Diffusive scaling limit of the Busemann process in Last Passage Percolation

TL;DR

This paper establishes that the rescaled Busemann process in exponential last passage percolation converges to a universal limiting process described as an ensemble of sticky Brownian lines, revealing deep connections in the KPZ universality class.

Contribution

It proves the convergence of the rescaled Busemann process to a universal limit described by sticky Brownian lines, linking it to known distributions and algorithms.

Findings

The rescaled Busemann process converges to a cadlag process G.

The limit G is described as an ensemble of sticky Brownian lines.

The result suggests universality of the Busemann process in KPZ class.

Abstract

In exponential last passage percolation, we consider the rescaled Busemann process $x\mapsto N^{-1/3}B^\rho_{0,[xN^{2/3}]e_1} \,\, (x\in\mathbb{R})$, as a process parametrized by the scaled density $\rho=1/2+\frac{\mu}{4} N^{-1/3}$, and taking values in $C(\mathbb{R})$. We show that these processes, as $N\rightarrow \infty$, have a c\`adl\`ag scaling limit $G=(G_\mu)_{\mu\in \mathbb{R}}$, parametrized by $\mu$ and taking values in $C(\mathbb{R})$. The limiting process $G$, which can be thought of as the Busemann process under the KPZ scaling, can be described as an ensemble of "sticky" lines of Brownian regularity. We believe $G$ is the universal scaling limit of Busemann processes in the KPZ universality class. Our proof provides insight into this limiting behaviour by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Sepp\"al\"ainen in arXiv:1808.09069, and a sorting algorithm of random walks introduced by O'Connell and Yor [32]