arXiv subjects

Dmitry Zaporozhets

Publications and source records attributed to Dmitry Zaporozhets.

39 records · Page 3Linked to original sources

arXiv2011

On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients

Consider a random polynomial $G_n(z)=ξ_nz^n+...+ξ_1z+ξ_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|ξ_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\to\infty$, to two centered circles and is uniform in the argument as $n\to\infty$. The radii of the circles are $|ξ_0/ξ_τ|^{1/τ}$ and $|ξ_τ/ξ_n|^{1/(n-τ)}$, where $ξ_τ$ denotes the coefficient with the maximum modulus.

math.PR↗

arXiv2011

On random surface area

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \emph{non-random} smooth field.

math.PR↗

arXiv2011

On distribution of zeros of random polynomials in complex plane

Let $G_n(z)=ξ_0+ξ_1z+...+ξ_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2π]$ asymptotically as $n\to\infty$. We also prove that the condition $\E\ln(1+|ξ_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.

math.PR↗