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Limit theorems for height fluctuations in a class of discrete space and time growth models. (English) Zbl 0989.82030
We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ${h}_{t}\left(x\right)$ with corner initialization. We prove, with one exception, that the limiting distribution function of ${h}_{t}\left(x\right)$ (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. 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F05 Central limit and other weak theorems 82C22 Interacting particle systems