Limit shapes for inhomogeneous corner growth models with exponential and geometric weights

TL;DR

This paper extends exactly solvable corner growth models by incorporating inhomogeneous exponential and geometric weights with ergodic random parameters, deriving explicit limit shape functions through variational analysis.

Contribution

It introduces a generalized inhomogeneous model with random parameters and explicitly characterizes the limit shape via a variational problem, expanding the understanding of corner growth models.

Findings

Derived explicit limit shape functions for the inhomogeneous models.

Identified special cases where the shape function can be explicitly solved.

Extended the class of exactly solvable growth models with random inhomogeneities.

Abstract

We generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution $a_i+b_j$ and the parameter of the geometric distribution $a_i b_j$ at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are jointly ergodic random sequences. We identify the shape function in terms of a simple variational problem, which can be solved explicitly in some special cases.