Universality of Random Matrices Statistics

随机矩阵是无序物理系统的模型，描述其能级或本征态的关键属性。这个概率领域现在覆盖了可积系统、增长模型、数论或多元统计的许多方面。在过去的十年里，基于不变模型的解析方法和具有潜在独立性的模型的概率方法，包括戴森布朗运动的分析，人们对这些模型的数学理解有了很大的提高。本研究项目首先讨论了这种统计的局域普适性，扩大了表示随机矩阵型相互作用的模型的类别；这包括对所谓的β系综的研究，证明了许多随机哈密顿系统的局域相互作用只取决于它们的不变性类型，并分析了出现在非厄米随机矩阵理论中的二维谱的普适性。这个项目的另一个部分涉及随机矩阵理论的介观尺度，推导出与解析数论和随机能量模型相关的新统计。实际上，令人惊讶的随机矩阵理论重叠了解析数论、统计物理和统计学中的基本问题。类似于描述许多具有潜在独立性的系统的涨落的高斯分布，随机矩阵理论统计出现在许多强相关系统的普遍情况下。物理和数值实验证实了这一点，这些实验涉及量子系统的能级、公共交通工具的等待时间、停放的汽车的间隙尺寸分布、液晶湍流增长的界面以及L函数零点之间的典型和极端间距。随机矩阵是统计出现在非常不同的领域的范例，概率论中越来越重要的部分致力于理解这些联系。

Random matrices are models for disordered physical systems, describing key properties of their energy levels or eigenstates. This probabilistic field now overlaps many aspects of integrable systems, growth models, number theory, or multivariate statistics. The mathematical understanding of these models has considerably improved over the past ten years, based on analytic methods for invariant models, and probabilistic methods for models with underlying independence, including the analysis of Dyson's Brownian motion. This research project first concerns the local universality of such statistics, enlarging the class of models presenting the random matrices type of interactions; this includes the study of the so-called beta-ensembles, proving that the local interactions of the energy levels of many random Hamiltonian systems only depend on the their invariance type, and the analysis of universality for bidimensional spectra, appearing in non-Hermitian random matrix theory. Another part of this project concerns the mesoscopic scale in random matrix theory, deriving new statistics relevant in analytic number theory and random energy models.Indeed, surprisingly random matrix theory overlaps fundamental problems in analytic number theory, statistical physics and statistics. Analogously to the Gaussian distribution, describing the fluctuations of many systems with underlying independence, random matrix theory statistics appear universally for many strongly correlated systems. This as confirmed by physical and numerical experiments concerning the energy levels of quantum systems, waiting times in public transports, the gap size distribution of parked cars, growing interfaces of liquid crystal turbulence, or typical and extreme spacings between zeros of L-functions. Random matrices are paradigms for statistics appearing in very distinct fields, and an increasingly important part of probability theory is devoted to understanding these connections.