Airy Kernel Research Articles

Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

AbstractWe study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher‐order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher‐order generalizations of the Tracy‐Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants. © 2009 Wiley Periodicals, Inc.

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson's model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is $\beta/2$ times the inverse of particle distance. When $\beta=2$, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson's model with $\beta=2$ and $N$ particles, $\X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N a_1 > ... > a_N$, $N \geq 2$. In order to properly control the effect of such initial confinement of particles in the negative region of $\R$, we put the drift term to each Brownian motion, which increases in time as a parabolic function : $Y_j(t) = X_j(t) + t^2/4 + \{d_1 + \sum_{\ell=1}^N (1/a_{\ell})\}t, 1 \leq j \leq N$, where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of $\Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty)$, we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of $\Ai(z)$ on the negative $\R$ is occupied by one particle, to the stationary state $\mu_{\Ai}$. The stationary state $\mu_{\Ai}$ is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on $\R$ and in which the Tracy-Widom distribution describes the rightmost particle position.

On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum.

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

Dyson's nonintersecting Brownian motions with a few outliers

AbstractConsider n nonintersecting Brownian particles on ℝ (Dyson Brownian motions), all starting from the origin at time t = 0 and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ±√2nt(1 − t). The Airy process 𝒜(τ) is defined as the motion of these nonintersecting Brownian motions for large n but viewed from the curve 𝒞 : y = √2nt(1 − t) with an appropriate space‐time rescaling.Assume now a finite number r of these particles are forced to a different target point, say a = ρ0√n/2 > 0. Does it affect the Brownian fluctuations along the curve 𝒞 for large n? In this paper, we show that no new process appears as long as one considers points (y, t) ∈ 𝒞 such that 0 < t < (1 + ρ)−1, which is the t‐coordinate of the point of tangency of the tangent to the curve passing through (ρ0√n/2, 1). At this point the fluctuations obey a new statistics, which we call the Airy process with r outliers 𝒜(r)(τ) (in short, r‐Airy process). The log of the probability that at time τ the cloud does not exceed x is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in x and τ, from which the asymptotic behavior of the process can be deduced for τ → −∞. This kernel is closely related to one found by Baik, Ben Arous, and Péché in the context of multivariate statistics. © 2008 Wiley Periodicals, Inc.

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.

Correlation function of the Schur process with a fixed final partition

We consider a generalization of the Schur process in which a partition evolves from the empty partition into an arbitrary fixed final partition. We obtain a double integral representation of the correlation kernel. For a special final partition with only one row, the edge scaling limit is also discussed by the use of the saddle point analysis. If we appropriately scale the length of the row, the limiting correlation kernel changes from the extended Airy kernel.

Asymptotics of non-intersecting Brownian motions and a [formula omitted] Riemann–Hilbert problem

We consider n one-dimensional Brownian motions, such that n / 2 Brownian motions start at time t = 0 in the starting point a and end at time t = 1 in the endpoint b and the other n / 2 Brownian motions start at time t = 0 at the point - a and end at time t = 1 in the point - b , conditioned that the n Brownian paths do not intersect in the whole time interval ( 0 , 1 ) . The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case ab < 1 / 2 . We find that the limiting mean density of the positions of the Brownian motions is supported on one or two intervals and that the correlation kernel has the usual scaling limits from random matrix theory, namely the sine kernel in the bulk and the Airy kernel near the edges.

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.

From Gumbel to Tracy-Widom

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( − exp( − x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp( − x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $$M=M_0+\sqrt{2S} V$$ , with M 0 diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.

Asymptotic corrections to the eigenvalue density of the GUE and LUE

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.

PDEs for the joint distributions of the Dyson, Airy and Sine processes

In a celebrated paper, Dyson shows that the spectrum of an n\times n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDEs enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t_1 and t_2, when t_2-t_1\to \infty, as illustrated in this paper for the Airy process. This paper also contains a rigorous proof that the extended Hermite kernel, governing the joint probabilities for the Dyson process, converges to the extended Airy and Sine kernels after the appropriate rescalings.

Large n Limit of Gaussian Random Matrices with External Source, Part II

We continue the study of the Hermitian random matrix ensemble with external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ where $A$ has two distinct eigenvalues $\pm a$ of equal multiplicity. This model exhibits a phase transition for the value $a=1$, since the eigenvalues of $M$ accumulate on two intervals for $a > 1$, and on one interval for $0 < a < 1$. The case $a > 1$ was treated in part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case $0 < a < 1$. As in part I we apply the Deift/Zhou steepest descent analysis to a $3 \times 3$-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.

Large n Limit of Gaussian Random Matrices with External Source, Part I

We consider the random matrix ensemble with an external source defined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n→∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ht(x) with corner initialization. We prove, with one exception, that the limiting distribution function of ht(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppalainen–Johansson model. Hence some of our results are not new, but the proofs are.

Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1 it is positive on the interior of a finite number of intervals, 2 it vanishes like a square root at endpoints, and 3 outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a “bulk,” in which there is universal behavior involving the sine kernel, and “edge effects,” in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this “regular” behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V/c, the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials, and integrable systems. © 2000 John Wiley & Sons, Inc.

Level spacing distributions and the Bessel kernel

Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel”\(\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}\). Rescaling the GUE at the “edge” of the spectrum leads to the kernel\(\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}\), where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painlevé transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJ α(z) is the Bessel function of order α. In the cases α=∓1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.

Level-spacing distributions and the Airy kernel

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Môri, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

Level-spacing distributions and the Airy kernel

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sin π ( x − y)/ π( x − y). Similarly a double scaling limit at the “edge of the spectrum” leads to the Airy kernel [Ai( x)Ai′( y) − Ai′( x)Ai( y)]/( x− y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE's found by Jimbo, Miwa, Môri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues.