Toda, Plancherel, and Hurwitz

Following Okounkov's "Toda equations and Hurwitz numbers" https://arxiv.org/pdf/math/0004128.pdf [4], we show that the generating series for certain ramified coverings of the Riemann sphere (arbitrary ramification type over 0 and infinity, simple ramifications elsewhere) is a tau function for the Toda lattice hierarchy. We use and recall basic fermionic Fock space techniques in the process. Using the same techniques and following Okounkov's "Infinite wedge and random partitions" https://arxiv.org/abs/math/9907127 [5], we compute the correlation functions for the Plancherel measure and show they satisfy the same Toda hierarchy. We further discuss the probabilistic interest in the Plancherel measure (and if time permits, its variants and generalizations), connecting longest increasing subsequences of random permutations to largest eigenvalues of GUE random matrices.