# Anomalous Geodesics in the Inhomogeneous Corner Growth Model

## TL;DR

This paper explores unusual paths in a mathematical model of growth with varying weights, revealing new behaviors due to inhomogeneity.

## Contribution

The study introduces novel phenomena in geodesics and competition interfaces caused by inhomogeneous weights in a growth model.

## Key findings

- Novel Busemann functions are associated with flat regions of the limit shape and thin rectangles.
- Semi-infinite geodesics exhibit intervals of asymptotic directions and non-trivial axis-directed paths.
- Competition interfaces show a new dichotomy where second-class customers either get trapped or move at positive speed.

## 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, semi-infinite geodesics with intervals of asymptotic directions, 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 service stations at strictly positive speed.