No abstract

We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, 2k-angulations), and constellations. These formulas are the fastest known way of computing these numbers.Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely the Pandharipande recursion for Hurwitz numbers (proven by Okounkov and simplified by Dubrovin-Yang-Zagier), as well as formulas for several models of maps (Goulden-Jackson, Carrell-Chapuy, Kazarian-Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.

We prove the local convergence of uniform bipartite maps with prescribed face degrees in the high genus regime. Unlike in the previous work [6] on the subject, we do not make any assumption on the tail of the face degrees, except that they remain finite in the limit.

We study large uniform random bipartite quadrangulations whose genus grows linearly with the number of faces. Their local convergence was recently established by Budzinski and the author [9, 10]. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar with high probability up to logarithmic radius, and we prove that there exist non-contractible cycles of constant length with positive probability.

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in [18]. The proof relies on a combinatorial argument and the Goulden-Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity.As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane T is weakly Markovian in the sense that the probability to observe a finite triangulation t around the root only depends on the perimeter and volume of t, then T is a mixture of PSHT.