Universiteit Leiden

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The parabolic Anderson model and long-range percolation

Promotor: Prof.dr. F. den Hollander, G.Maillard

D. Erhard
01 juli 2014
Thesis in Leiden Repository

This thesis has two parts. The first part deals with the parabolic Anderson model, which is a stochastic differential equation. It models the evolution of a field of particles performing independent simple random walks with binary branching. The focus of this work is on the exponential growth rate of the solution, where several basic properties are derived. The second part deals with two long-range percolation models. The occupied set of the first model is obtained by taking the union of a collection of independent Brownian motion running up to time t whose initial positions are distributed according to a Poisson Point process. Basic properties such as existence of a percolation phase transition and the uniqueness of the unbounded cluster are proven. The second model is the model of random interlacements. It is shown that the vacant set of random interlacement is transient for almost all values of the supercritical phase as the dimension tends to infinity.