Large deviations of geodesic midpoint fluctuations in last-passage percolation with general i.i.d. weights

TL;DR

This paper establishes the large deviation behavior of geodesic midpoint fluctuations in last-passage percolation models with general i.i.d. weights, providing a rate function linked to tail large deviations and shape functions.

Contribution

It proves the large deviation limit for midpoint fluctuations in general LPP models and verifies a conjecture for exponential weights regarding geodesic path probabilities.

Findings

Rate function expressed via tail large deviations and shape function.

Verification of a conjecture for exponential weights on geodesic path probability.

Asymptotic probability of corner-path geodesic in exponential case.

Abstract

The study of transversal fluctuations of the optimal path is a crucial aspect of the Kardar-Parisi-Zhang (KPZ) universality class. In this work, we establish the large deviation limit for the midpoint transversal fluctuations in a general last-passage percolation (LPP) model with mild assumption on the i.i.d. weights. The rate function is expressed in terms of the right tail large deviation rate function of the last-passage value and the shape function. When the weights are chosen to be i.i.d. exponential random variables, our result verifies a conjecture communicated to us by Liu [Liu'22], showing the asymptotic probability of the geodesic from $(0,0)$ to $(n,n)$ following the corner path $(0,0) \to (n,0) \to (n,n)$ is $({4}/{e^2})^{n+o(n)}$.