Alexandru Gatea | Searcharxiv

arXiv2022

Grid entropy in a "choose the best of D samples" model

Grid entropy is a deterministic quantity inherent to lattice models which captures the entropy of empirical measures along paths that converge weakly to a given target measure. In this paper, we study the limiting behaviour of empirical measures in a model consisting of repeatedly taking $D$ samples from a distribution and picking out one according to an omniscient "strategy." We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all past and future choices, and furthermore, that this set coincides with the set of measures with finite grid entropy. This set is convex and weakly compact; we characterize its extreme points as those given by a natural "greedy" deterministic strategy and we compute the grid entropy of said extreme points to be 0. This yields a description of the set of limit points of empirical measures as the closed convex hull of measures given by a density which is $D \cdot Beta(1,D)$ distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.

math.PR↗

arXiv2022

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

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.

math.PR↗