A Potts/Ising Correspondence on Thin Graphs

TL;DR

This paper demonstrates a correspondence between q-state Potts and Ising models on thin random graphs, revealing an isomorphism that extends beyond planar graphs and provides insights into critical phenomena on complex topologies.

Contribution

It introduces a bond vertex model that exhibits Potts criticality on thin random graphs and establishes an isomorphism with the Ising model in this context, expanding understanding of critical behavior on non-planar structures.

Findings

Potts criticality can be modeled on thin random graphs.

An isomorphism between Potts and Ising models is established on these graphs.

The correspondence extends beyond planar graph limitations.

Abstract

We note that it is possible to construct a bond vertex model that displays q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary topology, which we denote as ``thin'' random graphs in contrast to the fat graphs of the planar diagram expansion. Since the four vertex model in question also serves to describe the critical behaviour of the Ising model in field, the formulation reveals an isomorphism between the Potts and Ising models on thin random graphs. On planar graphs a similar correspondence is present only for q=1, the value associated with percolation.