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 weakly converges as $N\to\infty$ to a deterministic measure, depending on just the variance and covariance of the entries of the $A_j$. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where $d=1$. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name ``the petaloid law''. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.
We study the linearization of the random $p$-adic orthogonal matrix model. Let $A_n$ and $B_n$ be Haar-random $n\times n$ special orthogonal and alternating matrices over $\mathbb{Z}_p$, respectively. For an odd prime $p$, we prove that $\mathrm{cok}(A_n-I_n)$ and $\mathrm{cok}(B_n)$ have the same limiting distribution...
We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2\pi])\right]$, the expected number of zeros in $[0, 2\pi]$ of a random periodic signal $S_n$ of the form \[ S_n(t)=\sum_{k=1}^{n}a_k f(kt), \] where $f$ is a non-analytic $2\pi-$periodic function and the coefficients $(a_k)$ are i.i...
For any non-zero periodic function f\,: Z $\rightarrow$ C of period N ___ 1 and any non-zero algebraic number z\_0 that doesn't lie on a half-line [e^2i${\ell}$$\pi$/N , e^2i${\ell}$$\pi$/N $\infty$[, 0 ___ ${\ell}$ ___ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z\_0 )-vector space sp...
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
Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $\mu_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in(0,1)$, we prove that, as $n \to \infty$, the empirical zero measure of the $k_n$...
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Zhang has given a complete proof of this conjecture by a centered pseudoinverse and spe...
Yin-Jie Li· 0 citations
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