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Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

Saddle points in the chaotic analytic function and Ginibre characteristic polynomial
Saddle points in the chaotic analytic function and Ginibre characteristic polynomial
Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realizing the logarithmic derivative of these infinite polynomials as the electric field of a distribution of Coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as pi^-1 |z|^-4 from the origin.
This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel.
0305-4470
3379-3383
Dennis, M.R.
ff55cf66-eb8b-4eb9-83eb-230c2f223d61
Hannay, J.H.
90d8c251-4201-42a3-80df-2eaa42eaedd5
Dennis, M.R.
ff55cf66-eb8b-4eb9-83eb-230c2f223d61
Hannay, J.H.
90d8c251-4201-42a3-80df-2eaa42eaedd5

Dennis, M.R. and Hannay, J.H. (2003) Saddle points in the chaotic analytic function and Ginibre characteristic polynomial. Journal of Physics A: Mathematical and General, 36 (12), 3379-3383. (doi:10.1088/0305-4470/36/12/329).

Record type: Article

Abstract

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realizing the logarithmic derivative of these infinite polynomials as the electric field of a distribution of Coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as pi^-1 |z|^-4 from the origin.
This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel.

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Published date: 2003
Organisations: Applied Mathematics

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Local EPrints ID: 29381
URI: https://eprints.soton.ac.uk/id/eprint/29381
ISSN: 0305-4470
PURE UUID: fd38283a-7bec-4930-8e5a-16b8d0c00d42

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Date deposited: 12 May 2006
Last modified: 17 Jul 2017 15:58

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