Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 111–148 | Cite as

Determinantal Processes with Number Variance Saturation

  • Kurt JohanssonEmail author


Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N→∞. This limting determinantal process has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann ζ-function, [21, 2]. The process can also be constructed using non-intersecting paths and we consider several variants of this construction. One construction leads to a model which shows a transition from a non-universal behaviour with number variance saturation to a universal sine-kernel behaviour as we go up the line.


Neural Network Statistical Physic Complex System Brownian Motion Nonlinear Dynamics 
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  1. 1.
    Aizenman, M., Goldstein, S., Lebowitz, J.L.: Bounded fluctuations and translation symmetry breaking in one-dimensional particle systems. J. Stat. Phys. 103, 601–618 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Berry, M.V.: Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity 1, 399–407 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Berry, M.V., Keating, J.P.: The Riemann Zeros and Eigenvalue Asymptotics. SIAM Review 41, 236–266 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Boas, R.P.: Entire Functions. New York: Academic Press, 1954Google Scholar
  5. 5.
    Bohigas, O., Giannoni, M.J., Schmit, C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Breiman, L.: Probability. Reading, MA: Addison-Wesley, 1968Google Scholar
  8. 8.
    Coram, M., Diaconis, P.: New test of the correspondence between unitary eigenvalues and the zeros of Riemann’s zeta function. J. Phys. A:Math. Gen. 36, 2883–2906 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Dyson, F.J.: A Brownian-motion Model for the Eigenvalues of a Random Matrix. J. Math. Phys. 3, 1191–1198 (1962)zbMATHGoogle Scholar
  10. 10.
    Dyson, F.J., Mehta, M.L.: Statistical theory of energy levels of complex systems IV. J. Math. Phys. 4, 701–712 (1963)zbMATHGoogle Scholar
  11. 11.
    Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles and random matrices. Ann. Inst. H. Poincaré 35, 177–204 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Guhr, T.: Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss. Ann. Phys. 250, 145–192 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Hejhal, D.: On the Triple Correlation of the Zeros of the Zeta Function. IMRN 1994, pp. 293–302Google Scholar
  14. 14.
    Johansson, K.: Universality of the Local Spacing Distribution in Certain Ensembles of Hermitian Wigner Matrices. Commun. Math. Phys. 215, 683–705 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Karlin, S., McGregor, G.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959)zbMATHGoogle Scholar
  16. 16.
    Katz, N.M., Sarnak, P.: Zeroes of Zeta Functions ans Symmetry. Bull. AMS 36, 1–26 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Keating, J.P., Snaith, N.C.: Random matrix theory and ζ(1/2+it). Commun. Math. Phys. 214, 57–89 (2000)Google Scholar
  18. 18.
    Lawden, D.: Elliptic functions and applications. Appl. Math. Sci. 80, New York: Springer, 1989Google Scholar
  19. 19.
    Lenard, A.: States of Classical Statistical Mechanical systems of Infinitely Many Particles II. Characterization of Correlation Measures. Arch. Rat. Mech. Anal. 59, 241–256 (1975)Google Scholar
  20. 20.
    Montgomery, H.: The Pair Correlation of Zeros of the Zeta Function. Proc. Sym. Pure Math. 24, Providence, RI: AMS, 1973, pp. 181–193Google Scholar
  21. 21.
    Odlyzko, A.M.: The 1020:th Zero of the Riemann Zeta Function and 70 Million of its Neighbors. Preprint, A.T.T., 1989Google Scholar
  22. 22.
    Rudnick, Z., Sarnak, P.: Zeros of Principal L-functions and random matrix theory. A celebration of John F. Nash. Duke Math. J. 81, 269–322 (1996)zbMATHGoogle Scholar
  23. 23.
    Selberg, A.: Contributions to the theory of the Riemann zeta-function. Arch. Math. OG. Naturv. B 48, 89–155 (1946)zbMATHGoogle Scholar
  24. 24.
    Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Tracy, C.A., Widom, H.: Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices. J. Stat. Phys. 92, 809–835 (1998)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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