Abstract
In last passage percolation models lying in the Kardar–Parisi–Zhang (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 $(0,0)\in \mathbb{R}^{2}$ , and the other is varied horizontally, over $(z,1)$ , $z\in \mathbb{R}$ , the polymer weight profile as a function of $z\in \mathbb{R}$ is locally Brownian; indeed, by Hammond [‘Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation’, Preprint (2016), arXiv:1609.02971, 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 $p\in (1,\infty )$ , 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 $p\in (1,3)$ . This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].
Highlights
In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations
The 1 + 1 dimensional KPZ universality class includes a wide range of interface models suspended over a one-dimensional domain, in which growth in a direction normal to the surface competes with a smoothening surface tension in the presence of a local randomizing force that roughens the surface
The broad range of interface models that are rigorously known or expected to lie in the KPZ universality class includes many last passage percolation models, in which the interface models the maximum obtainable value of paths, where a path is assigned a value by integrating over its course weights specified by a product measure random environment
Summary
The infected region grows, one site at a time. As Figure 1 illustrates, this region is at any given moment the collection of vertices abutting the present set of transmission edges. Theorem 1.2, roughly asserts that the weight profiles y → Wgt(ny;(,1∗): f,0), viewed as functions of y in a given compact real interval, enjoy a uniformly strong similarity with Brownian motion, even as the parameters n and f are permitted to vary over all sufficiently high integer values and over the Downloaded from https://www.cambridge.org/core. The patchwork quilt formed by this fabric sequence and stitch point set is itself a random continuous function, defined under the law P, that we will denote by Quilt[F, S], (where the bar notation indicates a vector, in this case, indexed by N). There is a quilt description of the modified process with a stitch sewn at onehalf, and there is no means of undoing that stitch
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