Discrete Painlevé equations for a class of P_VI τ-functions given as U(N) averages

In a recent work, difference equations (Laguerre–Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle were derived. It is shown here that in the case m = 3, these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke, using the methods of the Toeplitz lattice and Virasoro constraints. Moreover, it is shown that this coupled system can be reduced to yield the discrete fifth Painlevé equation dP_V as it occurs in the theory of the sixth Painlevé system. Methods based on affine Weyl group symmetries of Bäcklund transformations have previously yielded the dP_V equation, but with different parameters for the same problem. We find an explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin–spin correlation function of the square lattice Ising model.