Abstract
We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit-translation-invariant correlations. Then we calculate the correlations between the straightened zeros of theO(1) random polynomial.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BS] A. T. Bharucha-Reid and M. Sambandham,Random polynomials, Academic Press, New York, 1986.
[BP] A. Bloch and G. Polya, On the roots of certain algebraic equations,Proc. London Math. Soc. (3)33:102–114 (1932).
[BBL1] E. Bogomolny, O. Bohigas, and P. Leboeuf, Distribution of roots of random polynomials,Phys. Rev. Lett.,68:2726–2729 (1992).
[BBL2] E. Bogomolny, O. Bohigas, and P. Lebouef, Quantum chaotic dynamics and random polynomials. Preprint, 1996.
[Dys] F. J. Dyson, Correlation between eigenvalues of a random matrix,Commun. Math. Phys. 19:235–250 (1970).
[EK] A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?Bull. Amer. Math. Soc. 32:1–37 (1995).
[EO] P. Erdős and A. Offord, On the number of real roots of random algebraic equation,Proc. London Math. Soc. 6:139–160 (1956).
[ET] P. Erdős and P. Turán, On the distribution of roots of polynomials.Ann. Math. 51:105–119 (1950).
[Han] J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state,J. Phys. A: Math. Gen. 29:101–105 (1996).
[IM] I. A. Ibragimov and N. B. Maslova, On the average number of real zeros of random polynomials. I, II,Teor. Veroyatn. Primen. 16:229–248, 495–503 (1971).
M. Kac, On the average number of real roots of a random algebraic equation,Bull. Amer. Math. Soc. 49:314–320 (1943).
[K2] M. Kac, On the average number of real roots of a random algebraic equation. II,Proc. London Math. Soc. 50:390–408 (1948).
[K3] M. Kac,Probability and related topics in physical sciences. Wiley (Interscience), New York, 1959.
[Kos] E. Kostlan, On the distribution of roots of random polynomials,From Topology to Computation: Proceedings of the Smalefest, M. W. Hirsh, J. E. Marsden, and M. Shub, eds. (Springer-Verlag, New York, 1993), pp. 419–431.
[Leb1] P. Leboeul, Phase space approach to quantum dynamics,J. Phys. A: Math. Gen. 24:4575 (1991).
[Leb2] P. Leboeuf, Statistical theory of chaotic wavefunctions: a model in terms of random analytic functions. Preprint, 1996.
[LS] P. Leboeuf and P. Shukla, Universal fluctuations of zeros of chaotic wavefunctions. Preprint, 1996.
[LO] J. Littlewood and A. Offord, On the number of real roots of random algebraic equation. I, II, III,J. London Math. Soc. 13:288–295 (1938);Proc. Cambr. Phil. Soc. 35:133–148 (1939);Math. Sborn. 12:277–286 (1943).
[LS] B. F. Logan and L. A. Shepp, Real zeros of a random polynomial. I, II,Proc. London Math. Soc. 18:29–35, 308–314 (1968).
[Mai] E. Majorana,Nuovo Cimento 9:43–50 (1932).
[Mas1] N. B. Maslova, On the variance of the number of real roots of random polynomials,Teor. Veroyatn. Primen. 19:36–51 (1974).
[Mas2] N. B. Maslova, On the distribution of the number of real roots of random polynomials,Teor. Veroyatn. Primen. 19:488–500 (1974).
[Meh] M. L. Mehta,Random matrices (Academic Press, New York, 1991).
[M-A] G. Mezincescu, D. Bessis, J. Fournier, G. Mantica, and F. D. Aaron, Distribution of roots of random polynomials. Preprint, 1996.
[SS] M. Shub and S. Smale, Complexity of Bezout's Theorem II: Volumes and Probabilities,Computational Algebraic Geometry (F. Eysette and A. Calligo, eds.), Progr. Math., vol. 109, Birkhauser, Boston, 1993, pp. 267–285.
[Ste] D. Stevens, Expected number of real zeros of polynomials,Notices Amer. Math. Soc. 13:79 (1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bleher, P., Di, X. Correlations between zeros of a random polynomial. J Stat Phys 88, 269–305 (1997). https://doi.org/10.1007/BF02508472
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02508472