Correlations and pairing between zeros and critical points of Gaussian Random polynomials

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Abstract

Let pN be a degree N random polynomial in a complex variable. We obtain an explicit asymptotic formula for the covariance between the counting measures of its zeros and critical points, which we denote CovN(z,w). This formula shows that the correlation between a zero at z and a critical point at w is short range, decaying like e-N|z-w|2. With |z-w| on the order of N-1/2, however, CovN(z,w) is sharply peaked near z= w, causing zeros and critical points to appear in pairs. We prove bounds on the expected distance and angular dependence between a critical point and its paired zero.

Original languageEnglish (US)
Pages (from-to)381-421
Number of pages41
JournalInternational Mathematics Research Notices
Volume2015
Issue number2
DOIs
StatePublished - Jan 1 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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