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Difference operators and determinantal point processes

We consider a family {P} of determinantal point processes arising in representation theory and random matrix theory. The processes live on the one-dimensional lattice and their correlation kernels correspond to projection operators in the l^2 Hilbert space on the lattice. Moreover, these projections are spectral projections associated to certain selfadjoint second order difference operators on the lattice. The aim of the note is to demonstrate that the difference operators in question can be efficiently employed in the study of limit transitions inside the family {P}.