Kerov(1993) proved Wigners semi-circular law in Gaussian unitary ensembles is the transition distribution of the omega curve discovered by Vershik and Kerov(1977) for the limit shape of random partitions under the Plancherel measure.This es tablishes a close link between random Plancherel partitions and Gaussian unitary ensembles.In this talk we will report a recent work on characterizing the transi tion distribution of the limit shape or random Youngdia-grams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrsovs par tition functions and was further studied by Nekrasov and Okounkov (2003) and Okounkov (2003,2006).We ?nd an associated matrix model for this transition dis tribution.Our argument is based on a purely geometric analysis on the relation between matrix models and Seiberg-Witten differentials.