Anomalous geodesics in the inhomogeneous corner growth model

TL;DR

This paper investigates the complex behavior of geodesics and competition interfaces in an inhomogeneous last-passage percolation model, revealing new phenomena caused by inhomogeneity and analyzing their implications in queueing systems.

Contribution

It introduces novel types of Busemann functions and geodesic behaviors specific to inhomogeneous environments, expanding understanding of geodesic structures in such models.

Findings

Discovery of new Busemann functions related to flat regions and thin rectangles

Identification of undirected and axis-directed semi-infinite geodesics

Observation of a new dichotomy in competition interface behavior

Abstract

We study Busemann functions, semi-infinite geodesics, and competition interfaces in the exactly solvable last-passage percolation with inhomogeneous exponential weights. New phenomena concerning geodesics arise due to inhomogeneity. These include novel Busemann functions associated with flat regions of the limit shape and thin rectangles, undirected semi-infinite geodesics, non-trivial axis-directed geodesics, intervals with no geodesic directions, and isolated geodesic directions. We further observe a new dichotomy for competition interfaces and second-class customers in a series of memoryless continuous-time queues with inhomogeneous service rates: a second class customer either becomes trapped or proceeds through the queue at strictly positive speed.