Let \(X_{1,n},\ldots,X_{d,n}\) be \(n\times n\) random matrices built from independent i.i.d. entry arrays, with centered entries, normalized by \(n^{-1/2}\). We prove that, if every entry law has finite fourth moment, then this tuple converges almost surely strongly in \(*\)-distribution to a free circular family with the matching variances. Equivalently, normalized traces and operator norms converge for every fixed noncommutative \(*\)-polynomial, including polynomials with fixed matrix coefficients. No assumption is imposed on the pseudo-variances of the complex entries. The bounded-entry argument applies the spectrum and moment universality estimates of Brailovskaya and van Handel to all self-adjoint linear pencils. The matching Gaussian pencils are reduced to independent Wigner matrices and identified by Anderson's strong convergence theorem. A fixed-level centered truncation, followed by the Bai--Yin norm bound, transfers the result to finite fourth moments.
We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely...
Let $d=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d$ independent uniformly random permutation matrices fail to form a basis with probability $(1+o(1))n^2(1-1/n)^d=(e^{3/2}+o(1))n^2e^{-n}$. The same asymptotic holds for a uniformly random $d$-element subset,...
We consider the probability that a discrete random matrix $M_n(\xi)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\i...
S. Mateo, John Urschel, Nicholas West· 2 citations
Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establis...
We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $\alpha \in [0,2]$. We prove that, outside the unit disk, the characteristic...
Quentin François, J. Heiny, Xuechun Hu· 0 citations
We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identica...
Ion Grama, Sebastian Mentemeier, Hui Xiao· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
