Random matrix central limit theorems for nonintersecting random walks

Baik, Jinho; Suidan, Toufic

doi:10.1214/009117906000001105

Cited by 29 publications

(42 citation statements)

References 54 publications

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“…Then we will conclude that (Ξ (n) (t), t ∈ N 0 , P δ 2aZ ) converges to (Ξ(t), t ∈ [0, ∞), P δ 2aZ ) in DMR, for tightness. Relations between the present convergence in DMR and the previous results concerning convergence to the Dyson model [26,3,44] will be discussed elsewhere.…”

Section: Convergence To the Dyson Modelsupporting

confidence: 64%

“…(From now on BM stands for one-dimensional standard Brownian motion and Dyson's BM model with β = 2 is simply called the Dyson model in this paper.) Then the noncolliding RW has been attracted much attention as a discretization of models associated with the Gaussian random-matrix ensembles [2,22,40,26,23,3,15,12]. Eigenvalue distributions of random-matrix ensembles provide important examples of determinantal point processes, in which any correlation function is given by a determinant specified by a single continuous function called the correlation kernel [50,49,4].…”

Section: Introductionmentioning

confidence: 99%

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Determinantal Martingales and Correlations of Noncolliding Random Walks

Katori

2015

J Stat Phys

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We study the noncolliding random walk (RW), which is a particle system of onedimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, N < ∞, this discrete process is constructed as an h-transform of absorbing RW in the N -dimensional Weyl chamber. We consider Fujita's polynomial martingales of RW with time-dependent coefficients and express them by introducing a complex Markov process. It is a complexification of RW, in which independent increments of its imaginary part are in the hyperbolic secant distribution, and it gives a discrete-time conformal martingale. The h-transform is represented by a determinant of the matrix, whose entries are all polynomial martingales. From this determinantal-martingale representation (DMR) of the process, we prove that the noncolliding RW is determinantal for any initial configuration with N < ∞, and determine the correlation kernel as a function of initial configuration. We show that noncolliding RWs started at infinite-particle configurations having equidistant spacing are well-defined as determinantal processes and give DMRs for them. Tracing the relaxation phenomena shown by these infiniteparticle systems, we obtain a family of equilibrium processes parameterized by particle density, which are determinantal with the discrete analogues of the extended sinekernel of Dyson's Brownian motion model with β = 2. Following Donsker's invariance principle, convergence of noncolliding RWs to the Dyson model is also discussed.

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Section: Convergence To the Dyson Modelsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Determinantal Martingales and Correlations of Noncolliding Random Walks

Katori

2015

J Stat Phys

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“…To conduct asymptotic analysis of orthogonal polynomials with asymptotic Freud-like weight, we make use of the Riemann-Hilbert approach first developed by Deift and Zhou 6 for modified KdV equations, and further applied to orthogonal polynomials, 7-10 random matrix theory, [11][12][13] and other integrable systems. [14][15][16] In the previous study on orthogonal polynomials, the weight functions are usually zero-free in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

Long

Dai

et al. 2019

Stud Appl Math

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We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann-Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given. K E Y W O R D S asymptotic approximation, asymptotic Freud-like weight, continuous dual Hahn polynomials, Riemann-Hilbert problem ∑ =0 ( ) ( 2 ) ( + ) − ( − + ) − !( − )! , Stud Appl Math. 2020;144:133-163. wileyonlinelibrary.com/journal/sapm

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“…For recent progress related to discrete random walks, random tilings and random matrices with external source see [3,4,5,6,7,8,47,50].…”

Section: Introductionmentioning

confidence: 99%

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

Kuijlaars

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

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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.

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Random matrix central limit theorems for nonintersecting random walks

Cited by 29 publications

References 54 publications

Determinantal Martingales and Correlations of Noncolliding Random Walks

Determinantal Martingales and Correlations of Noncolliding Random Walks

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

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

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