Critical points of random polynomials with independent identically distributed roots

By Zakhar Kabluchko. 2012

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Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\C$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)...(z-X_n)$. We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure $\mu_n:=\frac 1{n-1}\sum_{P_n'(z)=0} \delta_z$ counting the complex zeros of the derivative $P_n'$ converges in probability to $\mu$, as $n\to\infty$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ee5c3d91e11b81cb1415b7c3f5428c1714d4591f