Universality of Random Matrices Statistics

随机矩阵是无序物理系统的模型，描述了它们的能级或特征态的关键特性。这个概率领域现在与可积系统、增长模型、数论或多元统计的许多方面重叠。在过去十年中，基于对不变模型的分析方法和对具有潜在独立性的模型的概率方法，包括对戴森布朗运动的分析，对这些模型的数学理解有了很大的提高。本研究项目首先关注这些统计的局部普遍性，扩大了呈现随机矩阵型相互作用的模型类别；这包括所谓的β系综的研究，证明了许多随机哈密顿系统的能级的局部相互作用只依赖于它们的不变性类型，并分析了二维谱的普惠性，出现在非厄米随机矩阵理论中。本项目的另一部分涉及随机矩阵理论中的介观尺度，推导出与解析数论和随机能量模型相关的新统计。事实上，令人惊讶的是，随机矩阵理论与解析数论、统计物理和统计学中的基本问题重叠。与描述具有潜在独立性的许多系统的波动的高斯分布类似，随机矩阵理论统计量在许多强相关系统中普遍出现。关于量子系统的能级、公共交通的等待时间、停放汽车的间隙大小分布、液晶湍流界面的增长，或l函数零点之间的典型和极端间隔，这些物理和数值实验都证实了这一点。随机矩阵是统计学的范例，出现在非常不同的领域，概率论中越来越重要的一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.