van den Berg-Kesten--type correlation inequalities for disjoint polymers in the KPZ universality class

TL;DR

This paper establishes a van den Berg-Kesten-type correlation inequality for the KPZ line ensemble and continuum directed random polymer, leveraging integrability to analyze upper tail behaviors and convergence properties in the KPZ universality class.

Contribution

It introduces a novel BK inequality for positive temperature polymer models using integrability and geometric RSK, extending zero temperature results.

Findings

Proves a BK inequality for KPZ line ensemble and continuum directed polymer.

Uses integrability and geometric RSK to establish the inequality.

Highlights the importance of integrability through a counter-example for non-integrable models.

Abstract

In classical percolation theory, the van den Berg-Kesten (BK) inequality is a fundamental tool that shows that disjoint events induce negative conditionings on each other. The inequality also holds in the context of last passage percolation (LPP), which is the zero temperature limit of polymer models and an important subclass in the Kardar-Parisi-Zhang (KPZ) universality class. Recently, an analog of the BK inequality was discovered in the context of zero temperature line ensembles and the scaling limit of LPP, where it was used to study upper tail probabilities of the weight and the scaling limit of geodesics under such upper tail conditionings. However, while it has become apparent that such an inequality in the positive temperature setting would have a number of applications, it seems likely that a direct generalization of the zero temperature inequality would not hold. In this work we prove a version of the BK inequality for the KPZ line ensemble and the continuum directed random polymer. We do so by working with the log gamma polymer, making use of its integrability and the geometric RSK correspondence. Our inequality serves as a key input in analyzing the KPZ line ensemble and proving sharp upper tail estimates of the KPZ equation in arXiv:2208.08922, and proving convergence of the continuum directed random polymer to Brownian bridge under the upper tail event in arXiv:2311.12009. The crucial role of integrability in the validity of such an inequality is highlighted via a counter-example for a non-integrable model.