Universality in non-Hermitian matrix models

普遍性是我们理解自然的一个基本方面。这意味着许多物理系统表现出相同的行为，而不管它们的组成元素之间相互作用的细节是什么。例如，所有宏观物体都遵循热力学定律，而与此同时，物质是由原子构成的。将宏观热力学定律与原子物理学相调和，是科学家们长期面临的一项基本而艰巨的挑战。换句话说，在宏观尺度上，物理系统表现出普遍性。粗略地说，随机矩阵理论（RMT）可以被认为是线性代数和概率论的结合。每次物理或数学过程具有随机性质并由线性方程控制时，很可能可以用RMT来建模。事实上，随机矩阵模型在数学和物理的许多分支中都有重要的应用，如组合学、复杂系统、动力系统、增长问题、可积系统、数论、算子代数、概率论、量子混沌、量子场论、量子信息、统计学、统计力学、结构动力学和电信。RMT之所以能在如此广泛的应用中成为一个强大的工具，其主要特点还是它的通用性。在这种情况下，这意味着对于大矩阵维度，随机矩阵的特征值的局部统计量仅取决于矩阵的对称性，而与控制其随机行为的概率密度的选择无关。一大类厄密矩阵模型的普适性已经得到了证明，但就其全部的普适性而言，它仍然是一个猜想。这个项目的主要目标是证明在大量非厄米矩阵模型中的普适性。这种矩阵的集成可以应用于生长问题，Hele-Shaw问题，特别是在临界点附近，以及量子霍尔体系中电子液滴的半经典研究。非厄米矩阵没有任何对称约束，除了它们的元素必须分别是实数、复数或实数四元数。在这三种情况下，我们将研究随机矩阵谱的普适性。此外，普适性将研究在体积以及奇异区域的频谱。可以预料，在这两种情况下，局部谱统计量将表现得相当不同。然而，它们仍然是普适性的，因为它们只依赖于临界点的类型而不依赖于矩阵的概率分布。最后，研究非厄米矩阵模型通用性的主要工具之一将是利用dbar问题对复平面上的正交多项式进行渐近分析，这将对诸如无色散多维可积系统的研究和可积算子的渐近分析产生影响。

Universality is a fundamental aspect of our understanding of Nature. It means that many physical systems manifest the same behaviour independently of what the details of the interaction among their constituent elements are. For example, all macroscopic objects obey the laws of thermodynamics, while at the same time matter is built out of atoms. The conciliation of the macroscopic laws of thermodynamics with atomic physics has been a long-standing, fundamental and difficult challenge for scientists. In other words, on a macroscopic scale physical systems exhibit universality.Loosely speaking, Random Matrix Theory (RMT) can be thought of as a combination of linear algebra and the theory of probability. Each time a physical or mathematical process has a stochastic nature and its governed by linear equations, it is likely that it may be modelled by RMT. Indeed, random matrix models have fundamentally important applications in many branches of mathematics and physics such as combinatorics, complex systems, dynamical systems, growth problems, integrable systems, number theory, operator algebra, probability theory, quantum chaos, quantum field theory, quantum information, statistics, statistical mechanics, structural dynamics and telecommunications. The main feature that makes RMT a powerful tool in such wide range of applications is, once again, universality. In this context it means that for large matrix dimensions the local statistics of eigenvalues of random matrices depend only on the symmetries of the matrices, but are independent of the choice of the probability densities that govern their stochastic behaviour. Universality has been proved for a large class of Hermitian matrix models, but in its full generality it is still a conjecture.The main goal of this project is to prove universality in a vast class of non-Hermitian matrix models. Such ensembles of matrices find applications to growth problems, to the Hele-Shaw problem, especially in the vicinity of a critical point, and to semiclassical study of electronic droplets in the Quantum Hall regime. Non-Hermitian matrices do not have any symmetry constraints, with the exception that their elements must be real, complex or real quaternions respectively. The universality of the spectra of random matrices will be studied in all these three cases. Furthermore, universality will be investigated in the bulk as well in singular regions of the spectrum. It is expected that in these two cases the local spectral statistics will behave rather differently. However, they will still be universal, in the sense that they will depend only on the type of critical point but not on the probability distribution of the matrices.Finally, one of the main tools in the investigation of universality in non-Hermitian matrix models will be the asymptotic analysis of orthogonal polynomials in the complex plain using the dbar problem, which will have implications, for example, in the study of dispersionless multi-dimensional integrable systems and in the asymptotic analysis of integrable operators.