Physics – Mathematical Physics
Scientific paper
2004-12-20
Physics
Mathematical Physics
Companion work to CA/0412394
Scientific paper
10.1088/0951-7715/18/5/010
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^m_{j=1}(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 Painlev\'e equation \dPV as it occurs in the theory of the sixth Painlev\'e system. Methods based on affine Weyl group symmetries of B\"acklund 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.
Forrester Peter J.
Witte Nicholas S.
No associations
LandOfFree
Discrete Painlevé equations for a class of \PVI $τ$-functions given as U(N) averages does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Discrete Painlevé equations for a class of \PVI $τ$-functions given as U(N) averages, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrete Painlevé equations for a class of \PVI $τ$-functions given as U(N) averages will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-577209