Going back at least to the works of Witten and Kontsevich, it is known that (symplectic or Weil-Petersson) volumes of moduli spaces of Riemann surfaces share many features with the enumeration of maps. It is therefore natural to expect that the theory of random hyperbolic metrics sampled according to the Weil-Petersson measure on, say, punctured spheres is closely related to the theory of random planar maps. I will highlight some similarities and show that tree bijections, which are ubiquitous in the study of random planar maps, have analogues for hyperbolic surfaces. As an application, jointly with Nicolas Curien, we show that these random hyperbolic surfaces with properly rescaled metric admit a scaling limit towards the Brownian sphere when the number of punctures increases.[-]

In this talk I will discuss a bijection between the moduli space of genus-0 hyperbolic surfaces with a distinguished cusp and certain labeled trees, analogous to known tree bijections in the combinatorics of planar maps. The Weil-Petersson measure on the moduli space takes a simple form at the level of the trees, and gives a bijective interpretation to the coefficients in the Weil-Petersson volume polynomials. The labels on the trees give precise information about geodesic distances in the surface, which can be used to study the geometry of random hyperbolic surfaces sampled from the Weil-Petersson measure. In particular, the random genus-0 hyperbolic surface with $n$ cusps is shown to converge as a metric space, after rescaling by $n^{-1/4}$, to the Brownian sphere.This talk is based on work with Nicolas Curien and with Thomas Meeusen and Bart Zonneveld.[-]