Isomonodromy Transformations of Difference Equations

该项目由两个主要部分组成。首先，由于Birkhoff，我们给差分方程取了一个不太明显的“单调”定义，目的是发展一种保持单调的差分方程组变换的一般理论。这种结构与经典的微分方程式等差变形理论有一定的相似之处，并在一定的极限下退化为它。其次，我们将差分方程组的同余变换应用于计算随机矩阵型离散概率模型中的所谓间隙概率，这些模型出现在许多不同的领域，包括组合学、表示论、渗流理论、平铺模型等。有理系数常微分方程组的同余变换理论是Riemann，Schlesinger，Fuchs和Garnier在二十一世纪末和二十一世纪初发展的一个经典课题。从那时起，等单变形在数学和数学物理的许多不同领域都得到了广泛的应用，从几何和微分几何到随机矩阵和表示论。另一方面，近年来，人们对分析一类离散概率模型非常感兴趣，这些模型在适当的范围内收敛于著名的随机矩阵理论模型。这些模型的来源相当多样，它们包括组合学、表征理论、渗流理论、随机增长过程、瓦片模型等。这项研究的目的是发展有理系数线性微分方程组的“等差变换”的一般理论。在上面提到的这些概率模型中，我们经常看到两个看似互不相关的问题，甚至来自似乎没有任何重叠的数学部分，如何导致相同的最终结果。我们相信，该项目将为许多此类巧合提供一个明显的共同点，从而促进和加速不同研究人员小组积累的方法和思想的交流。

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.