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Invasion percolation on regular trees : structure, scaling limit and ponds Goodman, Jesse Alexander

Abstract

Invasion percolation is an infinite subgraph of an infinite connected graph with finite degrees, defined inductively as follows. To each edge of the underlying graph, attach a random edge weight chosen uniformly from [0,1], independently for each edge. Starting from a single vertex, a cluster is grown by adding at each step the boundary edge with least weight. Continue this process forever to obtain the invasion cluster. In the following, we consider the case where the underlying graph is a regular tree: starting from the root, each vertex has a fixed number of children. In chapter 2, we study the structure of the invasion cluster, considered as a subgraph of the underlying tree. We show that it consists of a single backbone, the unique infinite path in the cluster, together with sub-critical percolation clusters emerging at every point along the backbone. By studying the scaling properties of the sub-critical parameters, we obtain detailed results such as scaling formulas for the r-point functions, limiting Laplace transforms for the level sizes and volumes within balls, and mutual singularity compared to the incipient infinite cluster. Chapter 3 gives the scaling limit of the invasion cluster. This is a random continuous tree described by a drifted Brownian motion, with a drift that depends on a certain local time. This representation also yields a probabilistic interpretation of the level size scaling limit. Finally, chapter 4 studies the internal structure of the invasion cluster through its ponds and outlets. These are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. Tail asymptotics for fixed ponds are also derived.

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