The project has two major parts. First, we take a rather non-obvious definition of "monodromy" for difference equations due to Birkhoff and aim to develop a general theory of transformations of difference equations which preserve the monodromy. The resulting structure bears some similarity to the classical theory of isomonodromy deformations of differential equations and degenerates to it in a certain limit. Second, we apply the isomonodromy transformations of difference equations to evaluation of the so-called gap probabilities in discrete probabilistic models of random-matrix type which arise in many different domains including combinatorics, representation theory, percolation theory, tiling models, etc.
The theory of isomonodromy deformations of ordinary differential equations with rational coefficients is a classical subject developed in the end of the XIXth -- beginning of the XXth century by Riemann, Schlesinger, Fuchs, and Garnier. Since then the isomonodromy deformations have found numerous applications in very different domains of mathematics and mathematical physics, from algebraic and differential geometry to random matrices and representation theory. On the other hand, 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. The goal of the research presented in this proposal is to develop a general theory of "isomonodromy transformations" for linear systems of DIFFERENCE equations with rational coefficients. In these probabilistic models mentioned above, we often see how two problems which seem to be unrelated to each other, and which even come from parts of mathematics that do not seem to have any overlap, lead to the same final result. We believe that the project will provide a visible common ground for many of such coincidences and thus will promote and accelerate the exchange of methods and ideas accumulated by different groups of researchers.
