Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

TL;DR

This paper introduces a new probabilistic coupling method to derive optimal deviation bounds for exit points of geodesics in exponential last-passage percolation, improving upon previous integrable probability techniques.

Contribution

The authors develop a novel probabilistic approach for deviation estimates in directed planar models, achieving optimal bounds previously accessible mainly through integrable probability methods.

Findings

Optimal cubic-exponential bounds for exit point deviations

Cube-root bounds with logarithmic correction for Busemann limits

Enhanced probabilistic coupling technique for various models

Abstract

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature.