Limit theorems for height fluctuations in a class of discrete space and time growth models. (English) Zbl 0989.82030

Gravner, Janko; Tracy, Craig A.; Widom, Harold

J. Stat. Phys. 102, No. 5-6, 1085-1132 (2001).

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 Robinson-Schensted-Knuth 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. The model is equivalent to the Seppäläinen-Johansson model. Hence some of our results are not new, but the proofs are.

MSC:

82C41	Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
82C22	Interacting particle systems in time-dependent statistical mechanics

Keywords:

growth processes; shape fluctuations; limit theorems; Airy kernel; invariance principle; limiting distribution function; Fredholm determinant; saddle point analysis