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Beschreibung:
This thesis investigates the phase-transition phenomenon in a certain percolation model with long-range dependencies. More precisely, it is mainly concerned with the discrete version of the so-called Gaussian free field, a universal object in probability theory and mathematical physics, and it concentrates on the percolation model obtained by considering its level sets. In the first part, the focus is going to be on percolation questions related to level sets of the Gaussian free field defined on transient trees. We derive sufficient conditions to ensure the non-triviality of the phase transition for the above mentioned level-set percolation, although our results extend far beyond that. The spotlight in the second part will be turned towards the zero-average Gaussian free field: a suitably defined version of the Gaussian free field for finite graphs. We consider two specific classes of finite graphs for d larger equal 3: the first given by large discrete d-dimensional tori, and the second including d-regular expanders of large girth and typical realisations of random d-regular graphs. As we will show, the local picture of the zero-average Gaussian free field on these graphs is provided by the Gaussian free field on the corresponding infinite graph, which is the d-dimensional Euclidean lattice or the d-regular tree respectively. This allows us to relate percolative properties of the level sets of the zero-average Gaussian free field on the two classes of finite graphs to the phase transition for level-set percolation of the Gaussian free field on the corresponding infinite graph.