Percolation analysis of the two-dimensional Widom-Rowlinson lattice model

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.