Critical points of random polynomials with independent identically distributed roots

Kabluchko, Zakhar

doi:10.1090/S0002-9939-2014-12258-1

Critical points of random polynomials with independent identically distributed roots
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by Zakhar Kabluchko

Proc. Amer. Math. Soc. 143 (2015), 695-702

DOI: https://doi.org/10.1090/S0002-9939-2014-12258-1

Published electronically: September 19, 2014

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Abstract:

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with values in $\mathbb {C}$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)\ldots (z-X_n)$. We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds., Advances in Combinatorics, Waterloo Workshop in Computer Algebra, $2011$] 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 $n\to \infty$.

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