Upper tail large deviations of the directed landscape

TL;DR

This paper establishes a large deviation principle for the upper tail behavior of the directed landscape, linking metric measures to entropy and applying results to directed geodesics.

Contribution

It introduces a novel large deviation principle at the metric level for the directed landscape, connecting measures on paths with entropy-based rate functions.

Findings

Large deviation principle for the directed landscape's upper tail

Characterization of finite-rate metrics via Kruzhkov entropy

Large deviation principle for directed geodesics

Abstract

Starting from one-point tail bounds, we establish an upper tail large deviation principle for the directed landscape at the metric level. Metrics of finite rate are in one-to-one correspondence with measures supported on a set of countably many paths, and the rate function is given by a certain Kruzhkov entropy of these measures. As an application of our main result, we prove a large deviation principle for the directed geodesic.