weaker version of this theorem. Indeed, if there are no sum-free subsets of {A} of order {k+1}, then a fraction {\gg_k 1} of all pairs {a,b} in {A} must have their sum also in {A} (otherwise one could take {k+1} random elements of {A} and they would be sum-free in {A} with positive probability). From this and the Balog-Szemeredi theorem and Freiman’s theorem (in arbitrary abelian groups, as established by Green and Ruzsa), we see that {A} must be “commensurate” with a “coset progression” {H+P} of bounded rank. One can then eliminate the torsion-free component {P} of this coset progression by a number of methods (e.g. by using variants of the argument in Proposition 1), with the upshot being that one can locate a finite group {H_1} that has large intersection with {A}.

At this point it is tempting to simply remove {H_1} from {A} and iterate. But one runs into a technical difficulty that removing a set such as {H_1} from {A} can alter the quantity {\phi(A)} in unpredictable ways, so one has to still keep {H_1} around when analysing the residual set {A \backslash H_1}. A second difficulty is that the latter set {A \backslash H_1} could be considerably smaller than {A} or {H_1}, but still large in absolute terms, so in particular any error term whose size is only bounded by {\varepsilon |A|} for a small {\varepsilon} could be massive compared with the residual set {A\backslash H_1}, and so such error terms would be unacceptable. One can get around these difficulties if one first performs some preliminary “normalisation” of the group {H_1}, so that the residual set {A \backslash H_1} does not intersect any coset of {H_1} too strongly. The arguments become even more complicated when one starts removing more than one group {H_1,\dots,H_i} from {A} and analyses the residual set {A \backslash (H_1 \cup \dots \cup H_i)}; indeed the “epsilon management” involved became so fearsomely intricate that we were forced to use a nonstandard analysis formulation of the problem in order to keep the complexity of the argument at a reasonable level (cf. my previous blog post on this topic). One drawback of doing so is that we have no effective bounds for the implied constants in our main theorem; it would be of interest to obtain a more direct proof of our main theorem that would lead to effective bounds.

Random matrices: tail bounds for gaps between eigenvalues

3 April, 2015 in math.PR, math.SP, paper | Tags: , , , | by | 17 comments

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices {M_n} of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power {n^{-A}} of {n}. This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices {M_n} (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on {M_n}) a result of the form

\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq n^{-A} ) \leq n^{-B}

for any {B} and {i}, if {A} is sufficiently large depending on {B} (in a linear fashion), and {n} is sufficiently large depending on {B}. The point here is that {B} can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)

In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of {M_n} have unit variance, so that the eigenvalues of {M_n} are {O(\sqrt{n})} with high probability), giving the bound

\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta \ \ \ \ \ (1)

for {\delta \geq n^{-O(1)}} (one also has good results of this type for smaller values of {\delta}). This is only optimal in the regime {\delta \sim 1}; we expect to establish some eigenvalue repulsion, improving the RHS to {\delta^2} for real matrices and {\delta^3} for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form

\displaystyle {\bf P} (\lambda_{i+k}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta^{ck^2}

for any fixed {k \geq 1} and some absolute constant {c>0} (which we can asymptotically make to be {1/3} for large {k}, though it ought to be as large as {1}), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.

In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of {n^{o(1)}} on the RHS.

As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least {n^{-A}} for some {A} with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)

Random matrices have simple spectrum

4 December, 2014 in math.CO, math.PR, math.SP, paper | Tags: , , , | by | 22 comments

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an {n \times n} Hermitian matrix is said to have simple eigenvalues if all of its {n} eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all {n \times n} Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.

For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix {M_n} of an Erdös-Rényi graph – a graph on {n} vertices in which any pair of vertices has an independent probability {p} of being in the graph. For the purposes of this paper one should view {p} as fixed, e.g. {p=1/2}, while {n} is an asymptotic parameter going to infinity. In this context, our main result is the following (answering a question of Babai):

Theorem 1 With probability {1-o(1)}, {M_n} has simple eigenvalues.

Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap {\lambda_{i+1}(M)-\lambda_i(M)} did not vanish with probability {1-o(1)} (in fact {1-O(n^{-c})} for some absolute constant {c>0}), but because there are {n} different gaps that one has to simultaneously ensure to be non-zero, this did not give Theorem 1 as one is forced to apply the union bound.

Our argument in fact gives simplicity of the spectrum with probability {1-O(n^{-A})} for any fixed {A}; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).

The basic idea of argument can be sketched as follows. Suppose that {M_n} has a repeated eigenvalue {\lambda}. We split

\displaystyle M_n = \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix}

for a random {n-1 \times n-1} minor {M_{n-1}} and a random sign vector {X}; crucially, {X} and {M_{n-1}} are independent. If {M_n} has a repeated eigenvalue {\lambda}, then by the Cauchy interlacing law, {M_{n-1}} also has an eigenvalue {\lambda}. We now write down the eigenvector equation for {M_n} at {\lambda}:

\displaystyle \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix} \begin{pmatrix} v \\ a \end{pmatrix} = \lambda \begin{pmatrix} v \\ a \end{pmatrix}.

Extracting the top {n-1} coefficients, we obtain

\displaystyle (M_{n-1} - \lambda) v + a X = 0.

If we let {w} be the {\lambda}-eigenvector of {M_{n-1}}, then by taking inner products with {w} we conclude that

\displaystyle a (w \cdot X) = 0;

we typically expect {a} to be non-zero, in which case we arrive at

\displaystyle w \cdot X = 0.

In other words, in order for {M_n} to have a repeated eigenvalue, the top right column {X} of {M_n} has to be orthogonal to an eigenvector {w} of the minor {M_{n-1}}. Note that {X} and {w} are going to be independent (once we specify which eigenvector of {M_{n-1}} to take as {w}). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector {X} is unlikely to be orthogonal to any given vector {w} independent of {X}, unless the coefficients of {w} are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)

Local universality of zeroes of random polynomials

17 July, 2013 in math.PR, paper | Tags: , , , , , | by | Leave a comment

Van Vu and I have just uploaded to the arXiv our joint paper “Local universality of zeroes of random polynomials“. This paper is a sequel to our previous work on local universality of eigenvalues of (non-Hermitian) random matrices {M_n} with independent entries. One can re-interpret these previous results as a universality result for a certain type of random polynomial {f: {\bf C} \rightarrow {\bf C}}, namely the characteristic polynomial {f(z) = \hbox{det}(M_n-z)} of the random matrix {M_n}. In this paper, we consider the analogous problem for a different model of random polynomial, namely polynomials {f} with independent random coefficients. More precisely, we consider random polynomials {f = f_n} of the form

\displaystyle f(z) = \sum_{i=0}^n c_i \xi_i z^n

where {c_0,\ldots,c_n \in {\bf C}} are deterministic complex coefficients, and {\xi_0,\ldots,\xi_n \equiv \xi} are independent identically distributed copies of a complex random variable {\xi}, which we normalise to have mean zero and variance one. For simplicity we will ignore the technical issue that the leading coefficient {c_n \xi_n} is allowed to vanish; then {f} has {n} zeroes {\zeta_1,\ldots,\zeta_n \in {\bf C}} (counting multiplicity), which can be viewed as a random point process {\Sigma = \{\zeta_1,\ldots,\zeta_n\}} in the complex plane. In analogy with other models (such as random matrix models), we expect the (suitably normalised) asymptotic statistics of this point process in the limit {n \rightarrow \infty} to be universal, in the sense that it is largely independent of the precise distribution of the atom variable {\xi}.

Our results are fairly general with regard to the choice of coefficients {c_i}, but we isolate three particular choice of coefficients that are particularly natural and well-studied in the literature:

  • Flat polynomials (or Weyl polynomials) in which {c_i := \frac{1}{\sqrt{i!}}}.
  • Elliptic polynomials (or binomial polynomials) in which {c_i := \sqrt{\binom{n}{i}}}.
  • Kac polynomials in which {c_i := 1}.

The flat and elliptic polynomials enjoy special symmetries in the model case when the atom distribution {\xi} is a complex Gaussian {N(0,1)_{\bf C}}. Indeed, the zeroes {\Sigma} of elliptic polynomials with complex Gaussian coefficients have a distribution which is invariant with respect to isometries {T: {\bf C} \cup \{\infty\} \rightarrow {\bf C} \cup \{\infty\}} of the Riemann sphere {{\bf C} \cup \{\infty\}} (thus {T\Sigma} has the same distribution as {\Sigma}), while the zeroes of the limiting case {\sum_{i=0}^\infty \frac{1}{\sqrt{i!}} \xi_i z^i} of the flat polynomials with complex Gaussian coefficients are similarly invariant with respect to isometries {T: {\bf C} \rightarrow {\bf C}} of the complex plane {{\bf C}}. (For a nice geometric proof of this facts, I recommend the nice book of Hough, Krishnapur, Peres, and Virag.)

The global (i.e. coarse-scale) distribution of zeroes of these polynomials is well understood, first in the case of gaussian distributions using the fundamental tool of the Kac-Rice formula, and then for more general choices of atom distribution in the recent work of Kabluchko and Zaporozhets. The zeroes of the flat polynomials are asymptotically distributed according to the circular law, normalised to be uniformly distributed on the disk {B(0,\sqrt{n})} of radius {\sqrt{n}} centred at the origin. To put it a bit informally, the zeroes are asymptotically distributed according to the measure {\frac{1}{\pi} 1_{|z| \leq \sqrt{n}} dz}, where {dz} denotes Lebesgue measure on the complex plane. One can non-rigorously see the scale {\sqrt{n}} appear by observing that when {|z|} is much larger than {\sqrt{n}}, we expect the leading term {\frac{1}{\sqrt{n!}} \xi_n z^n} of the flat polynomial {\sum_{i=0}^n \frac{1}{\sqrt{i!}} \xi_i z^i} to dominate, so that the polynomial should not have any zeroes in this region.

Similarly, the distribution of the elliptic polynomials is known to be asymptotically distributed according to a Cauchy-type distribution {\frac{1}{\pi} \frac{1}{1+|z|^2/n} dz}. The Kac polynomials {\sum_{i=0}^n \xi_i z^i} behave differently; the zeroes concentrate uniformly on the unit circle {|z|=1} (which is reasonable, given that one would expect the top order term {\xi_i z^i} to dominate for {|z| > 1} and the bottom order term {\xi_0} to dominate for {|z| < 1}). In particular, whereas the typical spacing between zeroes in the flat and elliptic cases would be expected to be comparable to {1}, the typical spacing between zeroes for a Kac polynomial would be expected instead to be comparable to {1/n}.

In our paper we studied the local distribution of zeroes at the scale of the typical spacing. In the case of polynomials with continuous complex atom disribution {\xi}, the natural statistic to measure here is the {k}-point correlation function {\rho^{(k)}(z_1,\ldots,z_k)}, which for distinct complex numbers {z_1,\ldots,z_k} can be defined as the probability that there is a zero in each of the balls {B(z_1,\varepsilon),\ldots,B(z_k,\varepsilon)} for some infinitesimal {\epsilon >0}, divided by the normalising factor {(\pi \epsilon^2)^k}. (One can also define a {k}-point correlation function in the case of a discrete distribution, but it will be a measure rather than a function in that case.) Our first main theorem is a general replacement principle which asserts, very roughly speaking, that the asymptotic {k}-point correlation functions of two random polynomials {f, \tilde f} will agree if the log-magnitudes {\log |f(z)|, \log |\tilde f(z)|} have asymptotically the same distribution (actually we have to consider the joint distribution of {\log |f(z_1)|, \ldots \log |f(z_k)|} for several points {z_1,\ldots,z_k}, but let us ignore this detail for now), and if the polynomials {f, \tilde f} obeys a “non-clustering property” which asserts, roughly speaking, that not too many of the zeroes of {f} can typically concentrate in a small ball. This replacement principle was implicit in our previous paper (and can be viewed as a local-scale version of the global-scale replacement principle in this earlier paper of ours). Specialising the replacement principle to the elliptic or flat polynomials, and using the Lindeberg swapping argument, we obtain a Two Moment Theorem that asserts, roughly speaking, that the asymptotic behaviour of the {k}-point correlation functions depends only on the first two moments of the real and imaginary components of {\xi}, as long as one avoids some regions of space where universality is expected to break down. (In particular, because {f(0) = c_0 \xi_0} does not have a universal distribution, one does not expect universality to hold near the origin; there is a similar problem near infinity.) Closely related results, by a slightly different method, have also been obtained recently by Ledoan, Merkli, and Starr. A similar result holds for the Kac polynomials after rescaling to account for the narrower spacing between zeroes.

We also have analogous results in the case of polynomials with real coefficients (so that the coefficients {c_i} and the atom distribution {\xi} are both real). In this case one expects to see a certain number of real zeroes, together with conjugate pairs of complex zeroes. Instead of the {k}-point correlation function {\rho^{(k)}(z_1,\ldots,z_k)}, the natural object of study is now the mixed {(k,l)}-point correlation function {\rho^{(k,l)}(x_1,\ldots,x_k,z_1,\ldots,z_l)} that (roughly speaking) controls how often one can find a real zero near the real numbers {x_1,\ldots,x_k}, and a complex zero near the points {z_1,\ldots,z_l}. It turns out that one can disentangle the real and strictly complex zeroes and obtain separate universality results for both zeroes, provided that at least one of the polynomials involved obeys a “weak repulsion estimate” that shows that the real zeroes do not cluster very close to each other (and that the complex zeroes do not cluster very close to their complex conjugates). Such an estimate is needed to avoid the situation of two nearby real zeroes “colliding” to create a (barely) complex zero and its complex conjugate, or the time reversal of such a collision. Fortunately, in the case of real gaussian polynomials one can use the Kac-Rice formula to establish such a weak repulsion estimate, allowing analogues of the above universality results for complex random polynomials in the real case. Among other things, this gives universality results for the number {N_{\bf R}} of real zeroes of a random flat or elliptic polynomial; it turns out this number is typically equal to {\frac{2}{\pi} \sqrt{n} + O(n^{1/2-c})} and {\frac{n} + O(n^{1/2-c})} respectively. (For Kac polynomials, the situation is somewhat different; it was already known that {N_{\bf R} = \frac{2}{\pi} \log n + o(\log n)} thanks to a long series of papers, starting with the foundational work of Kac and culminating in the work of Ibragimov and Maslova.)

While our methods are based on our earlier work on eigenvalues of random matrices, the situation with random polynomials is actually somewhat easier to deal with. This is because the log-magnitude {\log |f(z)|} of a random polynomial with independent coefficients is significantly easier to control than the log-determinant {\log |\hbox{det}(M-z)|} of a random matrix, as the former can be controlled by the central limit theorem, while the latter requires significantly more difficult arguments (in particular, bounds on the least singular value combined with Girko’s Hermitization trick). As such, one could view the current paper as an introduction to our more complicated previous paper, and with this in mind we have written the current paper to be self-contained (though this did make the paper somewhat lengthy, checking in at 68 pages).

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

11 June, 2012 in math.PR, math.SP, paper | Tags: , , , , , | by | 4 comments

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

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

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

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

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

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

for {|z| < 1} and

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Random matrices: The Universality phenomenon for Wigner ensembles

2 February, 2012 in math.PR, paper | Tags: , , , , | by | 4 comments

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: The Universality phenomenon for Wigner ensembles“. This survey is a longer version (58 pages) of a previous short survey we wrote up a few months ago. The survey focuses on recent progress in understanding the universality phenomenon for Hermitian Wigner ensembles, of which the Gaussian Unitary Ensemble (GUE) is the most well known. The one-sentence summary of this progress is that many of the asymptotic spectral statistics (e.g. correlation functions, eigenvalue gaps, determinants, etc.) that were previously known for GUE matrices, are now known for very large classes of Wigner ensembles as well. There are however a wide variety of results of this type, due to the large number of interesting spectral statistics, the varying hypotheses placed on the ensemble, and the different modes of convergence studied, and it is difficult to isolate a single such result currently as the definitive universality result. (In particular, there is at present a tradeoff between generality of ensemble and strength of convergence; the universality results that are available for the most general classes of ensemble are only presently able to demonstrate a rather weak sense of convergence to the universal distribution (involving an additional averaging in the energy parameter), which limits the applicability of such results to a number of interesting questions in which energy averaging is not permissible, such as the study of the least singular value of a Wigner matrix, or of related quantities such as the condition number or determinant. But it is conceivable that this tradeoff is a temporary phenomenon and may be eliminated by future work in this area; in the case of Hermitian matrices whose entries have the same second moments as that of the GUE ensemble, for instance, the need for energy averaging has already been removed.)

Nevertheless, throughout the family of results that have been obtained recently, there are two main methods which have been fundamental to almost all of the recent progress in extending from special ensembles such as GUE to general ensembles. The first method, developed extensively by Erdos, Schlein, Yau, Yin, and others (and building on an initial breakthrough by Johansson), is the heat flow method, which exploits the rapid convergence to equilibrium of the spectral statistics of matrices undergoing Dyson-type flows towards GUE. (An important aspect to this method is the ability to accelerate the convergence to equilibrium by localising the Hamiltonian, in order to eliminate the slowest modes of the flow; this refinement of the method is known as the “local relaxation flow” method. Unfortunately, the translation mode is not accelerated by this process, which is the principal reason why results obtained by pure heat flow methods still require an energy averaging in the final conclusion; it would of interest to find a way around this difficulty.) The other method, which goes all the way back to Lindeberg in his classical proof of the central limit theorem, and which was introduced to random matrix theory by Chatterjee and then developed for the universality problem by Van Vu and myself, is the swapping method, which is based on the observation that spectral statistics of Wigner matrices tend to be stable if one replaces just one or two entries of the matrix with another distribution, with the stability of the swapping process becoming stronger if one assumes that the old and new entries have many matching moments. The main formalisations of this observation are known as four moment theorems, because they require four matching moments between the entries, although there are some variant three moment theorems and two moment theorems in the literature as well. Our initial four moment theorems were focused on individual eigenvalues (and later also to eigenvectors), but it was later observed by Erdos, Yau, and Yin that simpler four moment theorems could also be established for aggregate spectral statistics, such as the coefficients of the Greens function, and Knowles and Yin also subsequently observed that these latter theorems could be used to recover a four moment theorem for eigenvalues and eigenvectors, giving an alternate approach to proving such theorems.

Interestingly, it seems that the heat flow and swapping methods are complementary to each other; the heat flow methods are good at removing moment hypotheses on the coefficients, while the swapping methods are good at removing regularity hypotheses. To handle general ensembles with minimal moment or regularity hypotheses, it is thus necessary to combine the two methods (though perhaps in the future a third method, or a unification of the two existing methods, might emerge).

Besides the heat flow and swapping methods, there are also a number of other basic tools that are also needed in these results, such as local semicircle laws and eigenvalue rigidity, which are also discussed in the survey. We also survey how universality has been established for wide variety of spectral statistics; the {k}-point correlation functions are the most well known of these statistics, but they do not tell the whole story (particularly if one can only control these functions after an averaging in the energy), and there are a number of other statistics, such as eigenvalue counting functions, determinants, or spectral gaps, for which the above methods can be applied.

In order to prevent the survey from becoming too enormous, we decided to restrict attention to Hermitian matrix ensembles, whose entries off the diagonal are identically distributed, as this is the case in which the strongest results are available. There are several results that are applicable to more general ensembles than these which are briefly mentioned in the survey, but they are not covered in detail.

We plan to submit this survey eventually to the proceedings of a workshop on random matrix theory, and will continue to update the references on the arXiv version until the time comes to actually submit the paper.

Finally, in the survey we issue some errata for previous papers of Van and myself in this area, mostly centering around the three moment theorem (a variant of the more widely used four moment theorem), for which the original proof of Van and myself was incomplete. (Fortunately, as the three moment theorem had many fewer applications than the four moment theorem, and most of the applications that it did have ended up being superseded by subsequent papers, the actual impact of this issue was limited, but still an erratum is in order.)

Random matrices: Sharp concentration of eigenvalues

24 January, 2012 in math.PR, math.SP, paper | Tags: , , , | by | 3 comments

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.

Random matrices: The Four Moment Theorem for Wigner ensembles

12 December, 2011 in math.PR, math.SP, paper | Tags: , , , | by | 8 comments

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.

A central limit theorem for the determinant of a Wigner matrix

28 November, 2011 in math.PR, math.SP, paper | Tags: , , , , , | by | 13 comments

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.)

Random matrices: Universality of eigenvectors

16 March, 2011 in math.PR, math.SP, paper | Tags: , , | by | 12 comments

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of eigenvectors“, submitted to Random Matrices: Theory and Applications. This paper concerns an extension of our four moment theorem for eigenvalues. Roughly speaking, that four moment theorem asserts (under mild decay conditions on the coefficients of the random matrix) that the fine-scale structure of individual eigenvalues of a Wigner random matrix depend only on the first four moments of each of the entries.

In this paper, we extend this result from eigenvalues to eigenvectors, and specifically to the coefficients of, say, the {i^{th}} eigenvector {u_i(M_n)} of a Wigner random matrix {M_n}. Roughly speaking, the main result is that the distribution of these coefficients also only depends on the first four moments of each of the entries. In particular, as the distribution of coefficients eigenvectors of invariant ensembles such as GOE or GUE are known to be asymptotically gaussian real (in the GOE case) or gaussian complex (in the GUE case), the same asymptotic automatically holds for Wigner matrices whose coefficients match GOE or GUE to fourth order.

(A technical point here: strictly speaking, the eigenvectors {u_i(M_n)} are only determined up to a phase, even when the eigenvalues are simple. So, to phrase the question properly, one has to perform some sort of normalisation, for instance by working with the coefficients of the spectral projection operators {P_i(M_n) := u_i(M_n) u_i(M_n)^*} instead of the eigenvectors, or rotating each eigenvector by a random phase, or by fixing the first component of each eigenvector to be positive real. This is a fairly minor technical issue here, though, and will not be discussed further.)

This theorem strengthens a four moment theorem for eigenvectors recently established by Knowles and Yin (by a somewhat different method), in that the hypotheses are weaker (no level repulsion assumption is required, and the matrix entries only need to obey a finite moment condition rather than an exponential decay condition), and a slightly stronger conclusion (less regularity is needed on the test function, and one can handle the joint distribution of polynomially many coefficients, rather than boundedly many coefficients). On the other hand, the Knowles-Yin paper can also handle generalised Wigner ensembles in which the variances of the entries are allowed to fluctuate somewhat.

The method used here is a variation of that in our original paper (incorporating the subsequent improvements to extend the four moment theorem from the bulk to the edge, and to replace exponential decay by a finite moment condition). That method was ultimately based on the observation that if one swapped a single entry (and its adjoint) in a Wigner random matrix, then an individual eigenvalue {\lambda_i(M_n)} would not fluctuate much as a consequence (as long as one had already truncated away the event of an unexpectedly small eigenvalue gap). The same analysis shows that the projection matrices {P_i(M_n)} obeys the same stability property.

As an application of the eigenvalue four moment theorem, we establish a four moment theorem for the coefficients of resolvent matrices {(\frac{1}{\sqrt{n}} M_n - zI)^{-1}}, even when {z} is on the real axis (though in that case we need to make a level repulsion hypothesis, which has been already verified in many important special cases and is likely to be true in general). This improves on an earlier four moment theorem for resolvents of Erdos, Yau, and Yin, which required {z} to stay some distance away from the real axis (specifically, that {\hbox{Im}(z) \geq n^{-1-c}} for some small {c>0}).

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