The "volume functions" are generalizations of the "volume polynomials" studied by M. Mirzkhani. Given a topological type T of closed geodesic, they allow to count the average number of closed geodesics of length L and topological type T, on a compact hyperbolic surface picked at random according to the Weil-Petersson measure. For simple geodesics, Mirzakhani showed that the volume function is a polynomial in L, depending on the genus g. She also found a topological recursion that allows, in principle, to compute volume functions explicitly as functions of g and L. Mirzakhani & Zograf then gave asymptotic estimates as the genus g goes to infinity.
Volume functions are much less explicit for non-simple geodesics. With Laura Monk, we studied their asymptotics as g goes to infinity, showing the existence of an asymptotic expansion in powers of g^{-1} with rather precise information about the coefficients (as functions of L). Victor Le Guilloux, on the other hand, studied the limit when L goes to infinity for fixed g. My talk will give a survey of these asymptotic results.