The geometry of near ground states in Gaussian polymer models

TL;DR

This paper analyzes the structure and geometry of near ground states in Gaussian polymer models, revealing their similarity to Brownian motion and providing estimates on path fluctuations and energetic shortfalls.

Contribution

It develops new tools for analyzing the energy landscape of last passage percolation models, establishing strong similarity to Brownian motion and detailed geometric bounds.

Findings

The routed weight profile resembles Brownian motion of rate two.

Rare events where different routes have near-maximal energies are quantitatively estimated.

Bounds on the gradient and excursions of near ground states are established.

Abstract

The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in scaled coordinates that respect these exponents, this random field of paths may be viewed as a complex energy landscape. We investigate the structure of valleys and connecting pathways in this landscape. The routed weight profile $\mathbb{R} \to \mathbb{R}$ associates to $x \in \mathbb{R}$ the maximum scaled energy obtainable by a path whose scaled journey from $(0,0)$ to $(0,1)$ passes through the point $(x,1/2)$. Developing tools of Brownian Gibbs analysis from [Ham16] and [CHH19], we prove an assertion of strong similarity of this profile for Brownian last passage percolation to Brownian motion of rate two on the unit-order scale. A sharp estimate on the rarity that two macroscopically different routes in the energy landscape offer energies close to the global maximum results. We prove robust assertions concerning modulus of continuity for the energy and geometry of scaled maximizing paths, that develop the results and approach of [HS20], delivering estimates valid on all scales above the microscopic. The geometry of excursions of near ground states about the maximizing path is investigated: indeed, we estimate the energetic shortfall of scaled paths forced to closely mimic the geometry of the maximizing route while remaining disjoint from it. We also provide bounds on the approximate gradient of the maximizing path, viewed as a function, ruling out sharp steep movement down to the microscopic scale. Our results find application in a companion study [GH20a] of the stability, and fragility, of last passage percolation under a dynamical perturbation.