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Phase Coexistence for the Hard-Core Model on ℤ2

Part of: Discrete mathematics in relation to computer science Combinatorial probability

Published online by Cambridge University Press:  21 May 2018

ANTONIO BLANCA ,
YUXUAN CHEN ,
DAVID GALVIN ,
DANA RANDALL  and
PRASAD TETALI
ANTONIO BLANCA
Affiliation:
School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: ablanca@cc.gatech.edu, randall@cc.gatech.edu)
YUXUAN CHEN
Affiliation:
Computer Science Department, Columbia University, New York, NY 10027, USA (e-mail: yuxuan.chen@columbia.edu)
DAVID GALVIN
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46656, USA (e-mail: dgalvin1@nd.edu)
DANA RANDALL
Affiliation:
School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: ablanca@cc.gatech.edu, randall@cc.gatech.edu)
PRASAD TETALI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: tetali@math.gatech.edu)
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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 λ, and an independent set I arises with probability proportional to λ|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 ℤ2 this model has a critical value λc ≈ 3.796 with the property that if λ < λc then it exhibits uniqueness of phase, while if λ > λc 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, Štefankovič and Yin showing that there is a unique Gibbs measure for all λ < 2.538. Here we explore the other direction and prove that there are multiple Gibbs measures for all λ > 5.3506. We also show that with the methods we are using we cannot hope to replace 5.3506 with anything below 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 ℤ2.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 68R05: Combinatorics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 1 , January 2019 , pp. 1 - 22
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Research supported in part by NSF grants CCF-1420934, CCF-1555579 and CCF-1617306.

‡

Research supported in part by Simons Foundation grant 360240.

§

Research supported in part by Simons Foundation grant 360240 and by National Security Agency grant NSA H98230-13-1-0248.

Research supported in part by NSF grant CCF-1526900.

‖

Research supported in part by NSF grant DMS-1407657.

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