Lattice polytopes in mathematical physics

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

Abstract

The Tutte polynomial of graphs is deeply connected to the q-state Potts model in statistical mechanics. In this survey, we describe this connection and show how one can use lattice polytopes and transfer matrices to tackle a special case given by the zero-temperature anti-ferromagnetic Potts model, where computing the Tutte polynomial reduces to computing the chromatic polynomial. We apply the techniques earlier introduced by the authors to this special case. This article can be seen as a brief summary of this work with a view towards statistical mechanics. In particular, we illustrate this procedure by computing the chromatic polynomials of some (families of) graphs. As it turns out, counting integer points in lattice polytopes is one of the key ingredients of these computations.
Original languageEnglish
Title of host publicationAlgebraic and Geometric Combinatorics on Lattice Polytopes
Subtitle of host publicationProceedings of the Summer Workshop on Lattice Polytopes
EditorsTakayuki Hibi, Akiyoshi Tsuchiya
PublisherWORLD SCIENTIFIC
ISBN (Print)9789811200472
DOIs
Publication statusPublished - Jun 2019
MoE publication typeA3 Part of a book or another research book
EventSummer Workshop on Lattice Polytopes - Osaka, Japan
Duration: 23 Jul 201810 Aug 2018

Workshop

WorkshopSummer Workshop on Lattice Polytopes
CountryJapan
CityOsaka
Period23/07/201810/08/2018

Keywords

  • q-state Potts model
  • partition function
  • Tutte polynomial
  • chromatic polynomial
  • lattice polytope
  • transfer-matrix method
  • Ehrhart theory

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