We introduce a new method for studying universality of random matrices. Let T_n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, T_n converges to the Stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, our work leads to conjectured operator limits for the entire family of soft edge distributions.
This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich matrix model ("matrix Airy function", as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral ("matrix Bessel function") known in the literature as generalized Kontsevich matrix model.
In this note, it is shown that the stochastic Airy operator, which is the "Schrödinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to $+\infty$, can naturally be defined as a generalized Sturm-Liouville operator and that it is self-adjoint and has purely discrete spectrum with probability one. Thus "stochastic Airy spectrum" of Ramírez, Rider and Virág is the spectrum of an operator in the ordinary sense of the word.
In this article we review the ideas in Tsai (2018) toward proving the one-point, lower-tail large deviation principle for the Kardar--Parisi--Zhang equation.
The Airy$_β$ point process, originally introduced by Ramírez, Rider, and Virág, is defined as the spectrum of the stochastic Airy operator $\mathcal{H}_β$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet boundary condition. In this paper we study the coupled family of point processes defined as the eigenvalues of $\mathcal{H}_β$ acting on a subspace of $L^2[t,\infty)$. These point processes are coupled through the Brownian term of $\mathcal{H}_β$. We show that these point processes as a function of $t$ are differentiable with explicitly computable derivative. Moreover when recentered by …
Consider $n+m$ nonintersecting Brownian bridges, with $n$ of them leaving from 0 at time $t=-1$ and returning to 0 at time $t=1$, while the $m$ remaining ones (wanderers) go from $m$ points $a_i$ to $m$ points $b_i$. First, we keep $m$ fixed and we scale $a_i,b_i$ appropriately with $n$. In the large-$n$ limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation. Letting the number $m$ of wanderers tend to…
We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789--1858, 2017) for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-$β$ point process and the soft-edge limit of rank-$1$ perturbations of Gaussian $β$-Ensembles for any $β>0$, and solves an open problem mentioned in a previous work of Bufetov, Nikitin, and Qiu (Mosc. Math. J., 19(2):217--274, 2019). Our proof uses a combination of the semigroup theory of the stochastic Airy operator and the techniques for studying insertion and deletion to…
We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_β$ process, which describes the largest eigenvalues in the $β$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ramírez, Rider, and Virág. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times…
Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence a…
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy opera…
We studied the universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues. We studied the asymptotic limit when the number of both eigenvalues goes to infinity and obtained universality results. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. We obtained a necessary and sufficient condition on the parameters such that the limiting distribution is supported on 2 disjoint intervals and have computed the eigenvalue density in the limit. Furthermore, by using Riemann-Hilbert analysis, we have shown that under proper rescal…
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $μ_{A_n}=(1/n)\sum_{i}δ_{a_i}, μ_{B_n}=(1/n)\sum_i δ_{b_i}$ are discretization of probability measures $μ_A, μ_B$. Under regularity assumptions of $μ_A, μ_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
We studied universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues and the number of each of these eigenvalue goes to infinity in the asymptotic limit. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. In our previous work the case when the support consists of 2 intervals was studied. This paper complements our previous analysis and studied the case when the support consists of a single interval. By using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation k…
It was shown in [J. A. Ramírez, B. Rider and B. Virág. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $β$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/β$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R}$ of intensity $e^x dx$ and that the eigenfunctions converge to Dirac masses centered at IID points with expone…
In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of order $T^{1/3}$ and governed by the Tracy-Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $T^{2/3}$ and vertically by $T^{1/3}$, converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin-Bufetov-Wheeler between the stochastic six-vertex model and the ascending…
We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the positive half-line, where b_x' is white noise. In doing so we extend the definition of the Tracy-Widom(beta) distributions to all beta>0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality …
Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $μ$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random co…
This paper studies the relationship between the Lyapunov function of a macrosystem whose dynamics is governed by the laws of stochastic chemical kinetics and the invariant measure of this macrosystem arising at large times. A necessary and sufficient condition for the reduction of the search problem for the equilibrium of the macrosystem (the most probable macrostate of the invariant measure of this macrosystem) to an entropy-linear programming problem is given.
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary, orthogonal, and symplectic ensembles. The Riemann-Hilbert problem for orthogonal polynomials is one possible tool to derive universality in unitary random matrix ensembles. An overview is presented of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem in the one-cut regular case. Non-standard universality classes that arise at singular point…
We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $β$ tends to $0$. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom $β$ law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index $k$. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. W…
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same para…
We study the scaling limit of the spectrum of the β-Jacobi ensemble at the soft-edge and hard-edge for general values of β. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in Ramirez-Rider-Virag and Ramirez-Rider.
In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on weakly asymmetric simple exclusion processes, for which the density is preserved and the equilibrium fluctuations are shown to cross from the Edwards-Wilkinson universality class to the KPZ universality class. The crossover depends on the strength of the asymmetry. For the latter case, we will present an exclusion process with three species of particles, known as the ABC model, for which we aim to…
We introduce a stochastic process and functional that should describe the semigroup generated by the stochastic Bessel operator. Recently Gorin and Shkolnikov showed that the largest eigenvalues for certain random matrix ensembles with soft edge behavior can be understood by analyzing large powers of tridiagonal matrices, which converge to operators in the stochastic Airy semigroup. In this article we make some progress towards realizing Gorin and Shkolnikov's program at the random matrix hard edge. We analyze large powers of a suitable tridiagonal matrix model (a slight modification of the $β…
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of…
We establish the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on a three dimensional periodic domain to the dynamic $Φ^4_3$ model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular $C^9$ is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their $L^p$ norms in terms of the graphs of the standard $Φ^4_3$ stochastic terms.
We show that the set of entries generated by any finite set of doubly stochastic matrices is nowhere dense, in contrast to the cases of stochastic matrices or unitary matrices. In other words, there is no finite universal set of doubly stochastic matrices, even with the weakest notion of universality. Our proof is based on a theorem for topological semigroups with the convergent property. A topological semigroup is convergent if every infinite product converges. We show that in a compact and convergent semigroup, under some restrictions, the closure of every finitely generated subsemigroup can…
We establish universality at the hard edge for general beta ensembles provided that the background potential V is a polynomial such that x -> V(x^2) is uniformly convex and beta is larger than or equal to one. The method rests on the corresponding tridiagonal matrix models, showing that their appropriate continuum scaling limit is given by the Stochastic Bessel Operator. As conjectured by Edelman-Sutton and rigorously established by Ramirez-Rider, the latter characterizes the hard edge in the case of linear potential and all beta (the classical "beta-Laguerre" ensembles)
We consider $n$ particles $0\leq x_1<x_2< \cdots < x_n < +\infty$, distributed according to a probability measure of the form $$ \frac{1}{Z_n}\prod_{1\leq i <j \leq n}(x_j-x_i)\prod_{1\leq i <j \leq n}(x_j^θ-x_i^θ)\prod_{j=1}^nx_j^αe^{-x_j}\ud x_j, ~~ α>-1,~~ θ>0, $$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which all…
We use the notion of energy solutions of the stochastic Burgers equation to give a short proof of the Hairer-Quastel universality result for a class of stationary weakly asymmetric stochastic PDEs.
We present mathematically rigorous expressions for the residence-time and first-passage-time distributions of a periodically forced Brownian particle in a bistable potential. For a broad range of forcing frequencies and amplitudes, the distributions are close to periodically modulated exponential ones. Remarkably, the periodic modulations are governed by universal functions, depending on a single parameter related to the forcing period. The behaviour of the distributions and their moments is analysed, in particular in the low- and high-frequency limits.
The universal approximation properties with respect to $L ^p $-type criteria of three important families of reservoir computers with stochastic discrete-time semi-infinite inputs is shown. First, it is proved that linear reservoir systems with either polynomial or neural network readout maps are universal. More importantly, it is proved that the same property holds for two families with linear readouts, namely, trigonometric state-affine systems and echo state networks, which are the most widely used reservoir systems in applications. The linearity in the readouts is a key feature in supervise…
These notes are based on a talk given at the 2018 Arizona School of Analysis and Mathematical Physics. We give a comprehensive introduction to the KPZ universality class, a conjectured class of stochastic process with local interactions related to random growth processes in $1+1$ dimensions. We describe some of the characteristic properties of the KPZ universality class such as scaling exponents and limiting statistics. In particular, we aim to extract the characteristic properties of the KPZ universality class by understanding the KPZ stochastic partial differential equation by a special disc…
The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory Goncalves-Jara '14, Gubinelli-Jara '13, Gubinelli-Perkowski '18, Gubinelli-Perkowski '20. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution Gubinelli-Perkowski '17 plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, fir…
We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.
We prove the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on $\mathbb{R}^3$ to the dynamical $Φ^4_3$ model by paracontrolled distributions on weighted Besov space. Our approach depends on the delicate choice of the weight, the localization operator technique and a modification version of the maximal principle from [GH18].
We consider a class of stochastic reaction-diffusion equations on the three dimensional torus. The non-linearities are odd polynomials in the weakly non-linear regime, and the smoothing mechanisms are very general higher order perturbations of the Laplacian. The randomness is the space-time white noise without regularisation. We show that these processes converge to the dynamical $Φ^4_3 (λ)$ model, where the coupling constant $λ$ has an explicit expression involving non-trivial interactions between all details of the smoothing mechanism and the non-linearity.
Universality of high-dimensional scaling limits of stochastic gradient descent
Department of Mathematics, Northwestern University（西北大学数学系）；Department of Statistics and Actuarial Science, Department of Applied Mathematics, and Cheriton School of Computer Science, University of Waterloo（滑铁卢大学统计与精算科学系、应用数学系和切尔顿计算机科学学院）
We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions $N \gg 1$, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the F…
We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such a…
