*Two back-to-back presentations are scheduled.

In this talk, I will present two recent examples, based on a couple of joint works with Simon Barazer, Andrew Elvey-Price, Wenjie Fang, Lucas Teyssier and Michael Wallner, where random walk methods help us obtain results in combinatorics and geometry.

More precisely, both examples deal with Hurwitz numbers, a combinatorial model of geometry where objects are discretized surfaces, parametrized by a size n and a topological invariant, the genus g. They are part of a broader class of objects in the realm of enumerative geometry. The goal is to obtain asymptotics for Hurwitz numbers as both n and g go to infinity. This is a problem of bivariate asymptotics, notoriously more difficult than the univariate case. But sometimes’ one can make it work !

While both examples are quite different in their proof, the key ingredient comes each time from a random walk perspective. I will give a hint why this is the case, without going into technical details. No knowledge of combinatorics or geometry required !