Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

Details Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

2011-06-23

arxiv.org/abs/1106.4737v4

Mathematics

Complex Variables

55 pages. LaTeX. output.txt is the output of running heisenberg_simpler.mpl through maple. heisenberg_simpler.mpl can be run b

Scientific paper

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3pi^2) for small sqrt(N)|z1-z2|. 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.

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