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.
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.
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U2 - 10.4310/MRL.2015.v22.n1.a7
DO - 10.4310/MRL.2015.v22.n1.a7
M3 - Article
AN - SCOPUS:84927659319
SN - 1073-2780
VL - 22
SP - 111
EP - 140
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 1
ER -