Abstract: The study of random hyperbolic surfaces, especially in the asymptotic of large genus, is of increasing interest in recent years. Many geometrical questions have analogous formulations in the theory random graphs with a large number of vertices, and results obtained in one domain can inspire the other.

In this way, we get interested in the diameter of random surfaces, which is a basic measure of the geometry of the surface. In 2019, Budzinski, Curien and Petri studied this measure for a special model of surfaces, built from random graphs. We extend it to obtain a richer class of models of random surfaces and we compute the asymptotic of the diameter of these surfaces. The strategy of the proof relies on a detailed study of an exploration process which is the analog to the breadth-first search exploration of a random graph. Its analysis is notably based on subadditive and concentration techniques.