Phase coexistence and slow mixing for the hard-core model on ℤ2

Antonio Blanca Pimentel, David Galvin, Dana Randall, Prasad Tetali

Research output: Chapter in Book/Report/Conference proceedingConference contribution

10 Citations (Scopus)

Abstract

For the hard-core model (independent sets) on ℤ2 with fugacity λ, we give the first explicit result for phase coexistence by showing that there are multiple Gibbs states for all λ > 5.3646. Our proof begins along the lines of the standard Peierls argument, but we add two significant innovations. First, building on the idea of fault lines introduced by Randall [19], we construct an event that distinguishes two boundary conditions and yet always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ℤ2. We also extend our characterization of fault lines to show that local Markov chains will mix slowly when λ > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when λ > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of λ [19].

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings
Pages379-394
Number of pages16
DOIs
StatePublished - Oct 15 2013
Event16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013 - Berkeley, CA, United States
Duration: Aug 21 2013Aug 23 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8096 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013
CountryUnited States
CityBerkeley, CA
Period8/21/138/23/13

Fingerprint

Hard-core Model
Phase Coexistence
Boundary conditions
Fault
Gibbs States
Markov processes
Self-avoiding Walk
Line
Independent Set
Innovation
Periodic Boundary Conditions
Free Boundary
Walk
Markov chain

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Pimentel, A. B., Galvin, D., Randall, D., & Tetali, P. (2013). Phase coexistence and slow mixing for the hard-core model on ℤ2. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings (pp. 379-394). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8096 LNCS). https://doi.org/10.1007/978-3-642-40328-6_27
Pimentel, Antonio Blanca ; Galvin, David ; Randall, Dana ; Tetali, Prasad. / Phase coexistence and slow mixing for the hard-core model on ℤ2. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings. 2013. pp. 379-394 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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Pimentel, AB, Galvin, D, Randall, D & Tetali, P 2013, Phase coexistence and slow mixing for the hard-core model on ℤ2. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8096 LNCS, pp. 379-394, 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013, Berkeley, CA, United States, 8/21/13. https://doi.org/10.1007/978-3-642-40328-6_27

Phase coexistence and slow mixing for the hard-core model on ℤ2. / Pimentel, Antonio Blanca; Galvin, David; Randall, Dana; Tetali, Prasad.

Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings. 2013. p. 379-394 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8096 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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N2 - For the hard-core model (independent sets) on ℤ2 with fugacity λ, we give the first explicit result for phase coexistence by showing that there are multiple Gibbs states for all λ > 5.3646. Our proof begins along the lines of the standard Peierls argument, but we add two significant innovations. First, building on the idea of fault lines introduced by Randall [19], we construct an event that distinguishes two boundary conditions and yet always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ℤ2. We also extend our characterization of fault lines to show that local Markov chains will mix slowly when λ > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when λ > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of λ [19].

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Pimentel AB, Galvin D, Randall D, Tetali P. Phase coexistence and slow mixing for the hard-core model on ℤ2. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings. 2013. p. 379-394. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-40328-6_27