TY - JOUR

T1 - Pairing of zeros and critical points for random meromorphic functions on riemann surfaces

AU - Hanin, Boris

N1 - Publisher Copyright:
© 2015 International Press.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - We prove that zeros and critical points of a random polynomial pN of degree N in one complex variable appear in pairs. More precisely, suppose pN is conditioned to have pN(ξ)=0 for a fixed ξεC. For εε (0, 1/ 2) we prove that there is a unique critical point in the annulus {z ∈ C/N-1-ε-1+ε}and no critical points closer to ξ with probability at least 1-O(N-3/2+3ε). We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

AB - We prove that zeros and critical points of a random polynomial pN of degree N in one complex variable appear in pairs. More precisely, suppose pN is conditioned to have pN(ξ)=0 for a fixed ξεC. For εε (0, 1/ 2) we prove that there is a unique critical point in the annulus {z ∈ C/N-1-ε-1+ε}and no critical points closer to ξ with probability at least 1-O(N-3/2+3ε). We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

UR - http://www.scopus.com/inward/record.url?scp=84927659319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84927659319&partnerID=8YFLogxK

U2 - 10.4310/MRL.2015.v22.n1.a7

DO - 10.4310/MRL.2015.v22.n1.a7

M3 - Article

AN - SCOPUS:84927659319

VL - 22

SP - 111

EP - 140

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 1

ER -