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

TL;DR

This paper studies height fluctuations in a class of discrete stochastic growth models, showing their limiting distributions relate to random matrix theory, with new results in the critical regime and novel proof techniques.

Contribution

Introduces a class of growth models and proves their height fluctuation distributions relate to Fredholm determinants, including a new critical regime analysis and rigorous saddle point methods.

Findings

Limiting distribution equals a Fredholm determinant with Airy kernel in the universal regime.

Identifies a new limiting distribution in the critical regime.

Provides a Brownian motion representation in the large t, fixed x regime.

Abstract

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function $h_t(x)$ with corner initialization. We prove, with one exception, that the limiting distribution function of $h_t(x)$ (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 equivalent to the Sepp\"al\"ainen-Johansson model. Hence some of our results are not new, but the proofs are.