Isomonodromy Transformations of Difference Equations

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ABSTRACTThe project has two major parts. First, we take a rathernon-obvious definition of "monodromy" for difference equations dueto Birkhoff and aim to develop a general theory of transformationsof difference equations which preserve the monodromy. Theresulting structure bears some similarity to the classical theoryof isomonodromy deformations of differential equations anddegenerates to it in a certain limit. Second, we apply theisomonodromy transformations of difference equations to evaluationof the so-called gap probabilities in discrete probabilisticmodels of random-matrix type which arise in many different domainsincluding combinatorics, representation theory, percolationtheory, tiling models, etc.The theory of isomonodromy deformations of ordinary differentialequations with rational coefficients is a classical subjectdeveloped in the end of the XIXth -- beginning of the XXth centuryby Riemann, Schlesinger, Fuchs, and Garnier. Since then theisomonodromy deformations have found numerous applications in verydifferent domains of mathematics and mathematical physics, fromalgebraic and differential geometry to random matrices andrepresentation theory. On the other hand, in recent years therehas been considerable interest in analyzing a certain class ofdiscrete probabilistic models which in appropriate limits convergeto well-known models of random matrix theory. The sources of thesemodels are quite diverse, they include combinatorics,representation theory, percolation theory, random growthprocesses, tiling models and others. The goal of the researchpresented in this proposal is to develop a general theory of"isomonodromy transformations" for linear systems of DIFFERENCEequations with rational coefficients. In these probabilisticmodels mentioned above, we often see how two problems which seemto be unrelated to each other, and which even come from parts ofmathematics that do not seem to have any overlap, lead to the samefinal result. We believe that the project will provide a visiblecommon ground for many of such coincidences and thus will promoteand accelerate the exchange of methods and ideas accumulated bydifferent groups of researchers.