Discrete Painlevé equations for a class of \PVI $\tau$-functions given as U(N) averages

Abstract

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 $w(z)={\prod }_{j=1}^m(z-z_j(t){)}^{{\rho }_j}$ were derived.Here it is shown 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 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 \dPV 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 \dPV equation but with different parameters for the same problem. We find the 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.

Companion work to CA/0412394

References in corpus (4)

• Random Matrix Theory and Entanglement in Quantum Spin Chains
• Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons
• Distribution of the first particle in discrete orthogonal polynomial ensembles
• Discrete Painlevé equations and random matrix averages

Cited by in corpus (14)

• Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• Nonperturbative theory of power spectrum in complex systems
• Derivative moments for characteristic polynomials from the CUE
• Random Matrix Theory and the Sixth Painlevé Equation
• Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model
• Resurgence, Painleve Equations and Conformal Blocks
• Form factor expansions in the 2D Ising model and Painlevé VI
• Power spectrum of the circular unitary ensemble
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• Fuchs versus Painlevé
• Semi-classical Laguerre polynomials and a third order discrete integrable equation
• The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems
• Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations