Sparse non-Hermitian random matrices

本文的目的是研究和探索非厄米特随机矩阵统计分析中的几个公开问题。研究最多的非厄米特随机矩阵的例子是Ginibre系综及其对实和辛情形的推广。然而，具有重要意义的非埃尔米特随机矩阵的种类很多。这类矩阵至少出现在两种不同的环境中：它们描述通过复杂网络结构相互作用的非平衡系统的谱；它们对应于具有随机初始数据的可积系统的Lax矩阵。本文将研究稀疏非厄米特随机矩阵的谱及其渐近性质。工作映射是计算重要的非厄米特随机矩阵的谱，获得数值不确定式，形成新的猜想，并证明严格的结果。

The goal of this thesis is to study and explore several open problems in the statistical analysis of non-Hermitian random matrices. The most studied example of non-Hermitian random matrices is the Ginibre ensemble and its generalization to the real and symplectic case. However there are rich classes of non-Hermitian random matrices of significant interest.Such matrices occur in at least two distinct settings: they describe the spectra of non-equilibrium systems that interact through a complex network structure; they correspond to the Lax matrices of integrable systems with random initial data.In the thesis the spectrum of sparse non-Hermitian random matrices and its asymptotic properties will be studied.The work map is to compute the spectrum of significant classes of non-Hermitian random matrices, obtain numerical insghts, formulate new conjectures and prove rigorous results.