Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice
(2003) Communications in Mathematical Physics — Vol. 237, n° 3, p. 397-440 (2003)
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- Abstract
- In a discussion in spring 2001, Alexei Borodin showed us recursion relations for the Toeplitz determinants going with the symbols e(t(z + z-1)) and (1 - xiz)(alpha) (1 - xiz(-1))(beta). Borodin obtained these relations using Riemann-Hilbert methods; see the recent work of Borodin B and Baik Baik. The nature of Borodin's recursion relations pointed towards the Toeplitz lattice and its Virasoro algebra, introduced by us in [3]. In this paper, we take the Toeplitz lattice and Virasoro algebra approach for a fairly large class of symbols, leading to a systematic way of generating recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol e(t(z + z-1)). The Painleve analysis of the Toeplitz lattice enables us to show the "singularity confinement'' for these recursion relations.
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Adler, M., & Van Moerbeke, P. (2003). Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice. Communications in Mathematical Physics, 237(3), 397-440. https://doi.org/10.1007/s00220-003-0818-4 (Original work published 2003)