Last passage percolation in an exponential environment with discontinuous rates

TL;DR

This paper establishes a strong law of large numbers for last passage times in an inhomogeneous exponential environment with discontinuous rates, revealing complex shape properties and phase transitions.

Contribution

It introduces a variational formula for the limiting shape in an inhomogeneous setting with discontinuous rates, advancing understanding of shape properties and phase transitions.

Findings

Shape function may not be strictly concave

Shape function can have non-differentiability points

Phase transition observed in macroscopic optimisers

Abstract

We prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be discontinuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves. Using this result, we present two examples that allow for partial analytical tractability and show that the shape function may not be strictly concave, and it may exhibit points of non-differentiability, flat segments, and non-uniqueness of the optimisers of the variational formula. Finally, in a specific example, we analyse further the macroscopic optimisers and uncover a phase transition for their behaviour.