Appears in collection : Probability and Geometry in, on and of non-Euclidian spaces / Probabilités et géométrie dans, sur et des espaces non-euclidiens

I will announce the proof, with Thomas Budzinski and Baptiste Louf, of the following fact: a uniformly random triangulation of size n whose genus grows linearly with $n$, has diameter $O(log(n))$ with high probability. The proof is based on isoperimetric inequalities built from enumerative estimates strongly built on the (celebrated) previous work of my two coauthors. But before this, I will try to review a little bit the questions surrounding random maps on surfaces, in either fixed genus or high genus.