Models of percolation based on random walks

本文的研究对象是基于随机游走的渗流模型。这些模型表现出长程相关性，通常不适用于传统的渗流方法。提出的研究集中在两个示范性例子：(A)大环面上的随机游动，(B)格上的随机游动环汤。它的目的是深入研究结构相变和由关联引起的新现象。我们期待在此背景下开发的方法为更广泛类别的相关结构的分析提供新的见解。此外，这个程序的一个组成部分导致了关于格上单个随机游动的范围和随机交错模型的基本开放问题，这也是我们想要解决的。

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.