Asymptotics of Plancherel – Type Random Partitions

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 Z+ of nonnegative integers. This can be viewed as an edge limit transition. The limit process is determined by a correlation kernel on Z+ 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.