Grid entropy in last passage percolation -- a superadditive critical exponent approach

TL;DR

This paper introduces the concept of grid entropy in last-passage percolation, providing a new framework for understanding entropy as a critical exponent and linking it to large deviation principles and empirical measure limits.

Contribution

It develops a novel approach to grid entropy in last-passage percolation, connecting it with variational formulas and improving bounds through relative entropy, while relating to recent empirical measure results.

Findings

Defined grid entropy as a limit of path entropies.

Linked grid entropy to large deviation rate functions.

Extended results to both point-to-point and point-to-level cases.

Abstract

Working in the setting of i.i.d. last-passage percolation on $\mathbb{R}^D$ with no assumptions on the underlying edge\hyp{}weight distribution, we arrive at the notion of grid entropy - a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Sepp\"al\"ainen as a large deviation rate function of empirical measures along paths. In their 2014 paper arXiv:1202.2584, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a recent paper arXiv:2006.12580 by Bates. In addition, we partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates. Our results cover both the point-to-point and point-to-level scenarios.