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

By Janko Gravner, Craig A. Tracy, Harold Widom. 2000

216 citations recorded by the source corpus.

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 Seppäläinen–Johansson model. Hence some of our results are not new, but the proofs are.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3a53cf32aba1fd71f137711247573eadb957810f

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Littelmann paths and Brownian paths (2004, result): “The first step has been the result of Gravner, Tracy and Widom, [15] and of Baryshnikov [1] which states that the largest eigenvalue of a random n× n Hermitian matrix in the GUE is distributed as the random variable”
Paths in Weyl chambers and random matrices (2002, background): “Baryshnikov [3] and Gravner, Tracy & Widom [14] have given a path representation of the largest eigenvalue λ1 of the G.U.E. : Theorem 1.1 Let W1, · · · ,Wd be independent standard real Brownian motions, then λ1 has the same distribution as max 1≥t1≥···≥td−1≥0 [W1(1)−W1(t1) + W2(t1) + · · · −Wd−1(td−”
A GUE central limit theorem and universality of directed first andlast passage site percolation (2004, methodology): “By studying a special random variable, Baryshnikov [2] and Gravner, Tracy and Widom [8] (see also [14, 13]) showed that (17) P(D̂k(1) ≤ s) = 1 Zk ∫ s −∞ .”
Brownian analogues of Burke's theorem (2001, methodology): “We will not prove this here because it actually follows from a recent observation, due to Baryshnikov (2001) and Gravner et al. (2000), namely that the random variable Mn= sup 06s16···6sn−161 {B(1)(0; s1) + · · ·+ B (n) (sn−1 ;1)}; (28) has the same law as the largest eigenvalue of an n-dimensional…”
Application of the τ-function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits (2002, unknown): “52), one has [6], [34], [4] Pr ( B((M − 1, N − 1); q(2)) ≤ l ) (5.”
Extreme value statistics of correlated random variables: A pedagogical review (2019, unknown): “[193] J.”
Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices (2005, background): “See [16], for example, for a discussion of this type.”
Non-Colliding Random Walks, Tandem Queues, and DiscreteOrthogonal Polynomial Ensembles (2002, result): “The Brownian analogue is also presented in [OY01b], in an attempt to understand the recent observation, due to Baryshnikov [Ba01] and Gravner/Tracy/Widom [GTW01], that the random variable M = inf 0=t0 t1 ··· tk=1 k∑ i=1 [Bi(ti) −Bi(ti−1)], where B = (B1, . . . , Bk) is a standard k-dimensional Brown”
A Universality Property for Last-Passage Percolation Paths Close to the Axis (2004, unknown): “The random variable L (1 , k ) has the same distribution as the largest eigenvalue of a k × k GUE random matrix [4], [7], [15].”
Random matrices, non-colliding processes and queues (2002, unknown): “Let us concentrate on the following, more exact, connection between directed percolation and random matrices which was more recently observed by Baryshnikov [5] and Gravner, Tracy and Widom [26].”