Properties of the limit shape for some last passage growth models in random environments

TL;DR

This paper analyzes the shape of the growth limit in last passage percolation models with randomly varying weights, revealing universal behavior near the boundary and effects of distribution tails near the axes.

Contribution

It characterizes the limit shape in last passage models with random environments, highlighting universal properties and boundary effects.

Findings

Universal form of the time constant near the y-axis.

Influence of tail distributions on the x-axis.

Boundary behavior depends on local distribution properties.

Abstract

We study directed last passage percolation on the first quadrant of the planar square lattice whose weights have general distributions, or equivalently, ./G/1 queues in series. The service time distributions of the servers vary randomly which constitutes a random environment for the model. Equivalently, each row of the last passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random means attached to the rows.