Hurwitz numbers count ramified coverings of the complex projective line, and a particular family of maps. They can also be interpreted as partition functions for random integer partitions under deformations of the Plancherel measure. The same measures were studied by Diaconis and Shahshahani in the context of random walks on the symmetric group, and were related to tau-functions of the Toda lattice integrable hierarchy by Okounkov. We study these measures in new regimes, finding limit shapes of large random partitions and approximate asymptotics of unconnected Hurwitz numbers. From the corresponding model of random unconnected maps, we consider the asymptotic enumeration of high genus connected maps. Based on ongoing joint work with Guillaume Chapuy and Baptiste Louf.
