Probability Surveys

Combinatorics and cluster expansions

William G. Faris

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Abstract

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.

Article information

Source
Probab. Surveys, Volume 7 (2010), 157-206.

Dates
First available in Project Euclid: 7 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.ps/1275928347

Digital Object Identifier
doi:10.1214/10-PS159

Mathematical Reviews number (MathSciNet)
MR2684165

Zentralblatt MATH identifier
1191.82009

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B05: Classical equilibrium statistical mechanics (general) 05C30: Enumeration in graph theory 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

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

Citation

Faris, William G. Combinatorics and cluster expansions. Probab. Surveys 7 (2010), 157--206. doi:10.1214/10-PS159. https://projecteuclid.org/euclid.ps/1275928347


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References

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