# Improved bounds on the phase transition for the hard-core model in 2 dimensions

Juan C. Vera, E. Vigoda, L. Yang

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz [Proceedings of the $38$th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 140--149]. Weitz presented a fully polynomial-time approximation scheme for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$, where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$, where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. [Probab. Theory Related Fields, 156 (2013), pp. 75--99] improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.
Original language English 1895–1915 21 SIAM Journal on Discrete Mathematics 29 4 https://doi.org/10.1137/140976923 Published - 1 Oct 2015

### Fingerprint

Hard-core Model
Phase Transition
Lower bound
Fully Polynomial Time Approximation Scheme
Lattice Gas Model
Nonuniqueness
Independent Set
Maximum Degree
Partition Function
Annual
Refinement
Uniqueness
Upper bound
Integer
Computing
Graph in graph theory

### Keywords

• approximate counting
• MCMC
• strong spatial mixing
• branching matrices
• linear programming

### Cite this

@article{5f510c9da43a4a369fdb0ab4f24100af,
title = "Improved bounds on the phase transition for the hard-core model in 2 dimensions",
abstract = "For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz [Proceedings of the $38$th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 140--149]. Weitz presented a fully polynomial-time approximation scheme for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$, where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$, where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. [Probab. Theory Related Fields, 156 (2013), pp. 75--99] improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.",
keywords = "approximate counting, MCMC, strong spatial mixing, branching matrices, linear programming",
author = "Vera, {Juan C.} and E. Vigoda and L. Yang",
year = "2015",
month = "10",
day = "1",
doi = "10.1137/140976923",
language = "English",
volume = "29",
pages = "1895–1915",
journal = "SIAM Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

In: SIAM Journal on Discrete Mathematics, Vol. 29, No. 4, 01.10.2015, p. 1895–1915.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Improved bounds on the phase transition for the hard-core model in 2 dimensions

AU - Vera, Juan C.

AU - Vigoda, E.

AU - Yang, L.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz [Proceedings of the $38$th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 140--149]. Weitz presented a fully polynomial-time approximation scheme for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$, where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$, where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. [Probab. Theory Related Fields, 156 (2013), pp. 75--99] improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.

AB - For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/nonuniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz [Proceedings of the $38$th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 140--149]. Weitz presented a fully polynomial-time approximation scheme for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$, where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$, where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. [Probab. Theory Related Fields, 156 (2013), pp. 75--99] improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.

KW - approximate counting

KW - MCMC

KW - strong spatial mixing

KW - branching matrices

KW - linear programming

UR - http://epubs.siam.org/doi/abs/10.1137/140976923

U2 - 10.1137/140976923

DO - 10.1137/140976923

M3 - Article

VL - 29

SP - 1895

EP - 1915

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -