关于随机游走在冷却随机环境中 Ergodic 行为和缩放极限的综合论述
A cohesive account on the ergodic behaviours and scaling limits of Random Walks in Cooling Random Environments
AI总结：
本文综述了随机游走在冷却随机环境中的主要结果，包括复发准则、大数定律、大偏差和不同重采样制度下的波动现象，并提出了一种统一的相图观点。
摘要：
在无序介质中的输运是一个概率论和统计物理的核心主题，其中底层介质的随机性导致了局部化、异常缩放和慢弛豫等现象。随机游走在随机环境中（RWRE）是这一主题的典型模型，自70年代以来已被广泛研究。近年来，一些研究探索了旨在在静态无序和完全均质动力学之间进行插值的传输模型扰动。随机游走在冷却随机环境中（RWCRE）是在这种背景下引入的：环境在规定时间动态重新采样并在之间保持固定，从而产生一种微妙的“准ergodic”结构，在时间上允许在均质随机游走和经典RWRE之间进行插值。本文的目的是双重的。第一个目标是提供关于RWCRE在1d中的主要结果的原创综述：复发准则、大数定律、大偏差和不同重采样制度下的波动现象。这些结果是在一系列近期工作中得出的，并在此补充了若干新陈述，旨在呈现统一的相图观点。作为第二个目标，我们试图提取一个连贯的概念图景，突出结构机制——如ergodic极限、扰动下的持续性和替换原理——这些机制不仅限于RWCRE的具体设置，而且对可以以相同方式通过引入独立重置进行扰动的更广泛无序系统具有相关性。
英文摘要：
Transport in disordered media is a central theme in probability and statistical physics, where randomness in the underlying medium produces phenomena such as localization, anomalous scaling, and slow relaxation. A paradigmatic model for transport in disordered media is that of Random Walks in Random Environments (RWRE), which has been extensively studied since the 1970's and is by now well understood in one dimension. More recently, several works have explored perturbations of models of transport in disordered media aimed at interpolating between static disorder and fully homogenized dynamics. Random walks in cooling random environments (RWCRE), introduced in this context, constitute a key example: the environment is dynamically resampled at prescribed times and kept fixed in between, giving rise to a delicate ``quasi-ergodic'' structure in time allowing to interpolate between homogeneous random walks and classical RWRE. The purpose of this paper is twofold. A first goal is to offer an original survey on the main results for RWCRE in 1d: recurrence criteria, law of large numbers, large deviations, and fluctuation phenomena across different resampling regimes. Those results have been derived in a series of recent works and are complemented here with a number of new statements aiming at presenting a unified phenomenological picture. As a second goal, we try to extract a coherent conceptual picture that highlights structural mechanisms -- such as ergodic limits, persistence under perturbations and replacement principles -- that extend beyond the specific setting of RWCRE and are relevant for a broader class of disordered systems that can be perturbed by introducing independent resetting in the same fashion.
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