Independence property of the Busemann function in exactly solvable KPZ models

TL;DR

This paper proves the independence of Busemann functions in various exactly solvable KPZ models, advancing understanding of their structure and providing probabilistic bounds for related rare events.

Contribution

It establishes the independence property of Busemann functions across multiple KPZ models, a key step for analyzing their coupling and growth properties.

Findings

Disjoint Busemann increments are independent in the corner growth model.

The independence holds when associated geodesics intersect almost surely.

Derived a near-optimal probability bound for endpoint deviations in inverse-gamma polymers.

Abstract

The study of Kadar-Parsi-Zhang (KPZ) universality class has been a subject of great interest among mathematicians and physicists over the past three decades. A notably successful approach for analyzing KPZ models is the coupling method, which hinges on understanding random growth from stationary initial conditions defined by Busemann functions. To advance in this direction, we investigate the independence property of the Busemann function across multiple directions in various exactly solvable KPZ models. These models encompass the corner growth model, the inverse-gamma polymer, Brownian last-passage percolation, the O'Connell-Yor polymer, the KPZ equation, and the directed landscape. In the context of the corner growth model, our result states that disjoint Busemann increments in different directions along a down-right path are independent, as long as their associated semi-infinite geodesics have nonempty intersections almost surely. The proof for the independence utilizes the queueing representation of the Busemann process developed by Sepp\"al\"ainen et al. As an application, our independence result yields a near-optimal probability upper bound (missing by a logarithmic factor) for the rare event where the endpoint of a point-to-line inverse-gamma polymer is close to the diagonal.