### Abstract

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p
_{x} } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p
_{x} }. Furthermore, we show that the usual proof of the Lovász local lemma - which provides a sufficient condition for this to occur - corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer
^{(98)} and explicitly by Dobrushin.
^{(37,38)} We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for ''soft'' dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

Original language | English (US) |
---|---|

Pages (from-to) | 1151-1261 |

Number of pages | 111 |

Journal | Journal of Statistical Physics |

Volume | 118 |

Issue number | 5-6 |

DOIs | |

State | Published - Mar 2005 |

### Fingerprint

### Keywords

- Cluster expansion
- Graph
- Hard-core interaction
- Independent-set polynomial
- Lattice gas
- Lovász local lemma
- Mayer expansion
- Polymer expansion
- Probabilistic method

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*118*(5-6), 1151-1261. https://doi.org/10.1007/s10955-004-2055-4

**The repulsive lattice gas, the independent-set polynomial, and the lovász local lemma.** / Scott, Alexander D.; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 118, no. 5-6, pp. 1151-1261. https://doi.org/10.1007/s10955-004-2055-4

}

TY - JOUR

T1 - The repulsive lattice gas, the independent-set polynomial, and the lovász local lemma

AU - Scott, Alexander D.

AU - Sokal, Alan D.

PY - 2005/3

Y1 - 2005/3

N2 - We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma - which provides a sufficient condition for this to occur - corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer (98) and explicitly by Dobrushin. (37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for ''soft'' dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

AB - We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma - which provides a sufficient condition for this to occur - corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer (98) and explicitly by Dobrushin. (37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for ''soft'' dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

KW - Cluster expansion

KW - Graph

KW - Hard-core interaction

KW - Independent-set polynomial

KW - Lattice gas

KW - Lovász local lemma

KW - Mayer expansion

KW - Polymer expansion

KW - Probabilistic method

UR - http://www.scopus.com/inward/record.url?scp=17544380550&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17544380550&partnerID=8YFLogxK

U2 - 10.1007/s10955-004-2055-4

DO - 10.1007/s10955-004-2055-4

M3 - Article

VL - 118

SP - 1151

EP - 1261

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -