TY - JOUR

T1 - Correlations and pairing between zeros and critical points of Gaussian Random polynomials

AU - Hanin, Boris

N1 - Publisher Copyright:
© 2013 The Author(s).
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84927645423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84927645423&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnt192

DO - 10.1093/imrn/rnt192

M3 - Article

AN - SCOPUS:84927645423

VL - 2015

SP - 381

EP - 421

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 2

ER -