Mathematics – Probability
Scientific paper
2000-05-13
J. of Statistical Physics 102 (2001), 1085-1132
Mathematics
Probability
39 pages, 7 figures, 2 tables. The revised version eliminates the simulations and corrects a number of misprints. Version 3 ad
Scientific paper
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.
Gravner Janko
Tracy Craig A.
Widom Harold
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