Isomonodromy Transformations of Difference Equations

【摘要】本课题主要分为两个部分。首先，我们采用Birkhoff对差分方程的“单性”的一个相当不明显的定义，并旨在发展一个保持单性的差分方程变换的一般理论。由此得到的结构与经典的微分方程等同形变理论有一定的相似性，并在一定极限下退化。其次，我们将差分方程的同构变换应用于随机矩阵型离散概率模型中所谓的间隙概率的计算，这些模型出现在许多不同的领域，包括组合学、表示理论、渗透理论、平铺模型等。带理性系数的常微分方程的同构变形理论是19世纪末至20世纪初由Riemann，施莱辛格，富克斯和卡尼尔。从那时起，同构变形在数学和数学物理的不同领域中得到了大量的应用，从代数和微分几何到随机矩阵和表示理论。另一方面，近年来人们对分析一类离散概率模型产生了浓厚的兴趣，这些模型在适当的极限下收敛于众所周知的随机矩阵理论模型。这些模型的来源相当多样化，它们包括组合学、表示理论、渗透理论、随机生长过程、平铺模型等。本提案的研究目标是发展具有有理系数的线性差分方程系统的“同构变换”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无关的问题，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，该项目将为许多这样的巧合提供一个可见的共同点，从而将促进和加速不同研究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.