The six-vertex model on random planar maps revisited

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 tech-niques. We convert Kostov's work to a combinatorial argu-ment involving functional equations coming from recursive decompositions of the maps, which we solve rigorously us-ing 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 in-terest we rederive the formulae discovered by Bousquet-Melou and the first author.(c) 2023 Elsevier Inc. All rights reserved.