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

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 ε of each other is bounded above by ε(k2−1)/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 Hammond (Forum Math. Pi 7 (2019) e2, 69) 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 ε2/3 on short scales ε.

A Patchwork Quilt Sewn From Brownian Fabric: Regularity of Polymer Weight Profiles in Brownian Last Passage Percolation

“…The rerooting procedure to resolve the second difficulty in the rough guide will make use of an important aspect of polymer geometry: polymers have Hölder-2/3− regularity. We now quote two results from [Ham17a] to this effect. Some notational preparation is needed.…”

“…The present article is the culmination of a four-paper study of KPZ from such a viewpoint. The companion papers are [Ham16], [Ham17b] and [Ham17a]. Our main result, Theorem 1.2, draws on important technical ingredients from the other works in order to make a significant probabilistic inference about KPZ universality.…”

“…The first of the companion papers develops a theory of Brownian Gibbs line ensembles that applies to scaled Brownian LPP. This theory is hidden for the reader of the present article since it is not needed directly; it is used during the proofs of the central results in [Ham17b] and [Ham17a], which will be applied here and recalled presently. The basic idea of the proof in this article, which we will explain in Section 3, after presenting some notation and tools, is crucial to the four-paper study, since the task of implementing it rigorously has required the development of tools throughout the study.…”

Modulus of continuity of polymer weight profiles in Brownian last passage percolation

“…Actually, when we apply this corollary in the present article, it will be in the case that t 1 = 0 and t 2 = 1, so in fact the use of the scaling principle is unnecessary in this regard. It is useful, however, to have a general form for the corollary: for example, it is used in [Ham17b].…”

“…The proposition shows that the discrepancy between polymer weight and parabola is controlled uniformly as the polymer's endpoints are varied over compact intervals lying in a very broad region. This tool is needed in the present article and in [Ham17b]. For exponential or Poissonian last passage percolation, a similar tool has been developed, in [BSS16, Propositions 10.1 and 10.5].…”