After a brief historical review, I will explain how to pick a (uniform) random hyperbolic surface of genus g. We will then turn to the length spectrum, focusing in particular on the short closed geodesics on such a random surface. It turns out that, when the genus g is big, the lengths of these geodesics are distributed just like the short cycles in a large random metric map. This is a joint work with Simon Barazer and Alessandro Giacchetto.
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