Critical Behaviour of Spanning Forests on Random Planar Graphs

TL;DR

This paper investigates the critical behavior of random spanning forests on planar graphs, revealing connections with loop models, KPZ universality, and the Berker-Kadanoff phase, supported by exact solutions and numerical analysis.

Contribution

It establishes a novel link between spanning forests on planar graphs and KPZ universality, including explicit critical and multicritical loci analysis.

Findings

Identification of critical and multicritical points consistent with KPZ and Berker-Kadanoff phases

Explicit equations for the generating function at a special fugacity value

Numerical series analysis related to combinatorial problems

Abstract

As a follow-up of previous work of the authors, we analyse the statistical mechanics model of random spanning forests on random planar graphs. Special emphasis is given to the analysis of the critical behaviour. Exploiting an exact relation with a model of O(-2)-loops and dimers, previously solved by Kostov and Staudacher, we identify critical and multicritical loci, and find them consistent with recent results of Bousquet-M\'elou and Courtiel. This is also consistent with the KPZ relation, and the Berker-Kadanoff phase in the anti-ferromagnetic regime of the Potts Model on periodic lattices, predicted by Saleur. To our knowledge, this is the first known example of KPZ appearing explicitly to work within a Berker-Kadanoff phase. We set up equations for the generating function, at the value t=-1 of the fugacity, which is of combinatorial interest, and we investigate the resulting numerical series, a favourite problem of Tony Guttmann's.