Journal of Statistical Physics

, Volume 148, Issue 2, pp 250–279 | Cite as

Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces



In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z 1 and z 2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3π 2) for small \(\sqrt{N}|z_{1}-\nobreak z_{2}|\). The scaling limit is directly calculated using a general form of the Kac-Rice formula and formulas and theorems of Pavel Bleher, Bernard Shiffman, and Steve Zelditch.


Several complex variables Random sections 



I would like to thank my advisor Bernard Shiffman for his advice, patience, encouragement, and many ideas about solving this problem. I would also like to thank Renjie Feng for many useful conversations about this problem.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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