Asymptotics of Plancherel-type random partitions

Abstract

We present a solution to a problem suggested by Philippe Biane: We prove that a certain Plancherel-type probability distribution on partitions converges, as partitions get large, to a new determinantal random point process on the set {0,1,2,...} of nonnegative integers. This can be viewed as an edge limit ransition. The limit process is determined by a correlation kernel on {0,1,2,...} which is expressed through the Hermite polynomials, we call it the discrete Hermite kernel. The proof is based on a simple argument which derives convergence of correlation kernels from convergence of unbounded self-adjoint difference operators. Our approach can also be applied to a number of other probabilistic models. As an example, we discuss a bulk limit for one more Plancherel-type model of random partitions.