2002/01/30 by Mark Adler Pierre van Moerbeke, van Moerbeke, Mark Adler Pierre
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.CO #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.48550/arxiv.math-ph/0201063
57 pages
arxiv created 2002/01/30 · arxiv updated 2009/11/30
The Toeplitz determinants (of increasing size) associated with the symbols expt(z+z-1) or (1-ξz)α (1-ξz-1)β satisfy recursion relations, thus expressing all the Toeplitz determinants as a rational function of the first few determinants. A. Borodin found these relations using Riemann-Hilbert methods. The nature of Borodin's relations pointed towards the Toeplitz lattice and its Virasoro algebra, as developed by the authors. In this paper, we take the Toeplitz and Virasoro approach for a fairly large class of symbols, leading to a systematic and simple way of generating such recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Lax pair for 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). Moreover, these recursion relations define an invariant manifold for the Toeplitz lattice. This leads to a "discrete Painlevé property" (singularity confinement) for the rational recursion relations, as a consequence of the classical ``continuous Painlevé property" for the Toeplitz lattice.