A tree bijection for rigid quadrangulations

Abstract: We study the counting problem of rigid quadrangulations, recently introduced by Budd and proven to be in bijection with colorful quadrangulations. The generating function for the latter has been derived in an algebraic manner by Bousquet-M\'elou and Elvey Price, which therefore also counts rigid quadrangulations. In this paper we will provide a direct, bijective proof, for this generating function. We will relate the rigid quadrangulations to some naturally appearing trees, decorated with certain natural data. By some slight bijective manipulation of the data, we get a decorated tree, for which the generating function can be found. This result opens the door to better understand the geometry of random rigid quadrangulations (and maybe even of random colorful quadrangulations), by studying the corresponding decorated trees. These properties are relevant the formulation of UV complete JT-gravity, following the work of Ferrari.