  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.
