The zeros of random polynomials cluster uniformly near the unit circle

The existence of the scaling limit and its universality, for correlations between zeros of {\it Gaussian} random polynomials, or more generally, {\it Gaussian} random sections of powers of a line bundle over a compact manifold has been proved in a great generality in the works [BBL2], [Ha], [BD], [BSZ1]-[BSZ4], and others. In the present work we prove the existence of the scaling limit for a class of {\it non-Gaussian} random polynomials. Our main result is that away from the...

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$ tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin and Kabluchko to the setting where the roots are neither independent n...

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau$ satisfying the Bernstein-Markov property on a regular compact set $K \subset \mathbb{C}$. We show that if $\mathbb{P}(|\xi_0|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_n$ converges weakly in p...

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents...

Let $\{\varphi_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ \mu $. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\sum_{k=0}^{n}\eta_k\varphi_k(z), $$ where $\{\eta_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $\eta_k$ satisfies certain uniform bounds for the fractional and logarithmic mome...

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.

We explore the asymptotic behavior of the centroids of random polygons constructed from regular polygons with vertices on the unit circle by extending the rays so that their lengths form a random permutation of the first \(n\) integers. Surprisingly, this question has connections to diverse mathematical contexts, including random matrix theory and discrete Fourier transforms. Through rigorous analysis, we establish that the sequence of the suitably rescaled centroids converge...

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is $\frac{2}{\pi} \log n + o(\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\frac{2}{\pi}\log n + O(1)$. Prior to this paper, such estimate has b...

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ where $\eta_0,\dots,\eta_n$ are complex-valued i.i.d.~standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a ver...

In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same.