Multicanonical sampling of rare events in random matrices

椭圆Ginibre矩阵实特征值的中等至大偏差渐进行为

Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

研究椭圆变形实Ginibre集合中实特征值数量的统计特性，推导中等至大偏差情形下稀有事件的概率，连接高斯波动与大偏差理论。

Large deviations of extreme eigenvalues of generalized sample covariance matrices

We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail of the large deviations of the smallest eigenvalue. The technique improves upon past methods by not requiring the explicit law of the eigenvalues, and we apply it to a large class of random matrices that were previously out of reach. In particular, we solve an open problem related to the performance of principal components analysis on highly correlated data, …

Native ultrametricity of sparse random ensembles

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the distribution of linear chains is purely exponential. We analyze in detail the spectral density of ensembles of linear subgraphs, discuss its ultrametric nature and show that near the spectrum boundary, the tail of the spectral density exhibits a Lifshitz singularity typical for Anderson localization. We also discuss an intriguing connection of the spectral density to…

Quench Dynamics in Randomly Generated Extended Quantum Models

We analyze the thermalization properties and the validity of the Eigenstate Thermalization Hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a Z_2 symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of …

Top eigenvalue of a random matrix: large deviations and third order phase transition

We study the fluctuations of the largest eigenvalue $λ_{\max}$ of $N \times N$ random matrices in the limit of large $N$. The main focus is on Gaussian $β$-ensembles, including in particular the Gaussian orthogonal ($β=1$), unitary ($β=2$) and symplectic ($β= 4$) ensembles. The probability density function (PDF) of $λ_{\max}$ consists, for large $N$, of a central part described by Tracy-Widom distributions flanked, on both sides, by two large deviations tails. While the central part characterizes the typical fluctuations of $λ_{\max}$ -- of order ${\cal O}(N^{-2/3})$ --, the large deviations t…