Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram

Abstract

The Schur process is a time-dependent analog of the Schur measure on partitions studied by A. Okounkov in Infinite wedge and random partitions, Selecta Math., New Ser. 7 (2001), 57–81. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.

Inhomogeneous field theory inside the arctic circle

“…Our setup, however, based on the DWIS, makes it more natural for us to work with the diagonal-to-diagonal transfer matrix, and therefore, with a two-band model. Presumably, using the row-to-row transfer matrix instead would lead us to a framework that should resemble the one of Okounkov and Reshetikhin [21]. But let us not pursue this direction here, and come back to our purpose, which is to get the imaginary-time propagator of a model that satisfies the three assumptions above.…”

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