Limit theorems for height fluctuations in a class of discrete space and time growth models

We introduce a class of one-dimensional discrete space-discrete time stocha stic growth models described by a height function h(t)(x) with corner initi alization We prove, with one exception, that the limiting distribution func tion of h(t)(x) (suitably centered and normalized) equals a Fredholm determ inant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the F redholm determinant with Airy kernel. In the exceptional case, called the c ritical regime, the limiting distribution seems not to have previously occu rred. 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 Seppalainen Johansson model. Hence some of our results a re not new, but the proofs are.