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- W2547103690 abstract "The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter $lambda$, and an independent set $I$ arises with probability proportional to $lambda^{|I|}$. On infinite graphs a Gibbs measure is defined as a suitable limit with the correct conditional probabilities, and we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs measures. It has long been conjectured that on ${mathbb Z}^2$ this model has a critical value $lambda_c approx 3.796$ with the property that if $lambda lambda_c$ then there is phase coexistence. Much of the work to date on this problem has focused on the regime of uniqueness, with the state of the art being recent work of Sinclair, Srivastava, Stefankovic and Yin showing that there is a unique Gibbs measure for all $lambda 5.3506$. There is some potential for lowering this bound, but with the methods we are using we cannot hope to replace $5.3506$ with anything below about $4.8771$. Our proof begins along the lines of the standard Peierls argument, but we add two innovations. First, following ideas of Kotecký and Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ${mathbb Z}^2$." @default.
- W2547103690 created "2016-11-11" @default.
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- W2547103690 date "2016-11-03" @default.
- W2547103690 modified "2023-09-27" @default.
- W2547103690 title "Phase Coexistence for the Hard-Core Model on ${mathbb Z}^2$." @default.
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