Sparse non-Hermitian random matrices

本文的目的是研究和探讨非厄米随机矩阵统计分析中的几个开放性问题。非厄米随机矩阵研究最多的例子是Ginibre系综及其在实辛情况下的推广。然而，有丰富的非厄米随机矩阵的重要兴趣。这样的矩阵至少出现在两种不同的情况下：它们描述了通过复杂网络结构相互作用的非平衡系统的光谱；它们对应于具有随机初始数据的可积系统的Lax矩阵。本文研究了稀疏非厄米随机矩阵的谱及其渐近性质。工作图是计算非厄米随机矩阵的重要类别的谱，获得数值见解，提出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.