The Tracy–Widom laws govern the fluctuations of the largest eigenvalue at the soft edge of Gaussian random-matrix ensembles and recur across probability, statistics and physics. Domínguez-Molina (Statistics & Probability Letters 123 (2017) 56–60) proved that no β-Tracy–Widom law is infinitely divisible and that the largest eigenvalue of a finite N×N GOE or GUE matrix (N ≥ 2) is not infinitely divisible, and listed five structural questions that define this problem: (1) is the classical or β-Tracy–Widom law freely infinitely divisible in the sense of free additive convolution; (2) is the largest eigenvalue of a GSE matrix non-infinitely-divisible for each N ≥ 2; (3) is the Tracy–Widom distribution indecomposable, i.e. not expressible as a nontrivial convolution of two probability laws; (4) what is the infinite-divisibility status of the largest eigenvalue of the tridiagonal β-Hermite ensemble at finite N for general β > 0; (5) which members of the known interpolation families connecting Tracy–Widom to other laws (Gumbel–Tracy–Widom, Tracy–Widom–Gaussian, Tracy–Widom–Weibull) are infinitely divisible away from the Tracy–Widom endpoint.

Background

Random matrix ensembles and the Tracy–Widom law. The Gaussian orthogonal, unitary and symplectic ensembles (GOE, GUE, GSE) are probability measures on, respectively, real symmetric, complex Hermitian and quaternionic Hermitian N×N matrices whose independent entries are centered Gaussians; they are conventionally labelled by the Dyson index β = 1, 2, 4. The joint density of the ordered eigenvalues is the one-dimensional Coulomb-gas density proportional to ∏i<j∣λi−λj∣βexp⁡(−β4∑iλi2)\prod_{i<j}|\lambda_i-\lambda_j|^{\beta}\exp(-\frac{\beta}{4}\sum_i\lambda_i^2), so the eigenvalues are strongly dependent random variables. Tracy and Widom computed the limiting law of the largest eigenvalue: after centering at the spectral edge (of order 2N2\sqrt{N} under this normalization) and scaling by N1/6N^{1/6}, λmax⁡\lambda_{\max} converges to the Tracy–Widom (TW) law FβF_\beta, which is expressed through a solution of the Painlevé II nonlinear ordinary differential equation. Dumitriu and Edelman showed that for every β > 0 the same eigenvalue density is realized by a symmetric tridiagonal matrix HNβH_N^\beta with independent entries — Gaussian N(0,2/β)\mathcal{N}(0,2/\beta) on the diagonal and χkβ/β\chi_{k\beta}/\sqrt{\beta} on the subdiagonal, where χt\chi_t denotes the chi distribution with t degrees of freedom (the tridiagonal β-Hermite ensemble) — and Ramírez, Rider and Virág proved that for every β > 0 the largest eigenvalue, so centered and scaled, converges to a limit TWβ\mathrm{TW}_\beta (the β-Tracy–Widom law), realized as the top of the spectrum of the stochastic Airy operator (the random Schrödinger-type operator −d2dx2+x-\frac{d^2}{dx^2}+x on the half-line, perturbed by a white-noise potential); for β = 1, 2, 4 this is the classical TW law. The two-sided tails of TWβ\mathrm{TW}_\beta are 1−Fβ(x)=exp⁡(−2β3x3/2(1+o(1)))1-F_\beta(x)=\exp(-\frac{2\beta}{3}x^{3/2}(1+o(1))) and Fβ(−x)=exp⁡(−β24x3(1+o(1)))F_\beta(-x)=\exp(-\frac{\beta}{24}x^{3}(1+o(1))) as x→∞x\to\infty.

Infinite divisibility. A random variable X is infinitely divisible (ID) if for every n ≥ 1 there exist independent identically distributed X1,…,XnX_1,\dots,X_n with X=dX1+⋯+XnX\stackrel{d}{=}X_1+\cdots+X_n; equivalently, its characteristic function admits a Lévy–Khintchine representation, the canonical decomposition of the characteristic function of an infinitely divisible law into a Gaussian variance, a drift and a jump (Lévy) measure. ID laws are exactly the one-time marginals of Lévy processes (processes with stationary independent increments) and are the canonical limit laws for sums of independent terms. TW, by contrast, is a limit law for the maximum of strongly dependent variables, so whether it can be decomposed into independent parts, or serve as a Lévy marginal, is a natural structural question. Two classical facts from Steutel and van Harn's monograph drive the source paper: (i) a non-degenerate infinitely divisible random variable is Gaussian if and only if lim sup⁡x→∞−log⁡P(∣X∣>x)/(xlog⁡x)=∞\limsup_{x\to\infty}-\log P(|X|>x)/(x\log x)=\infty; consequently, a non-Gaussian law whose two-sided tails are bounded by ae−bxcae^{-bx^{c}} with a, b > 0 and c > 1 is not infinitely divisible; (ii) a nonnegative infinitely divisible random variable satisfies −log⁡P(X>x)≤axlog⁡x-\log P(X>x)\le a x\log x eventually.

The source paper. Domínguez-Molina combined fact (i) with the TW tail asymptotics to prove that TWβ\mathrm{TW}_\beta is not infinitely divisible for any β > 0, and combined it with Gaussian-type large-deviation inequalities for the largest eigenvalue of GOE (Ben Arous–Dembo–Guionnet) and GUE (Ledoux) to prove that λmax⁡\lambda_{\max} of an N×N GOE/GUE matrix is not infinitely divisible for any N ≥ 2 (the N = 2 case having been settled by skewed-symmetric arguments in Domínguez-Molina–Rocha-Arteaga). Using fact (ii) the same note observes that the Wigner surmise (the 2×2 GOE level-spacing law with density π2s e−π2s2/4\frac{\pi}{2}s\,e^{-\pi^2 s^2/4}), ∣TWβ∣|\mathrm{TW}_\beta| and the one-sided truncations of TWβ\mathrm{TW}_\beta are not infinitely divisible on the positive half-line. The GSE case and all general-β finite-N cases were left open, as were the deeper structural questions below.

Free probability. In Voiculescu's free probability, classical independence is replaced by free independence and convolution of probability measures by free additive convolution ⊞\boxplus; a measure is freely infinitely divisible (FID) if for each n it is the n-fold ⊞\boxplus-power of some measure, i.e. infinitely divisible in the noncommutative world. Just as classical cumulants linearize ordinary convolution, free cumulants linearize ⊞\boxplus, and the Voiculescu transform (the free counterpart of the logarithm of the characteristic function) turns free-divisibility questions into analytic ones. The semicircle law (the free analogue of the Gaussian), the Marchenko–Pastur law (free Poisson), the q-Gaussian family (a one-parameter deformation of the Gaussian interpolating between the Gaussian at q = 1 and the semicircle law at q = 0), and many beta, gamma, Student-t and ultraspherical distributions are known to be freely infinitely divisible, while others are provably not. Whether TWβ\mathrm{TW}_\beta — the most prominent law born from random matrix theory beyond the semicircle — is FID is unknown.

Indecomposability and interpolation families. A probability law is indecomposable if it cannot be written as the distribution of Y1+Y2Y_1+Y_2 for independent, non-degenerate Y1,Y2Y_1,Y_2; indecomposability is strictly stronger than non-infinite-divisibility. Johansson constructed a one-parameter family of determinantal point processes (point processes whose correlation functions are determinants built from a kernel) whose edge statistics interpolate between the Gumbel extreme-value law (Poisson edge) and Tracy–Widom (Airy edge), and a related deformed-GUE family whose edge law transitions from Tracy–Widom to Gaussian. Bohigas, de Carvalho and Pato built parametric deformations of the Gaussian ensembles in which the largest-eigenvalue law moves continuously from Tracy–Widom to Gaussian (an external random source superimposed on the ensemble) and from Tracy–Widom to Weibull (random removal of a fraction of the correlated eigenvalues). Nothing is known about the divisibility of the intermediate members of any of these families.

Problem Statement

The common objective is to locate the Tracy–Widom laws and their finite-N Gaussian and β-Hermite ancestors within the divisibility and decomposability hierarchy of probability theory: which of these laws (or families of laws) can be built as sums or free sums of independent pieces, and which are irreducible. The source paper leaves five linked questions, each of which remains unanswered:

  1. Free infinite divisibility. Is the β-Tracy–Widom law TWβ\mathrm{TW}_\beta freely infinitely divisible, i.e. does there exist, for every n ≥ 1, a probability measure μn\mu_n with μn⊞n=TWβ\mu_n^{\boxplus n}=\mathrm{TW}_\beta, for all β > 0 (with the classical cases β = 1, 2, 4 as the primary instances)?

  2. GSE at finite N. For each N ≥ 2, is the largest eigenvalue of an N×N Gaussian symplectic ensemble matrix not infinitely divisible (classically), paralleling the settled GOE/GUE cases?

  3. Indecomposability. Is the Tracy–Widom distribution indecomposable, i.e. is it impossible to write TWβ=dY1+Y2\mathrm{TW}_\beta\stackrel{d}{=}Y_1+Y_2 with Y1,Y2Y_1,Y_2 independent non-degenerate random variables (for all β > 0, with the classical β = 1, 2, 4 as the primary instances)?

  4. β-Hermite at finite N. Determine the classical infinite-divisibility status of λmax⁡(HNβ)\lambda_{\max}(H_N^\beta) for finite N ≥ 2 and general β > 0 in the tridiagonal β-Hermite ensemble. The GOE/GUE cases correspond to β = 1, 2 and are settled negatively; item 2 is the β = 4 instance of this item. The small-deviation estimates of Ledoux–Rider were suggested by the source author as a possible tool.

  5. Divisible interpolation families. It is not known whether the members of the known interpolation families connecting the Tracy–Widom law to other laws — the Gumbel–Tracy–Widom family of determinantal edge processes (Johansson), and the Tracy–Widom–Gaussian and Tracy–Widom–Weibull deformations (Bohigas–de Carvalho–Pato) — are infinitely divisible at parameter values other than the Tracy–Widom endpoint, where the law is provably not infinitely divisible. Determine which members of these families are infinitely divisible away from that endpoint. Since the class of infinitely divisible laws on the real line is closed under weak convergence while the Tracy–Widom law is not infinitely divisible, no family that converges weakly to the Tracy–Widom law can have infinitely divisible members at every parameter approaching the endpoint; the open question is the set of parameter values at which the members are infinitely divisible.

The five questions are independent sub-questions that jointly define the objective: each can be settled separately, and the objective is complete when the divisibility or decomposability status of every listed law or family has been determined. Items 2 and 4 are matrix-level (finite N) facets of the same divisibility question, items 1, 3 and 5 are law-level facets (classical, free and parametric respectively).

References

  1. J. Armando Domínguez-Molina, The Tracy–Widom distribution is not infinitely divisible, Statistics & Probability Letters 123 (2017) 56–60, DOI: 10.1016/j.spl.2016.11.029, https://doi.org/10.1016/j.spl.2016.11.029; preprint arXiv:1601.02898, https://arxiv.org/abs/1601.02898
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