The six-vertex model on random planar maps revisited

TL;DR

This paper rigorously analyzes the six-vertex model on random planar maps by converting previous matrix integral solutions into combinatorial functional equations, leading to new insights and simplified formulas in special cases.

Contribution

It provides a rigorous combinatorial derivation of the six-vertex model on random maps, building on Kostov's non-rigorous solution and exploring modular properties for simplification.

Findings

Rigorous combinatorial solution of the model

Re-derivation of known formulas in special cases

Identification of modular properties simplifying the solution

Abstract

We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-M\'elou and the first author.