Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
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DOI：
10.1112/plms.12292
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发表时间：
2017-09
影响因子：
1.8
通讯作者：
A. Hammond
中科院分区：
文献类型：
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作者：
A. Hammond
在KPZ普适类中的最后一次通过渗流模型中，长最大化路径与其端点的线性插值有典型的偏差，由插值距离的三分之二次幂控制。这三分之二的幂决定了缩放坐标的选择，在这些坐标中，这些最大化者，现在称为聚合物，跨越单位距离，具有单位阶波动。在这篇文章中，我们考虑了在这些标度坐标中的布朗最后一次通过渗流，并证明了存在k个不相交的聚合物穿过一个单位阶区域而开始和结束在彼此的短距离ε内的概率由ε（k2−1）/2+o（1）上界。我们推测这一结果是尖锐的，它产生了对聚合物聚结结构的均匀性的理解，并在哈蒙德（Forum Math. Pi 7（2019）e2，69）中发挥了基础作用，证明了从一般初始数据中聚合物重量分布的单位阶尺度与布朗运动的比较。本文还包含聚合物几何结构的三分之二幂定律的尺度上的表达：聚合物在短尺度ε上波动ε2/3。
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 .