arXiv subjects
Publications and source records attributed to Jorge Garza Vargas.
We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it. This allows new, streamlined proofs of the gap labeling and Aomoto index theorems. We prove a version of this new formula for the Anderson model on trees.
Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $γ\in (0,1)$. We show that with probability $0.99$, $A + γG_n$ has all of its eigenvalue condition numbers bounded by $O\left(n^{5/2}/γ^{3/2}\right)$ and eigenvector condition number bounded by $O\left(n^3 /γ^{3/2}\right)$. Furthermore, we show that for any $s > 0$, the probability that $A + γG_n$ has two eigenvalues within distance at most $s$ of each other is $O\left(n^4 s^{1/3}/γ^{5/2}\right).$ In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [Banks et al. 2019] which proved an eigenvector condition number bound of $O\left(n^{3/2} / γ\right)$ for the simpler case of {\em complex} i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts $z-(A+γG_n)$ which recover the tail behavior of the complex Ginibre ensemble when $\Im z\neq 0$. This yields sharp control on the area of the pseudospectrum $Λ_ε(A+γG_n)$ in terms of the pseudospectral parameter $ε>0$, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.
The $k$-th moment of the mean empirical spectral distribution of the squared unimodular random matrix of dimension $N$ can be expressed in the form $N^{-2k-1} Q_k(N)$, where $Q_k(x)$ is a polynomial of degree $k+1$ with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchala, K. Zyczkowski (2014).
We study the max-convolution and max-stable laws for Boolean independence and prove that these are Dagum distributions (also known as log-logistical distributions).