Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

TL;DR

This paper establishes the probability bounds for multiple disjoint polymers in Brownian last passage percolation, revealing their coalescence structure and fluctuation behavior, which advances understanding of the KPZ universality class.

Contribution

It proves the probability decay rate for $k$ disjoint polymers crossing a region, providing a foundational result for polymer coalescence and fluctuation analysis in KPZ models.

Findings

Probability of $k$ disjoint polymers scales as $ ext{epsilon}^{(k^2 - 1)/2 + o(1)}$.

Polymers fluctuate by $ ext{epsilon}^{2/3}$ on short scales.

Results support the conjecture of sharpness of the bounds.

Abstract

In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of $k$ disjoint polymers crossing a unit-order region while beginning and ending within a short distance $\epsilon$ of each other is bounded above by $\epsilon^{(k^2 - 1)/2 \, + \, o(1)}$. This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in [Ham17c] in proving comparison on unit-order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on-scale articulation of the two-thirds power law for polymer geometry: polymers fluctuate by $\epsilon^{2/3}$ on short scales $\epsilon$.