The multivariate Tutte polynomial (alias Potts model) for graphs and matroids

@misc{citeulike:141971, abstract = {The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics.}, author = {Sokal, Alan D. }, citeulike-article-id = {141971}, comment = {* p.23 Any model can be mapped to hard-core local gas "polymer expansion" or "cluster expansion" (section V.7 of Simon) * p.6 2.16, spanning-tree generating polynomial (spanning forest for disconnected graphs) * p.22 Z non-vanishing in D means no phase-transition in D * p.23 5.5, independent set polynomial. Fugacities inside l-inf ball of lambda\_c radius=>polynomial non-zero=>unique measure * p.25 Laplacian in terms of incidence matrix BXB' where X is diag.mat. of edge weights. Matrix-tree theorem, proof through Cauchy-Binet? * p.26 half-plane property -- positive weights make spanning tree polynomial non-zero}, eprint = {math.CO/0503607}, keywords = {ising, resistance}, month = {March}, posted-at = {2007-11-29 02:03:18}, priority = {3}, title = {The multivariate Tutte polynomial (alias Potts model) for graphs and matroids}, url = {http://arxiv.org/abs/math.CO/0503607}, year = {2005} }

TY - ELEC ID - citeulike:141971 L3 - citeulike-article-id:141971 TI - The multivariate Tutte polynomial (alias Potts model) for graphs and matroids N2 - The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics. KW - ising KW - resistance N1 - * p.23 Any model can be mapped to hard-core local gas "polymer expansion" or "cluster expansion" (section V.7 of Simon) * p.6 2.16, spanning-tree generating polynomial (spanning forest for disconnected graphs) * p.22 Z non-vanishing in D means no phase-transition in D * p.23 5.5, independent set polynomial. Fugacities inside l-inf ball of lambda_c radius=>polynomial non-zero=>unique measure * p.25 Laplacian in terms of incidence matrix BXB' where X is diag.mat. of edge weights. Matrix-tree theorem, proof through Cauchy-Binet? * p.26 half-plane property -- positive weights make spanning tree polynomial non-zero AU - Sokal, Alan PY - 2005/03/25/ UR - http://arxiv.org/abs/math.CO/0503607 ER -