Determinantal Processes and Stochastic Domination

TL;DR

This paper establishes stochastic domination results for determinantal processes with finite rank kernels, extending Lyons' discrete results to a more general setting and applying these to last passage percolation models.

Contribution

It provides a new proof of stochastic domination for determinantal processes that avoids matroid machinery and applies to continuous settings, also linking to last passage percolation.

Findings

Proved stochastic domination for finite rank determinantal processes.

Extended Lyons' theorem to general measure settings.

Applied results to last passage percolation models.

Abstract

We prove the stochastic domination for determinantal processes associated with finite rank projection kernels. The result was first proved by Lyons in discrete setting. We avoid the machinery of matroids in order to obtain a proof that works in a general setting. We prove another result on the stochastic domination of two determinantal processes where the kernels are represented with respect to different measures. Combining this result with Lyons' theorem on the Stochastic domination we obtain a result on the stochastic domination for the last passage time in a directed last passage percolation on $\mathbb{Z}^2$ with i.i.d. geometric weights.