  We study the asymptotic behaviour of random integer partitions under a new
probability law that we introduce, the Plancherel-Hurwitz measure. This
distribution, which has a natural definition in terms of Young tableaux, is a
deformation of the classical Plancherel measure which appears naturally in the
context of Hurwitz numbers, enumerating certain transposition factorisations in
symmetric groups.
  We study a regime in which the number of factors in the underlying
factorisations grows linearly with the order of the group, and the
corresponding topological objects, Hurwitz maps, are of high genus. We prove
that the limiting behaviour exhibits a new, twofold, phenomenon: the first part
becomes very large, while the rest of the partition has the standard
Vershik-Kerov-Logan-Shepp limit shape. As a consequence, we obtain asymptotic
estimates for unconnected Hurwitz numbers with linear Euler characteristic,
which we use to study random Hurwitz maps in this regime. This result can also
be interpreted as the return probability of the transposition random walk on
the symmetric group after linearly many steps.
