Abstract
We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper (Macdonald in J. Stat. Phys. 136(5):807, 2009), the author used the Poincaré-Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac-Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.
Similar content being viewed by others
References
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)
Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000)
Bogomolny, E., Bohigas, O., Leboeuf, P.: Distribution of roots of random polynomials. Phys. Rev. Lett. 68(18), 2726–2729 (1992)
Bogomolny, E., Bohigas, O., Leboeuf, P.: Quantum chaotic dynamics and random polynomials. J. Stat. Phys. 85(5), 639–679 (1996)
Douglas, M., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua I. Commun. Math. Phys. 252(1), 325–358 (2004)
Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Am. Math. Soc. 32(1), 1–37 (1995)
Hannay, J.: Chaotic analytic zero points: exact statistics for those of a random spin state. J. Phys. A, Math. Gen. 29(5), L101–L105 (1996)
Ibragimov, I., Zeitouni, O.: On roots of random polynomials. Trans. Am. Math. Soc. 349(6), 2427–2441 (1997)
Kac, M.: On the average number of real roots of a random algebraic equation (II). Proc. Lond. Math. Soc. 2(6), 401 (1948)
Macdonald, B.: Density of complex zeros of a system of real random polynomials. J. Stat. Phys. 136(5), 807 (2009)
Prosen, T.: Exact statistics of complex zeros for Gaussian random polynomials with real coefficients. J. Phys. A, Math. Gen. 29(15), 4417–4423 (1996)
Rice, S.: Mathematical Analysis of Random Noise. Selected Papers on Noise and Stochastic Processes, pp. 133–294 (1954)
Shepp, L., Vanderbei, R.: The complex zeros of random polynomials. Trans. Am. Math. Soc. 347(11), 4365–4384 (1995)
Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Commun. Math. Phys. 200(3), 661–683 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Macdonald, B. Density of Complex Critical Points of a Real Random SO(m+1) Polynomial. J Stat Phys 141, 517–531 (2010). https://doi.org/10.1007/s10955-010-0057-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-0057-y