Abstract

We have considered three different “one-body” statistical systems involving Brownian excursions, which possess for fluctuations Kardar–Parisi–Zhang scaling with the critical exponent \(\nu = \tfrac{1}{3}\). In all models imposed external constraints push the underlying stochastic process to a large deviation regime. Specifically, we have considered fluctuations for: (i) Brownian excursions on non-uniform finite trees with linearly growing branching originating from the mean-field approximation of the Dumitriu–Edelman representation of matrix models, (ii) (1+1)D “magnetic” Dyck paths within the strip of finite width, (iii) inflated ideal polymer ring with fixed gyration radius. In the latter problem cutting off the long-ranged spatial fluctuations and leaving only the “typical” modes for stretched paths, we ensure the KPZ-like scaling for bond fluctuations. To the contrary, summing up all normal modes, we get the Gaussian behavior. In all considered models, KPZ fluctuations emerge in presence of two complementary conditions: (i) the trajectories are pushed to a large deviation region of a phase space, and (ii) the trajectories are leaning on an impenetrable boundary.

中文翻译：

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具有KPZ标度的均衡均值类似统计模型

摘要

我们考虑了涉及布朗漂移的三种不同的“单体”统计系统，它们具有随临界指数\（\ nu = \ tfrac {1} {3} \）波动的Kardar–Parisi–Zhang标度。。在所有模型中，施加的外部约束都会将潜在的随机过程推向较大的偏差范围。具体来说，我们考虑了以下波动：（i）线性增长分支的非均匀有限树上的布朗漂移，源自矩阵模型的Dumitriu–Edelman表示的平均场近似，（ii）（1 + 1）D“有限宽度的条带内的“ Dyck磁”路径，（iii）具有固定回转半径的理想聚合物环膨胀。在后一个问题中，可以消除远距离的空间波动，而仅保留“典型”模式用于拉伸路径，我们确保对键波动采用类似KPZ的缩放比例。相反，总结所有正常模式，我们得到高斯行为。在所有考虑的模型中，KPZ波动在两个互补条件下出现：