Abstract
The average density of zeros for monic generalized polynomials,\(P_n (z) = \phi (z) + \sum\nolimits_{k = 1}^n {c_k f_k } (z)\), with real holomorphic ϕ,f k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form\(P_n (z) = z^n + \sum\nolimits_{k = 1}^n {c_k z^{n - k} }\) with real independent, identically distributed Gaussian coefficients having zero mean and dispersion\(\delta = 1/\sqrt {n\lambda }\). The average density tends to a simple,universal function of ξ=2nlog|z| and λ in the domain ξcoth(ξ/2)≪n|sin arg(z)|, where nearly all the roots are located for largen.
Similar content being viewed by others
References
A. Bloch and G. Polya,Proc. Lond. Math. Soc. 33:102 (1932).
J. Littlewood and A. Offord,J. Lond. Math. Soc. 13:288 (1938).
J. Littlewood, and A. Offord,Proc. Cambr. Phil. Soc. 35:133 (1939).
M. Kac,Probability and Related Topics in Physical Sciences (Wiley-Interscience, New York, 1959).
A. Bharucha-Reid and M. Sambadham,Random Polynomials (Academic Press, New York, 1986).
A. Edelman and E. Kostlan,Bull. Am. Math. Soc. 32:1 (1995).
M. Girschick,Ann. Math. Stat. 13:235 (1942).
J. Hammersley, InProceedings of 3rd Berkeley Symposium on Mathematical Statistics and Probability,2:89 (1956).
E. Bogomolny, O. Bohigas, and P. LebœufPhys. Rev. Lett. 68:2726 (1992).
L. A. Shepp and R. J. Vanderbei,Trans. Am. Math. Soc. 347:4365 (1995).
D. Shparo and M. Shur,Vestn. Mosk. Univ. Ser. I Mat. Mekh. 1962:40.
I. Ibragimov and O. Zeitouni, On roots of random polynomials, preprint Technion.
G. Szegö,Orthogonal Polynomials (American Mathematical Society, Providence, Rhode Island, 1939).
N. Wiener,Extrapolation, Interpolation and Smoothing of Stationary Time Series with an Appendix by N. Levinson (Technology Press of MIT, Cambridge, Massachusetts, and Wiley, New York, 1949).
L. Pakula,IEEE Trans. Information Theory IT 33:569 (1987).
W. Jones, O. Njastad, and E. Saff,J. Comput. Appl. Math. 32:387 (1990).
W. Jones and O. Njastad,Rocky Mt. J. Math. 21:387 (1991).
J.-D. Fournier, G. Mantica, G. A. Mezincescu, and D. Bessis,Europhys. Lett. 22: 325 (1993).
J.-D. Fourier, G. Mantica, G. A. Mezincescu, and D. Bessis, pp. 257–271, InChaos and Diffusion in Hamiltonian Systems, Proceedings of the Chamonix Winter School, D. Benest and C. Froeschlé, eds. Editions Frontières, Gif-sur-Yvette, 1996.
S. O. Rice,Bell Syst. Tech. J. 23:282 (1944); 242;45 (1945).
L. Arnold,Random Power Series (Michigan State University, E. Lansing, Michigan, 1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mezincescu, G.A., Bessis, D., Fournier, JD. et al. Distribution of roots of random real generalized polynomials. J Stat Phys 86, 675–705 (1997). https://doi.org/10.1007/BF02199115
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02199115