Unicellular maps and hyperbolic surfaces in high genus

In the past few years, the study of the geometric properties of random maps has been extended to a new regime, the "high genus regime", where we are interested in maps whose size and genus tend to infinity at the same time, at the same rate.
We consider here a slightly different case, where the genus also tends to infinity, but less rapidly than the size, and we study the law of simple cycles (with a well-chosen rescaling of the graph distance) in unicellular maps (maps with one face), thanks to a powerful bijection of Chapuy, Féray and Fusy.
The interest of this work is that we obtain exactly the same law as Mirzakhani and Petri who counted closed geodesics on a model of random hyperbolic surfaces in large genus (the Weil-Petersson measure). This leads us to conjecture that these two models are somehow "the same" in the limit. This is joint work with Svante Janson.