Advertisement

Journal of Statistical Physics

, Volume 123, Issue 6, pp 1219–1230 | Cite as

Crystallization of Random Trigonometric Polynomials

  • David W. Farmer
  • Mark Yerrington
Article

Abstract

We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.

Random polynomial crystallization pair correlation of zeros 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. T. Bharucha-Reid and M. Sambandham, Random Polynomials (Academic Press, 1986).Google Scholar
  2. 2.
    E. Bogomolny, O. Bohigas and P. Lebœuf, Quantum chaotic dynamics and random polynomials, J. Statist. Phys. 85(5-6):639679 (1996).CrossRefGoogle Scholar
  3. 3.
    J. E. A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. 16:53 (1966).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Edelman and E. Kostlan, How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. 32:1 (1995).Google Scholar
  5. 5.
    D. W. Farmer and S. Gonek, The pair correlation of zeros of the derivative of the Riemann ξ-function, in preparation.Google Scholar
  6. 6.
    D. W. Farmer and R. C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357(9):3789 (2005).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Kac, On the average number of real roots of a random algebraic equation, Bull. Am. Math. Soc. 49:314 (1943).MATHCrossRefGoogle Scholar
  8. 8.
    S. O. Rice, The distribution of maxima of a random curve, Amer. Jour. Math 61:409 (1939).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.American Institute of MathematicsPalo AltoUSA

Personalised recommendations