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].

中文翻译：

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用布朗织物缝制的拼布被子：布朗最后一次渗透中聚合物重量分布的规律性

在位于 Kardar-Parisi-Zhang (KPZ) 普适性类的最后一段渗透模型中，可以研究长能量最大化路径的能量作为路径的端点位置对的函数。可以引入比例坐标，以便这些最大化路径或聚合物现在通过单位级波动跨越单位距离，并且具有单位级的缩放能量或重量。在本文中，我们在这些缩放坐标中考虑布朗最后通道渗透。在窄楔形情况下，当此类聚合物的一个端点固定时，例如在 $(0,0)\in \mathbb{R}^{2}$ , 另一个是水平变化的, 在 $(z,1)$ , $z\in \mathbb{R}$ ，聚合物重量分布作为函数 $z\in \mathbb{R}$ 是局部布朗式；事实上，Hammond ['Airy line ensemble 的 Brownian 规律性和 Brownian last pass percolation 中的多聚合物西瓜'，预印本（2016 年），arXiv:1609.02971, 定理 2.11 和命题 2.5]，已知轮廓定律在给定的紧区间上与布朗桥有非常强的比较，每个都有 Radon-Nikodym 导数 $L^{p}$ 空间 $p\in (1,\infty)$ ，在缩放参数中一致，前提是在进行比较之前对权重曲线进行仿射调整。在本文中，我们概括了这种窄楔形案例，并从一个非常一般的初始条件开始研究聚合物重量分布。我们证明了紧区间上的轮廓在统一的意义上类似于布朗桥：将紧区间分成随机但受控数量的块，仿射调整后每个块中的轮廓具有 Radon-Nikodym 导数，位于每个 $L^{p}$ 空间 $p\in (1,3)$ . 通过利用对 Hammond 开发的聚合物领域中均匀聚结结构的理解来证明这一结果 ['控制布朗最后通道渗透中不相交聚合物稀有性的指数'，预印本 (2017)一种),arXiv:1709.04110] 使用来自 Hammond (2016) 和 ['Brownian last pass percolation 中聚合物重量分布的连续性模量'，预印本 (2017) 的技术b),arXiv:1709.04115]。