Influence of the initial condition in equilibrium last-passage percolation models

TL;DR

This paper investigates how initial conditions influence the fluctuations of last-passage times in an equilibrium percolation model with a Poisson environment, providing a new formula linking initial measures to passage times.

Contribution

It introduces an $ ext{L}^2$-formula connecting initial measures with last-passage times, aiding the analysis of fluctuations in equilibrium percolation models.

Findings

Derived an $ ext{L}^2$-formula relating initial measure and passage time

Analyzed fluctuations along non-characteristic directions

Provided tools for studying equilibrium last-passage percolation

Abstract

In this paper we consider an equilibrium last-passage percolation model on an environment given by a compound two-dimensional Poisson process. We prove an $\LL^2$-formula relating the initial measure with the last-passage percolation time. This formula turns out to be a useful tool to analyze the fluctuations of the last-passage times along non-characteristic directions.