Models of percolation based on random walks

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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.