Journal of Statistical Physics

, Volume 148, Issue 2, pp 250–279

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

• John Baber
Article

## Abstract

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.

## Keywords

Several complex variables Random sections

## Notes

### Acknowledgements

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.

## References

1. 1.
Baber, J.: Scaled correlations of critical points of random sections on Riemann surfaces (2012). url:http://arxiv.org/abs/1106.4737
2. 2.
Bleher, P., Shiffman, B., Zelditch, S.: Poincaré-Lelong approach to universality and scaling of correlations between zeros. Commun. Math. Phys. 208(3), 771–785 (2000). doi:
3. 3.
Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000). doi:
4. 4.
Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of zeros on symplectic manifolds. In: Random Matrix Models and Their Applications. Math. Sci. Res. Inst. Publ., vol. 40, pp. 31–69. Cambridge University Press, Cambridge (2001) Google Scholar
5. 5.
Bloom, T.: Random polynomials and Green functions. Int. Math. Res. Not. 28, 1689–1708 (2005). doi:
6. 6.
Bloom, T., Shiffman, B.: Zeros of random polynomials on ℂm. Math. Res. Lett. 14(3), 469–479 (2007)
7. 7.
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. I. Commun. Math. Phys. 252(1–3), 325–358 (2004). doi:
8. 8.
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. II. Asymptotics and extremal metrics. J. Differ. Geom. 72(3), 381–427 (2006). url:http://projecteuclid.org/getRecord?id=euclid.jdg/1143593745
9. 9.
Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. III. String/M models. Commun. Math. Phys. 265(3), 617–671 (2006). doi:
10. 10.
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 original
11. 11.
Hammersley, J.M.: The zeros of a random polynomial. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, pp. 89–111. University of California Press, Berkeley and Los Angeles (1956) Google Scholar
12. 12.
Hannay, J.H.: Chaotic analytic zero points: exact statistics for those of a random spin state. J. Phys. A 29(5), L101–L105 (1996). doi:
13. 13.
Kac, M.: On the average number of real roots of a random algebraic equation. II. Proc. Lond. Math. Soc. (2) 50, 390–408 (1949)
14. 14.
Rice, S.O.: The distribution of the maxima of a random curve. Am. J. Math. 61(2), 409–416 (1939). doi:
15. 15.
Rice, S.O.: Mathematical analysis of random noise. Bell Syst. Tech. J. 23, 282–332 (1944)
16. 16.
Shiffman, B., Zelditch, S.: Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18(4), 1422–1475 (2008). doi:
17. 17.
Sodin, M.: Zeroes of Gaussian analytic functions. In: European Congress of Mathematics, pp. 445–458. Eur. Math. Soc., Zürich (2005) Google Scholar
18. 18.
Sodin, M., Tsirelson, B.: Random complex zeroes. I. Asymptotic normality. Isr. J. Math. 144, 125–149 (2004). doi:
19. 19.
Sodin, M., Tsirelson, B.: Random complex zeroes. III. Decay of the hole probability. Isr. J. Math. 147, 371–379 (2005). doi:
20. 20.
Sodin, M., Tsirelson, B.: Random complex zeroes. II. Perturbed lattice. Isr. J. Math. 152, 105–124 (2006). doi: