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Phase Coexistence for the Hard-Core Model on ℤ2

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

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: [email protected], [email protected])
YUXUAN CHEN
Affiliation:
Computer Science Department, Columbia University, New York, NY 10027, USA (e-mail: [email protected])
DAVID GALVIN
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46656, USA (e-mail: [email protected])
DANA RANDALL
Affiliation:
School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: [email protected], [email protected])
PRASAD TETALI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: [email protected])
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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.

References

[1] Alm, S. (1993) Upper bounds for the connective constant of self-avoiding walks. Combin. Probab. Comput. 2 115–136.Google Scholar
[2] Alm, S. and Parviainen, R. (2004) Bounds for the connective constant of the hexagonal lattice. J. Phys. A: Math. Gen. 37 549–560.Google Scholar
[3] Baxter, R. J., Enting, I. G. and Tsang, S. K. (1980) Hard-square lattice gas. J. Statist. Phys. 22 465–489.Google Scholar
[4] Beffara, V. and Duminil-Copin, H. (2012) The self-dual point of the two-dimensional random cluster model is critical above q = 1. Probab. Theory Rel. Fields 153 511–542.Google Scholar
[5] van den Berg, J. and Steif, J. E. (1994) Percolation and the hard-core lattice model. Stochastic Process. Appl. 49 179–197.Google Scholar
[6] Blanca, A., Chen, E., Galvin, D., Randall, D. and Tetali, P. Taxi walks computations. http://nd.edu/~dgalvin1/TD/Google Scholar
[7] Blanca, A., Galvin, D., Randall, D. and Tetali, P. (2013) Phase coexistence and slow mixing for the hard-core model on ℤ2 In Approximation, Randomization, and Combinatorial Optimization, Vol. 8096 of Lecture Notes in Computer Science, Springer, pp. 379–394.Google Scholar
[8] Borgs, C. Personal communication.Google Scholar
[9] Borgs, C., Chayes, J. T., Frieze, A., Kim, J. H., Tetali, P., Vigoda, E. and Vu, V. H. (1999) Torpid mixing of some MCMC algorithms in statistical physics. In FOCS: 40th IEEE Annual Symposium on Foundations of Computer Science, IEEE, pp. 218–229.Google Scholar
[10] Brightwell, G. and Winkler, P. (1999) Graph homomorphisms and phase transitions. J. Combin. Theory Ser. B 77 221–262.Google Scholar
[11] Dobrushin, R. L. (1968) The problem of uniqueness of a Gibbs random field and the problem of phase transitions. Funct. Anal. Appl. 2 302–312.Google Scholar
[12] Dobrushin, R. L. (1968) The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 197–224.Google Scholar
[13] Galvin, D. (2008) Sampling independent sets on the discrete torus. Random Struct. Alg. 33 356–376.Google Scholar
[14] Galvin, D. and Kahn, J. (2004) On phase transitions in the hard-core model on Zd. Combin. Probab. Comput. 13 137–164.Google Scholar
[15] Georgii, H.-O. (1988) Gibbs Measures and Phase Transitions, De Gruyter.Google Scholar
[16] Goulden, I. and Jackson, D. M. (1979) An inversion theorem for cluster decompositions of sequences with distinguished subsequences. J. London Math. Soc. 20 567–576.Google Scholar
[17] Guttmann, A. J. and Joyce, G. S. (1972) On a new method of series analysis in lattice statistics. J. Phys. A: Gen. Phys. 5 L81.Google Scholar
[18] Kesten, H. (1963) On the number of self-avoiding walks. J. Math. Phys. 4 960–969.Google Scholar
[19] Kotecký, R. Personal communication.Google Scholar
[20] Noonan, J. and Zeilberger, D. (1999) The Goulden–Jackson cluster method: Extensions, applications, and implementations. J. Diff. Equ. Appl. 5 355–377.Google Scholar
[21] Onsager, L. (1944) Crystal statistics, I: A two-dimensional model with an order-disorder transition. Phys. Rev. Lett. 65 117–149.Google Scholar
[22] Pantone, J. Personal communication.Google Scholar
[23] Radulescu, D. C. and Styer, D. F. (1987) The Dobrushin–Shlosman phase uniqueness criterion and applications to hard squares. J. Statist. Phys. 49 281–295.Google Scholar
[24] Randall, D. (2006) Slow mixing of Glauber dynamics via topological obstructions. In SODA: 17th ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 870–879.Google Scholar
[25] Restrepo, R., Shin, J., Tetali, P., Vigoda, E. and Yang, L. (2013) Improved mixing condition on the grid for counting and sampling independent sets. Probab. Theory Rel. Fields 156 75–99.Google Scholar
[26] Sinclair, A., Srivastava, P., Štefankovič, D. and Yin, Y. (2017) Spatial mixing and the connective constant: Optimal bounds. Probab. Theory Rel. Fields 168 153–197.Google Scholar
[27] Steele, J. M. (1997) Probability Theory and Combinatorial Optimization, SIAM.Google Scholar
[28] Vera, J. C., Vigoda, E. and Yang, L. (2013) Improved bounds on the phase transition for the hard-core model in 2-dimensions. In Approximation, Randomization, and Combinatorial Optimization, Vol. 8096 of Lecture Notes in Computer Science, Springer, pp. 699–713.Google Scholar
[29] Weitz, D. (2005) Combinatorial criteria for uniqueness of Gibbs measures. Random Struct. Alg. 27 445–475.Google Scholar
[30] Weitz, D. (2006) Counting independent sets up to the tree threshold. In STOC: 38th ACM Symposium on the Theory of Computing, ACM, pp. 140–149.Google Scholar
[31] Wilf, H. (2005) Generatingfunctionology, A. K. Peters.Google Scholar