  We study a class of corner growth models in which the weights are either all
exponentially or all geometrically distributed. The parameter of the
distribution at site $(i, j)$ is $a_i+b_j$ in the exponential case and $a_ib_j$
in the geometric case, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are
themselves drawn randomly at the outset from ergodic distributions. These
models are inhomogeneous generalizations of the much studied exactly solvable
models in which the parameters are the same for all sites. Our motivation is to
understand how inhomogeneity influences the limit shape and the corresponding
limit fluctuations. We obtain a simple variational formula for the shape
function and prove that it is strictly concave inside a cone (possibly the
entire quadrant) but is linear outside. This is in contrast with the situation
in the models with i.i.d. weights in which the shape function is expected to be
strictly concave under mild assumptions. For the directions inside the cone, we
show that the limit fluctuations are governed by the Tracy-Widom GUE
distribution and derive bounds for the deviations of the last-passage times
above the shape function. To obtain the shape result, we couple the model with
an explicit family of stationary versions of it. For the fluctuation and large
deviation results, we perform steepest-descent analysis on an available
Fredholm determinant formula for the one-point distribution of the last-passage
time. We also develop a detailed appendix on the steepest-descent curves of
harmonic functions of two real variables and approximate the contour integral
of an arbitrary meromorphic function along such curves.
