A motivic approach to phase transitions in Potts models

TL;DR

This paper introduces a novel algebraic geometric approach to analyze phase transitions in Potts models by estimating the complexity of the zero locus of the partition function using Grothendieck classes, with explicit calculations for specific graph families.

Contribution

It develops a new method linking algebraic geometry and statistical mechanics to study phase transitions in Potts models, providing explicit calculations for certain graph structures.

Findings

Explicit calculations for linked polygons and dual graphs.

A deletion-contraction formula for Grothendieck classes.

Generation of functions for edge splitting and doubling.

Abstract

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion-contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.