Mathematics – Combinatorics
Scientific paper
2010-05-29
Adv. in Appl. Math., 47 (2011) 772-782
Mathematics
Combinatorics
This e-print is an extended version, including additional background information and more detailed proofs, of a paper of the s
Scientific paper
10.1016/j.aam.2011.02.004
The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Nobel and Welsh's W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion-contraction reduction and Fortuin-Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.
Ellis-Monaghan Joanna A.
Moffatt Iain
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