Time-time covariance for last passage percolation in half-space

TL;DR

This paper investigates the two-time covariance in half-space last passage percolation, revealing that for close endpoints, the correction matches the stationary model, with new comparison inequalities and geodesic localization results.

Contribution

It introduces comparison inequalities for last passage increments and establishes tightness and localization results specific to half-space models.

Findings

First order covariance correction matches stationary model for close endpoints.

Derived new comparison inequalities for last passage increments.

Proved tightness and geodesic localization in half-space setting.

Abstract

This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we have to overcome the difficulty that the point-to-point model in half-space with generic start and end points is not known.