Difference Operators and Determinantal Point Processes

The paper deals with a family $\{P\}$ of determinantal point processes arising in representation theory and random matrix theory. The processes $P$ live on a one-dimensional lattice and have a number of special properties. One of them is that the correlation kernel $K(x,y)$ of each of the processes is a projection kernel: it determines a projection $K$ in the Hilbert $\ell^2$ space on the lattice. Moreover, the projection $K$ can be realized as the spectral projection onto the positive part of the spectrum of a self-adjoint difference second-order operator $D$. The aim of the paper is to show that the difference operators $D$ can be efficiently used in the study of limit transitions within the family $\{P\}$.