Actions for Phase Coexistence for the Hard-Core Model on Z^2 : Improved Bounds

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Phase Coexistence for the Hard-Core Model on Z^2 : Improved Bounds

Author
Liang, Leo
Published
[University Park, Pennsylvania] : Pennsylvania State University, 2021.
Physical Description
1 electronic document
Additional Creators
Pimentel, Antonio Blanca and Schreyer Honors College
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Open Access.
Summary
The hard-core model is a very useful model studied in multiple disciplines, as it represents lattice gases in statistical physics and independent sets in discrete mathematics and computer science. Researchers are interested in finding the critical points at which the hard-core model undergoes a uniqueness/non-uniqueness phase transition on the integer lattice graph Z^2. With much of the work focused on the regime of uniqueness, it has been proved that there is a unique Gibbs state for the hard-core model for all [lambda]<2.3882, which also means that there is rapid mixing for the local Markov chains with parameter [lambda]<2.3882. Blanca et al. consider the other regime, trying to find the critical point for when there is non-uniqueness (also known as phase coexistence region) for the hard-core model, which has long been conjectured to be as [lambda]>3.795. The method is based on counting a type of self-avoiding walks called taxi walks; the greater the taxi walk length that can be counted, the better the result. Current bounds obtained from this method use the number of taxi walks of length 60. With careful analysis of the properties of taxi walks, and a series of optimizations to our algorithms for computing the number of taxi walks, we are able to count taxi walks of length 70 and higher, obtaining an improved bound for the non-uniqueness region. In particular, our results imply that on Z^2 with the infinite setting, there is a phase coexistence for the hard-core model when [lambda]>=5.3485, and in the finite setting that local Markov chains will mix slowly for all [lambda]>=5.3485.
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Dissertation Note
B.S. Pennsylvania State University 2021.
Technical Details
The full text of the dissertation is available as an Adobe Acrobat .pdf file ; Adobe Acrobat Reader required to view the file.
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