2006/01/10 by Mark Adler, M. Adler, Adler, M. +6
Mathematics · Medicine · Physics and Astronomy · #35Q58 #37K60 #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Epidemiology and Ecology Models #Numerical methods for differential equations #math-ph #math.MP #msc:35Q58 #msc:37K60
paper · pdf · doi:10.48550/arxiv.math-ph/0601020
arxiv created 2006/01/10 · openalex publication_date 2006/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a recent publication, it was shown that a large class of integrals over the unitary group U(n) satisfy difference equations over n, involving a finite number of steps; special cases are generating functions appearing in questions of longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property; roughly speaking, this means that, after a finite number of steps, the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (``singularity confinement''). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the ``Painlevé property'' for the discrete relations is inherited from the ``Painlevé property'' of the (continuous) Toeplitz lattice.