Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-17T21:45:07.742Z Has data issue: false hasContentIssue false
  • English
  • Français

An Improvement of the Lovász Local Lemma via Cluster Expansion

Published online by Cambridge University Press:  20 June 2011

RODRIGO BISSACOT ,
ROBERTO FERNÁNDEZ ,
ALDO PROCACCI  and
BENEDETTO SCOPPOLA
RODRIGO BISSACOT
Affiliation:
Departamento de Matemática–ICEx, Universidade Federal de Minas Gerais, CP 702 Belo Horizonte, MG, 30161-970Brazil, and Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, 76801, France (e-mail: rodrigo.bissacot@gmail.com)
ROBERTO FERNÁNDEZ
Affiliation:
Department of Mathematics, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands (e-mail: R.Fernandez1@uu.nl)
ALDO PROCACCI
Affiliation:
Departamento de Matemática–ICEx, Universidade Federal de Minas Gerais, CP 702 Belo Horizonte, MG, 30161-970Brazil (e-mail: aldo@mat.ufmg.br)
BENEDETTO SCOPPOLA
Affiliation:
Dipartimento di Matematica, Universita Tor Vergata di Roma, 00133 Roma, Italy (e-mail: scoppola@mat.uniroma2.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 709 - 719
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alon, N. and Spencer, J. (2008) The Probabilistic Method, third edition, Wiley-Interscience.CrossRefGoogle Scholar
[2]Böttcher, J., Kohayakawa, Y. and Procacci, A. Properly coloured copies and rainbow copies of large graphs with small maximum degree. Random Struct. Alg., to appear. Preprint. arXiv.org/abs/1007.3767Google Scholar
[3]Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.CrossRefGoogle Scholar
[4]Dobrushin, R. L. (1996) Perturbation methods of the theory of Gibbsian fields. In Lectures on Probability Theory and Statistics: Ecole d'Etude Probabilité de Saint-Flour XXIV, 1994 (Bernard, P., ed.), Vol. 1648 of Lecture Notes in Mathematics, Springer, pp. 1–66.CrossRefGoogle Scholar
[5]Erdős, P. and Lovász, L. (1975) Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and Finite Sets II, Vol. 10 of Colloq. Math. Soc. János Bolyai, North-Holland, pp. 609–627.Google Scholar
[6]Erdős, P. and Spencer, J. (1991) Lopsided Lovász local lemma and Latin transversals. Discrete Appl. Math. 30 151–154.CrossRefGoogle Scholar
[7]Fernandez, R. and Procacci, A. (2007) Cluster expansion for abstract polymer models: New bounds from an old approach. Comm. Math. Phys. 274 123–140.CrossRefGoogle Scholar
[8]Haxell, P. E. (2001) A note on vertex list colouring. Combin. Probab. Comput. 10 345–347.CrossRefGoogle Scholar
[9]Molloy, M. (1998) The probabilistic method. In Probabilistic Methods for Algorithmic Discrete Mathematics (Habib, M., McDiarmid, C., Ramirez-Alfonsin, J. and Reed, B., eds), Springer.Google Scholar
[10]Moser, R. and Tardos, G. (2010) A constructive proof of the general Lovász local lemma. J. Assoc. Comput. Mach. 57 #11.CrossRefGoogle Scholar
[11]Ndreca, S., Procacci, A. and Scoppola, B. Improved bounds on coloring of graphs. Preprint. arXiv.org/abs/1005.1875Google Scholar
[12]Pegden, W. An extension of the Moser-Tardos algorithmic local lemma. Preprint. arXiv.org/abs/1102.2853Google Scholar
[13]Ruelle, D. (1969) Statistical Mechanics: Rigorous Results, W. A. Benjamin.Google Scholar
[14]Scott, A. and Sokal, A. (2005) The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Statist. Phys. 118 1151–1261.CrossRefGoogle Scholar
[15]Scott, A. and Sokal, A. (2006) On dependency graphs and the lattice gas. Combin. Probab. Comput. 15 253–279.CrossRefGoogle Scholar
[16]Shearer, J. B. (1985) On a problem of Spencer. Combinatorica 5 241–245.CrossRefGoogle Scholar
[17]Spencer, J. (1977) Asymptotic lower bounds for Ramsey functions. Discrete Math. 20 69–76.CrossRefGoogle Scholar