Random surfaces

How does a random (closed compact orientable) surface look like? The answer depends of course on the kind of geometric structure we endow the surface with and the geometric property we want to study. I will present several results in this vein for some of the pairs (geometry, property) with "geometry" in {hyperbolic, flat with conical singularities, triangulation with the graph distance} and "property" in {diameter, systole, Cheeger constant, global/local scaling limit, length spectrum}. I will also explain the tools and ideas involved in the proofs.