Enumeration of meanders and volumes of moduli spaces of quadratic differentials

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversely. In physics, meanders provide a model of polymer folding, and their enumeration is directly related to the entropy of the associated dynamical systems. We combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero with a fact that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated to study the asymptotic enumeration of meanders. As a result, we get an explicit asymptotic formula for the number of closed meanders with fixed number of minimal arcs when the number of crossings goes to infinity (after a joint work with V.Delecroix, E.Goujard and A.Zorich)