The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. The most famous example perhaps is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of one dimensional discrete Schroedinger operators. This talk will review recent work (partly joint with Maurice Duits) elucidating and exploiting this connection in the context of universality, laws of large numbers and central limit theorems.