Carstens, Sebastian Maurice (2012): Percolation analysis of the twodimensional WidomRowlinson lattice model. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
We consider the twodimensional WidomRowlinson 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 WidomRowlinson 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 WidomRowlinson 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 twodimensional 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.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  dependent site percolation, Zhang’s argument, infinite cluster, infinite path, bounded energy, TwoDimensional WidomRowlinson Lattice Model 
Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  1. June 2012 
1. Referee:  Georgii, HansOtto 
Persistent Identifier (URN):  urn:nbn:de:bvb:19145341 
MD5 Checksum of the PDFfile:  944f5fef518ba9c3e4e6640b5e5d6752 
Signature of the printed copy:  0001/UMC 20491 
ID Code:  14534 
Deposited On:  06. Aug 2012 14:22 
Last Modified:  16. Oct 2012 09:02 