Point processes and the infinite symmetric group. Part V: Analysis of the matrix Whittaker kernel

Abstract

The matrix Whittaker kernel has been introduced by A. Borodin in Part IV of the present series of papers. This kernel describes a point process -- a probability measure on a space of countable point configurations. The kernel is expressed in terms of the Whittaker confluent hypergeometric functions. It depends on two parameters and determines a $J$-symmetric operator $K$ in $L^2(R_+)\oplus L^2(R_+)$. It turns out that the operator $K$ can be represented in the form $L(1+L)^{-1}$, where $L$ is a rather simple integral operator: the kernel of $L$ is expressed in terms of elementary functions only. This is our main result; it elucidates the nature of the matrix Whittaker kernel and makes it possible to directly verify the existence of the associated point process. Next, we show that the matrix Whittaker kernel can be degenerated to a family of kernels expressed through the Bessel and Macdonald functions. In this way one can obtain both the well-known Bessel kernel (which arises in random matrix theory) and certain interesting new kernels.