The Annals of Probability

Gaussian Limit for Determinantal Random Point Fields

Alexander Soshnikov

Full-text: Open access

Abstract

We prove that, under fairly general conditions, a properly rescaled determinantal random point field converges to a generalized Gaussian random process.

Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 171-187.

Dates
First available in Project Euclid: 29 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1020107764

Digital Object Identifier
doi:10.1214/aop/1020107764

Mathematical Reviews number (MathSciNet)
MR1894104

Zentralblatt MATH identifier
1033.60063

Subjects
Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems

Keywords
Determinantal random point fields central limit theorem self-similar random processes

Citation

Soshnikov, Alexander. Gaussian Limit for Determinantal Random Point Fields. Ann. Probab. 30 (2002), no. 1, 171--187. doi:10.1214/aop/1020107764. https://projecteuclid.org/euclid.aop/1020107764


Export citation

References

  • [1] BASOR, E. (1997). Distribution functions for random variables for ensembles of positive Hermitian matrices. Comm. Math. Phys. 188 327-350.
  • [2] BASOR, E. and WIDOM, H. (1999). Determinants of Airy operators and applications to random matrices. J. Statist. Phys. 96 1-20.
  • [3] COSTIN, O. and LEBOWITZ, J. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69-72.
  • [4] DALEY, D. J. and VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • [5] DEIFT, P. (1999). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Amer. Math Soc., Alexandria, VA.
  • [6] DIACONIS, P. and EVANS, S. N. (2000). Immanants and finite point processes. J. Combin. Theory Ser. A 91 305-321.
  • [7] DIACONIS, P. and EVANS, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615-2633.
  • [8] DIACONIS, P. and SHAHSHAHANI, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A (Special vol.) 49-62.
  • [9] DOBRUSHIN, R. L. (1978). Automodel generalized random fields and their renorm-group. In Multicomponent Random Systems (R. L. Dobrushin and Ya. G. Sinai, eds.). Dekker, New York.
  • [10] DOBRUSHIN, R. L. (1979). Gaussian and their subordinated self-similar random generalized fields. Ann. Probab. 7 1-28.
  • [11] GELFAND, I. M. and VILENKIN, N. Ya. (1964). Generalized Functions IV: Some Applications of Harmonic Anaysis. Academic Press, New York.
  • [12] HIDA, T. (1980). Brownian Motion. Springer, New York.
  • [13] JOHANSSON, K. (1998). On fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151-204.
  • [14] JOHANSSON, K. On random matrices from the compact classical groups. Ann. Math. 145 519- 545
  • [15] KARAMATA, J. (1930). Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4 38-53.
  • [16] KARAMATA, J. (1933). Sur un mode de croissance régulière. Théorèmes fondamentaux. Bull. Soc. Math. France 61 55-62.
  • [17] LENARD, A. (1973). Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys. 30 35-44.
  • [18] LENARD, A. (1975). States of classical statistical mechanical system of infinitely many particles I. Arch. Rational Mech. Anal. 59 219-239.
  • [19] LENARD, A. (1975). States of classical statistical mechanical systems of infinitely many particles II. Arch. Rational Mech. Anal. 59 240-256.
  • [20] LUKACS, E. (1970). Characteristic Functions, 2nd. ed. Griffin, London.
  • [21] MACCHI, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7 82-122.
  • [22] MACCHI, O. (1977). The fermion process-a model of stochastic point process with repulsive points. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians A 391-398. Reidel, Dordrecht.
  • [23] REED, M. and SIMON, B. (1975-1980). Methods of Modern Mathematical Physics 1-4. Academic Press, New York.
  • [24] SENETA, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. Springer, New York.
  • [25] SINAI, Ya. (1976). Automodel probability distributions. Theory Probab. Appl. 21 273-320.
  • [26] SOSHNIKOV, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923- 975. Available via xxx.lanl.gov/abs/math/0002099.
  • [27] SOSHNIKOV, A. (2000). Central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353-1370.
  • [28] SOSHNIKOV, A. (2000). Gaussian fluctuations in Airy, Bessel, sine and other determinantal random point fields. J. Statist. Phys. 100 491-522.
  • [29] SPOHN, H. (1987). Interacting Brownian particles: A study of Dyson's model. In Hydrodynamic Behavior and Interacting Particle Systems (G. Papanicolau, ed.). Springer, New York.
  • [30] SHIRAI, T. and TAKAHASHI, Y. (2000). Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties. Preprint.
  • [31] WIEAND, K. (1998). Eigenvalue distribution of random matrices in the permutation group and compact Lie groups. Ph. D. dissertation, Dept. Mathematics, Harvard Univ.
  • DAVIS, CALIFORNIA 95616 E-MAIL: soshniko@math.ucdavis.edu