### 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 [28] and explicitly by Dobrushin [12, 13]. We also present a generalization of the Lovász Local Lemma that allows for 'soft' dependencies. The paper aims to provide an accessible discussion of these results, which are drawn from a longer paper [26] that has appeared elsewhere.

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

Pages (from-to) | 253-279 |

Number of pages | 27 |

Journal | Combinatorics Probability and Computing |

Volume | 15 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2006 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Statistics and Probability

### Cite this

*Combinatorics Probability and Computing*,

*15*(1-2), 253-279. https://doi.org/10.1017/S0963548305007182

**On dependency graphs and the lattice gas.** / Scott, Alexander D.; Sokal, Alan D.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 15, no. 1-2, pp. 253-279. https://doi.org/10.1017/S0963548305007182

}

TY - JOUR

T1 - On dependency graphs and the lattice gas

AU - Scott, Alexander D.

AU - Sokal, Alan D.

PY - 2006/1

Y1 - 2006/1

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 [28] and explicitly by Dobrushin [12, 13]. We also present a generalization of the Lovász Local Lemma that allows for 'soft' dependencies. The paper aims to provide an accessible discussion of these results, which are drawn from a longer paper [26] that has appeared elsewhere.

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 [28] and explicitly by Dobrushin [12, 13]. We also present a generalization of the Lovász Local Lemma that allows for 'soft' dependencies. The paper aims to provide an accessible discussion of these results, which are drawn from a longer paper [26] that has appeared elsewhere.

UR - http://www.scopus.com/inward/record.url?scp=29844456285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29844456285&partnerID=8YFLogxK

U2 - 10.1017/S0963548305007182

DO - 10.1017/S0963548305007182

M3 - Article

VL - 15

SP - 253

EP - 279

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1-2

ER -