Critical points of random polynomials with independent identically distributed roots

@article{Kabluchko2012CriticalPO,
  title={Critical points of random polynomials with independent identically distributed roots},
  author={Zakhar Kabluchko},
  journal={arXiv: Probability},
  year={2012},
  pages={695-702}
}
  • Zakhar Kabluchko
  • Published 2012
  • Mathematics
  • arXiv: Probability
  • 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 $n\to\infty$. 

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