Joint distribution of Busemann functions in the exactly solvable corner growth model

TL;DR

This paper characterizes the joint distribution of Busemann functions in the exactly solvable corner growth model, providing insights into geodesics and their distributions within the KPZ universality class.

Contribution

It offers a comprehensive description of the joint distribution of Busemann functions in all directions, advancing understanding of geodesics in the model.

Findings

Derived a marked point process representation for Busemann functions

Calculated marginal distributions of Busemann functions

Analyzed semi-infinite geodesics in the model

Abstract

The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. This paper describes the joint distribution of the Busemann functions, simultaneously in all directions of growth. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and calculate some marginal distributions of Busemann functions and semi-infinite geodesics.