On Transfer Operators for Markovian Products of Invertible Random Matrices

Asymptotic Analysis for Markovian Queues with Two Types of Nonpersistent Retrial Customers

We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number…

Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

We consider three different models of sparse random graphs:~undirected and directed Erdős-Rényi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we show that if the edge connectivity probability $p \in (0,1)$ satisfies $n p \ge \log n + k(n)$ with $k(n) \to \infty$ as $n \to \infty$, then the adjacency matrix is invertible with probability approaching one (here $n$ is the number of vertices in the two former cases and the number of left and right vertices in the latter case). If $np \le \log n -k(n)$ then these matrices are invertible with pro…

稀疏矩阵的良可逆列子集是罕见的

Well-invertible column subsets of sparse matrices are rare

研究在特定条件下，常数行稀疏矩阵S不是\((r,\alpha)\)-遗忘子空间注入（OSI）。通过将概率问题转化为非概率问题，证明Bourgain - Tzafriri定理保证存在的良可逆子矩阵是罕见的。

Invertibility of random matrices: unitary and orthogonal perturbations

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur and Zeitouni.

Hitting time theorems for random matrices

Starting from an n-by-n matrix of zeros, choose uniformly random zero entries and change them to ones, one-at-a-time, until the matrix becomes invertible. We show that with probability tending to one as n tends to infinity, this occurs at the very moment the last zero row or zero column disappears. We prove a related result for random symmetric Bernoulli matrices, and give quantitative bounds for some related problems. These results extend earlier work by Costello and Vu [arXiv:math/0606414].

Precise Large Deviation Results for Products of Random Matrices

The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur-Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev-Guivarch me…

矩阵乘积压力函数的解析性

Analyticity of the pressure function for products of matrices

本文研究非可逆矩阵乘积的压力函数，在不可约与压缩假设下证明其解析性，并建立变分原理以推广已有结果。

Markovian lifts of positive semidefinite affine Volterra type processes

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multi…

The probability of rectangular unimodular matrices over $\F_q[x]$

In this note, we compute the probability that a $k\times n$ matrix can be extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's density theorem on the co-prime integers and its various generalizations are also presented.