Last Passage Percolation in Macroscopically Inhomogeneous Media

TL;DR

This paper studies last passage percolation in media with large-scale inhomogeneity, deriving a variational framework for the limiting shape and conditions for solution uniqueness, with implications for related stochastic processes.

Contribution

It introduces a variational approach to analyze last passage percolation in inhomogeneous media and establishes conditions for the uniqueness of the limiting shape.

Findings

Derived an ODE for the deterministic limiting shape.

Provided a sufficient condition for the uniqueness of the variational problem's solution.

Discussed implications for the totally asymmetric simple exclusion process.

Abstract

In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed.