Diffusions on the space of interval partitions with PoissonDirichlet stationary distributions


Douglas Rizzolo (U. Delaware)
We construct a pair of related diffusions on a space of partitions of the unit interval whose stationary distributions are the complements of the zero sets of Brownian motion and Brownian bridge respectively. The processes of ranked interval lengths of our partitionvalued diffusions are members of a two parameter family of infinitely many neutral allele diffusion models introduced by Ethier and Kurtz (1981) and Petrov (2009). Our construction is a step towards describing a diffusion on the space of real trees, stationary with respect to the law of the Brownian CRT, whose existence has been conjectured by Aldous. Based on joint work with N. Forman, S. Pal, and M. Winkel.
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The point processes at turning points of large lozenge tilings 

Sevak Mkrtchyan (U. Rochester)
In the thermodynamic limit of the lozenge tiling model the frozen boundary develops special points where the liquid region meets with two different frozen regions. These are called turning points. It was conjectured by Okounkov and Reshetikhin that in the scaling limit of the model the local point process near turning points should converge to the GUE corners process. We will discuss various results showing that the point process at a turning point is the GUE corner process and that the GUE corner process is there in some form even when at the turning point the liquid region meets two semifrozen regions of arbitrary rational slope. The last regime arises when weights in the model are periodic in one direction with arbitrary fixed finite period.
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Integrability of Periodic ASEP 

Eric Brattain (SUNY New Paltz)
The asymmetric simple exclusion process (ASEP) for N particles on a ring with L sites may be analyzed using the Bethe ansatz. We provide a rigorous proof that the Bethe ansatz is complete for the periodic ASEP. More precisely, we show that for all but finitely many values of the hopping rate, the solutions of the Bethe ansatz equations do indeed yield all \binom{N}{L} eigenstates. The proof follows ideas of Langlands and SaintAubin, which draw upon a range of techniques from algebraic geometry, topology and enumerative combinatorics. We also indicate how to extend these methods to open boundary conditions. Joint work with Axel Saenz and Norman Do.
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Universality beyond mean field models 

Paul Bourgade (Courant)
We will explain universality of local spectral statistics for a class of random band matrices. By relying on a meanfield reduction technique, we will show that strong forms of eigenvector delocalization implies eigenvalues statistics of GOE or GUE type. Hence, quantum unique ergodicity of the eigenstates implies repulsion of the eigenvalues.
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Zhipeng Liu (Courant)
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Zsolt PajorGyulai (Courant)
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