tailieunhanh - Đề tài " Isomonodromy transformations of linear systems of difference equations"

We introduce and study “isomonodromy” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial A(z). Our main result is construction of an isomonodromy action of Zm(n+1)−1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. . | Annals of Mathematics Isomonodromy transformations of linear systems of difference equations By Alexei Borodin Annals of Mathematics 160 2004 1141 1182 Isomonodromy transformations of linear systems of difference equations By Alexei Borodin Abstract We introduce and study isomonodromy transformations of the matrix linear difference equation Y z 1 A z Y z with polynomial A z . Our main result is construction of an isomonodromy action of zm ra 1 -1 on the space of coefficients A z here m is the size of matrices and n is the degree of A z . The birational action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes. Similarly to the continuous case for m n 2 the difference Schlesinger equations and their ợ-analogs yield discrete Painleve equations examples include dPII dPIV dPv and Ợ-PVI. Introduction In recent years there has been considerable interest in analyzing a certain class of discrete probabilistic models which in appropriate limits converge to well-known models of random matrix theory. The sources of these models are quite diverse they include combinatorics representation theory percolation theory random growth processes tiling models and others. One quantity of interest in both discrete models and their random matrix limits is the gap probability - the probability of having no particles in a given set. It is known due to works of many people see JMMS Me TW P HI BD that in the continuous random matrix type setup these probabilities can be expressed through solution of an associated isomonodromy problem for a linear system of differential equations with rational coefficients. The goal of this paper is to .