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Combinatorics, Complexity, and ChanceA Tribute to Dominic Welsh$
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Geoffrey Grimmett and Colin McDiarmid

Print publication date: 2007

Print ISBN-13: 9780198571278

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198571278.001.0001

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FLOWS AND FERROMAGNETS

FLOWS AND FERROMAGNETS

Chapter:
(p.130) 8 FLOWS AND FERROMAGNETS
Source:
Combinatorics, Complexity, and Chance
Author(s):

Geoffrey Grimmett

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198571278.003.0008

The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety of objects defined on graphs. This chapter shows that they permit a representation of the two-point correlation function of a ferromagnetic Potts model on a graph G in terms of the flow polynomials of certain related random graphs. This representation extends to general Potts models the so-called random-current expansion for Ising models, and it amplifies the links between the Potts partition function and the Tutte polynomial.

Keywords:   Tutte polynomial, Ising model, ferromagnetic Potts model, random-cluster model, Potts partition function

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