Wednesday, February 8th, 2012
9:30am - 10:30am, in Campus Center 2-2540

Mirjana Vuletic

Brown University

Asymptotics of large random strict plane partitions, their combinatorics and generalized MacMahon's formula

Abstract: We introduce a measure on strict plane partitions that is an analog of the uniform measure on plane partitions. We describe this measure in terms of a Pfaffian point process and compute its bulk and edge limits when partitions become large. We show that height fluctuations around the limit shape are governed by a Gaussian free field. The above measure is a special case of the shifted Schur process, which generalizes the shifted Schur measure introduced by Tracy and Widom and is an analog of the Schur process introduced by Okounkov and Reshetikhin. We use the Fock space formalism to prove that the shifted Schur process is a Pfaffian point process and calculate its correlation kernel. We derive generating functions for strict plane partitions using various methods such as nonintersecting paths, RSK type algorithms and symmetric functions. We show that strict plane partitions are in bijection with domino tilings and we give some basic properties of this correspondence. We also obtain a generalization of MacMahon's formula for the generating function of plane partitions. We give a 2-parameter generalization related to Macdonald's symmetric functions. The formula is especially simple in the Hall-Littlewood case.