The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model

2012 ◽  
Vol 173 (1) ◽  
pp. 1377-1386 ◽  
Author(s):  
U. A. Rozikov ◽  
R. M. Khakimov
Keyword(s):  
Gibbs Measure ◽  
Hard Core ◽  
Core Model ◽  
Uniqueness Condition ◽  
Periodic Gibbs Measure ◽  
Hard Core Model ◽  
Weakly Periodic Gibbs Measure

Phase Coexistence for the Hard-Core Model on ℤ2

2018 ◽  
Vol 28 (1) ◽  
pp. 1-22 ◽  
Author(s):  
ANTONIO BLANCA ◽  
YUXUAN CHEN ◽  
DAVID GALVIN ◽  
DANA RANDALL ◽  
PRASAD TETALI
Keyword(s):  
Gibbs Measure ◽  
Gibbs Measures ◽  
Hard Core ◽  
Independent Set ◽  
Phase Coexistence ◽  
Infinite Graphs ◽  
Lattice Gases ◽  
Core Model ◽  
Discrete Mathematics ◽  
Hard Core Model

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.

scholarly journals Percolation games, probabilistic cellular automata, and the hard-core model

2018 ◽  
Vol 174 (3-4) ◽  
pp. 1187-1217 ◽  
Author(s):  
Alexander E. Holroyd ◽  
Irène Marcovici ◽  
James B. Martin
Keyword(s):  
Cellular Automata ◽  
Hard Core ◽  
Core Model ◽  
Probabilistic Cellular Automata ◽  
Hard Core Model

``Hard‐Core'' Model of Liquids

1966 ◽  
Vol 45 (1) ◽  
pp. 378-383 ◽  
Author(s):  
Neil S. Snider
Keyword(s):  
Hard Core ◽  
Core Model ◽  
Hard Core Model

scholarly journals Liquid-solid transitions in the three-body hard-core model

2015 ◽  
Vol 109 (2) ◽  
pp. 20003
Author(s):  
Tommaso Comparin ◽  
Sebastian C. Kapfer ◽  
Werner Krauth
Keyword(s):  
Hard Core ◽  
Core Model ◽  
Three Body ◽  
Hard Core Model

scholarly journals Tunneling of the hard-core model on finite triangular lattices

2018 ◽  
Vol 55 (1) ◽  
pp. 215-246
Author(s):  
Alessandro Zocca
Keyword(s):  
Hard Core ◽  
Core Model ◽  
Triangular Lattices ◽  
Hard Core Model

Phase Transitions in 2:1 and 3:1 Hard-Core Model Electrolytes

2002 ◽  
Vol 88 (4) ◽  
Author(s):  
Athanassios Z. Panagiotopoulos ◽  
Michael E. Fisher
Keyword(s):  
Phase Transitions ◽  
Hard Core ◽  
Core Model ◽  
Hard Core Model

scholarly journals The Widom–Rowlinson model, the hard-core model and the extremality of the complete graph

2017 ◽  
Vol 62 ◽  
pp. 70-76 ◽  
Author(s):  
Emma Cohen ◽  
Péter Csikvári ◽  
Will Perkins ◽  
Prasad Tetali
Keyword(s):  
Complete Graph ◽  
Hard Core ◽  
Core Model ◽  
Hard Core Model

Entropy of the hard-core model

2010 ◽  
Vol 82 (2) ◽  
Author(s):  
A. Winkler ◽  
G. Alsmeyer ◽  
O. Rubner ◽  
A. Heuer
Keyword(s):  
Hard Core ◽  
Core Model ◽  
Hard Core Model

On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$

2004 ◽  
Vol 13 (2) ◽  
pp. 137-164 ◽  
Author(s):  
DAVID GALVIN ◽  
JEFF KAHN
Keyword(s):  
Phase Transition ◽  
Hard Core ◽  
Core Model ◽  
Hard Core Model

Approximations to hard-core models and their application to statistical analysis

1982 ◽  
Vol 19 (A) ◽  
pp. 281-292 ◽  
Author(s):  
Mark Westcott
Keyword(s):  
Statistical Analysis ◽  
Statistical Inference ◽  
Lower Bounds ◽  
Hard Core ◽  
Distribution Functions ◽  
Core Model ◽  
Upper And Lower Bounds ◽  
Nearest Neighbour ◽  
Hard Core Model

This paper derives upper and lower bounds to the distribution functions of nearest-neighbour and minimum nearest-neighbour distances between N points generated by a hard-core model on the surface of a sphere. The use of these bounds in statistical inference is discussed.

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