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.