Open Access
2010 Combinatorics and cluster expansions
William G. Faris
Probab. Surveys 7: 157-206 (2010). DOI: 10.1214/10-PS159


This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on species that is mapped to a pointwise multiplication of the exponential generating functions. The statistical mechanics side deals with a probability model of an equilibrium gas. The cluster expansion that gives the density of the gas is the exponential generating function for the species of rooted connected graphs. The main results of the theory are simple criteria that guarantee the convergence of this expansion. It turns out that other problems in combinatorics and statistical mechanics can be translated to this gas setting, so it is a universal prescription for dealing with systems of high dimension.


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William G. Faris. "Combinatorics and cluster expansions." Probab. Surveys 7 157 - 206, 2010.


Published: 2010
First available in Project Euclid: 7 June 2010

zbMATH: 1191.82009
MathSciNet: MR2684165
Digital Object Identifier: 10.1214/10-PS159

Primary: 60K35 , 82B20
Secondary: 05A15 , 05C30 , 82B05

Keywords: cluster expansion , connected graph , Equilibrium lattice gas , exponential generating function , polymer system , species of structures

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.7 • 2010
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