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Zbl 0989.82030
Gravner, Janko; Tracy, Craig A.; Widom, Harold
Limit theorems for height fluctuations in a class of discrete space and time growth models. (English)
[J] J. Stat. Phys. 102, No.5-6, 1085-1132 (2001). ISSN 0022-4715; ISSN 1572-9613/e

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 2000:: *82C41 Dynamics of random walks, etc.
60K35 Interacting random processes
60F05 Weak limit theorems
82C22 Interacting particle systems

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

Cited in: Zbl 1037.15019 Zbl 1021.60083 Zbl 1020.15024

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