Journal of Theoretical Probability

, Volume 32, Issue 2, pp 907–933 | Cite as

Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps

  • Yosuke Kawamoto
  • Hirofumi OsadaEmail author


The distributions of N-particle systems of Gaussian unitary ensembles converge to Sine\( _2\) point processes under bulk scaling limits. These scalings are parameterized by a macro-position \( \theta \) in the support of the semicircle distribution. The limits are always Sine\( _{2}\) point processes and independent of the macro-position \( \theta \) up to the dilations of determinantal kernels. We prove a dynamical counterpart of this fact. We prove that the solution to the N-particle system given by a stochastic differential equation (SDE) converges to the solution of the infinite-dimensional Dyson model. We prove that the limit infinite-dimensional SDE (ISDE), referred to as Dyson’s model, is independent of the macro-position \( \theta \), whereas the N-particle SDEs depend on \( \theta \) and are different from the ISDE in the limit whenever \( \theta \not = 0 \).


Gaussian unitary ensembles Dyson’s model Bulk scaling limit Interacting Brownian motion Infinite-dimensional stochastic differential equation 

Mathematics Subject Classification (2010)

60J60 60J70 15A52 60F17 60J65 



H.O. thanks Professor H. Spohn for a useful comment at RIMS in Kyoto University in 2002.


  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    Bourgade, P., Erdös, L., Yau, H.-T.: Universality of general \(\beta \)-ensembles. Duke Math. J. 163, 1127–1190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn. North-Holland, Amsterdam (1989)zbMATHGoogle Scholar
  4. 4.
    Inukai, K.: Collision or non-collision problem for interacting Brownian particles. Proc. Jpn. Acad. Ser. A 82, 66–70 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Katori, M., Tanemura, H.: Non-equilibrium dynamics of Dyson’s model with an infinite number of particles. Commun. Math. Phys. 136, 1177–1204 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Katori, M., Tanemura, H.: Markov property of determinantal processes with extended sine. Airy Bessel Kernels Markov Process. Relat. Fields 17, 541–580 (2011)zbMATHGoogle Scholar
  7. 7.
    Kawamoto, Y.: Density preservation of unlabeled diffusion in systems with infinitely many particles. In: Stochastic Analysis on Large Scale Interacting Systems, vol. B59, pp. 337–350. RIMS Kyoto (2016)Google Scholar
  8. 8.
    Kawamoto, Y., Osada, H.: Finite-particle approximations for interacting Brownian particles with logarithmic potentials. to appear in J. Math. Soc. Jpn. arXiv:1607.06922v2 [math.PR]
  9. 9.
    Landon, B., Sosoe, P., Yau, H.-T.: Fixed energy universality of Dyson Brownian motion. arXiv:1609.09011
  10. 10.
    Mehta, M.L.: Random Matrices, 3rd edn. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  11. 11.
    Osada, H.: Non-collision and collision properties of Dyson’s model in infinite dimensions and other stochastic dynamics whose equilibrium states are determinantal random point fields. In: Funaki T., Osada H. (eds.) Stochastic Analysis on Large Scale Interacting Systems. Advanced Studies in Pure Mathematics, vol. 39, pp. 325–343 (2004)Google Scholar
  12. 12.
    Osada, H.: Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Relat. Fields 153, 471–509 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41, 1–49 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: airy random point field. Stoch. Process. Appl. 123, 813–838 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Osada, H., Osada, S.: Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality. J. Stat. Phys. 170(2), 421–435 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Osada, H., Tanemura, H.: Strong Markov property of determinantal processes with extended kernels. Stoch. Process. Appl. 126(1), 186–208 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Osada, H., Tanemura, H.: Infinite-dimensional stochastic differential equations and tail \( \sigma \)-fields. arXiv:1412.8674v5 [math.PR]
  18. 18.
    Osada, H., Tanemura, H.: Infinite-dimensional stochastic differential equations related to Airy random point fields. arXiv:1408.0632
  19. 19.
    Plancherel, M., Rotach, W.: Sur les valeurs asymptotiques des polynomes dH́ermite \(H_{n}(x)=(-1)^{n} e^{x^{2}/2}\frac{d^{n }}{dx^{n }}(e^{-x^{2}/2})\). Comment. Math. Helv. 1, 227–254 (1929)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point process. J. Funct. Anal. 205, 414–463 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Spohn, H.: Interacting Brownian particles: a study of Dyson’s model. In: Papanicolaou, G. (ed.) Hydrodynamic Behavior and Interacting Particle Systems, IMA Volumes in Mathematics and its Applications, vol. 9, pp. 151–179. Springer, Berlin (1987)Google Scholar
  23. 23.
    Tsai, Li-Cheng: Infinite dimensional stochastic differential equations for Dyson’s model. Probab. Theory Relat. Fields. 166(3–4), 801–850 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan

Personalised recommendations