arXiv subjects
Publications and source records attributed to Mihai Stoiciu.
We study the $ε$-pseudospectra $σ_ε(A)$ of square matrices $A \in \mathbb{C}^{N \times N}$. We give a complete characterization of the $ε$-pseudospectrum of any $2 \times 2$ matrix and describe the asymptotic behavior (as $ε\to 0$) of $σ_ε(A)$ for any square matrix $A$. We also present explicit upper and lower bounds for the $ε$-pseudospectra of bidiagonal matrices, as well as for finite rank operators.
We study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β^{-1} appears as the square of the coupling constant.
We consider polynomials on the unit circle defined by the recurrence relation Φ_{k+1}(z) = z Φ_{k} (z) - \barα_{k} Φ_k^{*}(z) for k \geq 0 and Φ_0=1. For each n we take α_0, α_1, ...,α_{n-2} i.i.d. random variables distributed uniformly in a disk of radius r < 1 and α_{n-1} another random variable independent of the previous ones and distributed uniformly on the unit circle. The previous recurrence relation gives a sequence of random paraorthogonal polynomials \{Φ_n\}_{n \geq 0}. For any n, the zeros of Φ_n are n random points on the unit circle. We prove that, for any point p on the unit circle, the distribution of the zeros of Φ_n in intervals of size O(1/n) near p is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.