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

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.