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

Abstract

In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at (0, 0) ∈ R 2 , and the other is varied horizontally, over (z, 1), z ∈ R, the polymer weight profile as a function of z ∈ R is locally Brownian; indeed, by [Ham16, Theorem 2.11 and Proposition 2.5], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon-Nikodym derivative in every L p space for p ∈ (1, ∞), uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon-Nikodym derivative that lies in every L p space for p ∈ (1, 3). This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in [Ham17a] using techniques from [Ham16] and [Ham17b].

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

“…Noting that C k is increasing, c k is decreasing, C k C and c k c, we verify (24). This completes the proof of Proposition 6.3(2).…”

“…Next is [, Lemma A.1]. Since this result is drawn the final article in our four‐paper study, it is worth mentioning that the result has a fairly short and self‐contained proof in [, Appendix A].…”

“…Next is [, Lemma A.1]. Since this result is drawn the final article in our four‐paper study, it is worth mentioning that the result has a fairly short and self‐contained proof in [, Appendix A]. Lemma Let and let and , , be a collection of pairs of points in such that is non‐empty.…”

“…Were to be adapted to Brownian last passage percolation, the present article's theorems and consequences might become applicable to universal KPZ structure. In particular, this outcome should be rather direct in the case of the consequence [, Theorem 1.2] concerning the Brownian nature of general initial condition profiles.…”

“…(4) The exponent . In its key application in , Theorem is applied when , for a triple of polymers. The form of the exponent is important in this application, dictating that a certain Radon–Nikodym derivative that describes the strength of Brownian comparison lies in .…”