Transition distributions of young diagrams under periodically weighted plancherel measures

Abstract

Kerov[16,17] proved that Wigner’s semi-circular law in Gaussian unitary ensembles is the transition distribution of the omega curve discovered by Vershik and Kerov[34] for the limit shape of random partitions under the Plancherel measure. This establishes a close link between random Plancherel partitions and Gaussian unitary ensembles. In this paper we aim to consider a general problem, namely, to characterize the transition distribution of the limit shape of random Young diagrams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrasov’s partition functions and is further studied by Nekrasov and Okounkov[25] and Okounkov[28,29]. We also find an associated matrix model for this transition distribution. Our argument is based on a purely geometric analysis on the relation between matrix models and Seiberg-Witten differentials.