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Carstens, Sebastian Maurice (2012): Percolation analysis of the two-dimensional Widom-Rowlinson lattice model. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

We consider the two-dimensional Widom-Rowlinson lattice model. This discrete spin model describes a surface on Which a one to one mixture of two gases is sprayed. These gases shall be strongly repelling on short distances. We indicate the amount of gas by a positive parameter, the so called activity. The main result of this thesis states that given an activity larger than 2, there are at most two ergodic Widom-Rowlinson measures if the underlying graph is the star lattice. This falls naturally into two parts: The first part is quite general and establishes a new sufficient condition for the existence of at most two ergodic Widom-Rowlinson measures. This condition demands the existence of 1*lassos, i.e., 1*circuits 1*connected to the boundary, with probability bounded away from zero. Our approach is based upon the infinite cluster method. More precisely, we prevent the (co)existence of infinite clusters of certain types. To this end, we first have to improve the existing results in this direction, which will be done in a general setting for two-dimensional dependent percolation. The second part is devoted to verify the sufficient condition of the first part for activities larger than 2. To this end, we have to compare the probabilities of configurations exhibiting 1*lassos to the ones exhibiting 0lassos. This will be done by constructing an injection that fills certain parts of 0circuits with 1spins and, hereby, forms a 1*lasso.