Critical points of random polynomials and characteristic polynomials of random matrices

2014/12/15 by O'Rourke, Sean
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1412.4703

Abstract

Let p_n be the characteristic polynomial of an n × 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 nor identically distributed.

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