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

TL;DR

This dissertation investigates the properties of the limit shape in last passage percolation models with random environments, establishing existence, universality, and boundary behavior of the limiting time constant across various weight distributions.

Contribution

It proves the existence of the limiting time constant for general distributions and explores its properties, including universality near the boundary and effects of tail behavior.

Findings

Existence of the limiting time constant for general weight distributions.

Universality of the time constant near the y-axis, similar to homogeneous models.

Boundary effects of the environment's tail on the time constant near the x-axis.

Abstract

We study directed last-passage percolation on the planar square lattice whose weights have general distributions, or equivalently, queues in series with general service distributions. Each row of the last passage model has its own randomly chosen weight distribution. We first show the existence of the limiting time constant and list its properties. Next we study the problem for models with Bernoulli and exponential weights, for which we already have more precise results. We then present some universality results about 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 environment. In particular we will give some estimates of the upper bound in this case.