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Scaling limits for a dynamic model of 2D Young diagrams

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Stochastic Partial Differential Equations

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik (Func. Anal. Appl., ‘96), are uniform measures under conditioning on their area. We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper space-time scaling. The stationary solution of the limit equation is identified with the so-called Vershik curve. We also discuss the corresponding dynamic fluctuation problem under a non-equilibrium situation, and derive stochastic partial differential equations in the limit. We study both uniform and restricted uniform statistics for the Young diagrams.

This is a joint work with Makiko Sasada (Univ Tokyo) and the paper on the part of the hydrodynamic limit is available:

arXiv:0909.5482.

This talk is part of the Isaac Newton Institute Seminar Series series.

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