  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.
