Models of percolation based on random walks

本研究建议的对象是基于随机游动的渗流模型。这些模型具有长程相关性，通常不适用于传统的渗流方法。所提出的研究集中在两个示范性的例子：（a）大环面上的随机游动，（B）随机游动回路汤格。它的目的是深入研究结构相变和相关性引起的新现象。我们希望在这种情况下开发的方法提供了新的见解，分析更广泛的一类依赖结构。此外，该计划的一个组成部分导致基本的开放问题的范围内的一个单一的随机行走的格子和随机交错的模型，我们也打算解决。

The objects of this research proposal are models of percolation based on random walks. These models exhibit long-range correlations and are typically not amenable to traditional methods of percolation. The proposed research focuses on two demonstrative examples: (a) random walks on large tori, (b) random walk loop soups on lattices. It aims to investigate in depth structural phase transitions and new phenomena induced by correlations. We expect that the methods developed in this context provide novel insights into analysis of a broader class of dependent structures. Furthermore, an integral part of this programme leads to fundamental open questions about the range of a single random walk on a lattice and the model of random interlacements, which we also intend to solve.