Journal of the Mathematical Society of Japan

Finite-particle approximations for interacting Brownian particles with logarithmic potentials

Yosuke KAWAMOTO and Hirofumi OSADA

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Abstract

We prove the convergence of $N$-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to $\mathrm{Airy}_\beta$ interacting Brownian motion with $\beta=1, 2, 4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

Note

The first author is supported by Grant-in-Aid for JSPS JSPS Research Fellowships (No. 15J03091). The second author is supported in part by a Grant-in-Aid for Scenic Research (KIBAN-A, No. 24244010; KIBAN-A, No. 16H02149; KIBAN-S, No. 16H06338) from the Japan Society for the Promotion of Science.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 921-952.

Dates
Received: 23 July 2016
Revised: 29 November 2016
First available in Project Euclid: 18 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1529309020

Digital Object Identifier
doi:10.2969/jmsj/75717571

Mathematical Reviews number (MathSciNet)
MR3830792

Zentralblatt MATH identifier
06966967

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
random matrix theory infinite-dimensional stochastic differential equations interacting Brownian motions Airy point processes the Ginibre point process logarithmic potential finite-particle approximations

Citation

KAWAMOTO, Yosuke; OSADA, Hirofumi. Finite-particle approximations for interacting Brownian particles with logarithmic potentials. J. Math. Soc. Japan 70 (2018), no. 3, 921--952. doi:10.2969/jmsj/75717571. https://projecteuclid.org/euclid.jmsj/1529309020


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References

  • G. W. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cambridge university press, 2010.
  • Peter J. Forrester, Log-gases and Random Matrices, London Mathematical Society Monographs, Princeton University Press, 2010.
  • J. Fritz, Gradient Dynamics of Infinite Point Systems, Ann. Probab., 15 (1987), 478–514.
  • M. Fukushima, et al., Dirichlet forms and symmetric Markov processes, 2nd ed., Walter de Gruyter, 2011.
  • R. Honda and H. Osada, Infinite-dimensional stochastic differential equations related to Bessel random point fields, Stochastic Process. Appl., 125 (2015), 3801–3822.
  • K. Inukai, Collision or non-collision problem for interacting Brownian particles, Proc. Japan Acad. Ser. A, 82 (2006), 66–70.
  • K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields, 123 (2002), 225–280.
  • K. Johansson, Discrete polynuclear growth and determinantal processes, Commun. Math. Phys., 242 (2003), 277–329.
  • M. Katori and H. Tanemura, Noncolliding Brownian motion and determinantal processes, J. Stat. Phys., 129 (2007), 1233–1277.
  • M. Katori and H. Tanemura, Markov property of determinantal processes with extended sine, Airy, and Bessel kernels, Markov processes and related fields, 17 (2011), 541–580.
  • M. Katori and H. Tanemura, Noncolliding square Bessel processes, J. Stat. Phys., 142 (2011), 592–615.
  • R. Lang, Unendlich-dimensionale Wienerprocesse mit Wechselwirkung I, Z. Wahrschverw. Gebiete, 38 (1977), 55–72.
  • R. Lang, Unendlich-dimensionale Wienerprocesse mit Wechselwirkung II, Z. Wahrschverw. Gebiete, 39 (1978), 277–299.
  • Z.-M. Ma and M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, 1992.
  • M. L. Mehta, Random Matrices, 3rd edition, Amsterdam, Elsevier, 2004.
  • H. Osada, Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions, Commun. Math. Phys., 176 (1996), 117–131.
  • H. Osada, 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 Stochastic Analysis on Large Scale Interacting Systems, (eds. T. Funaki and H. Osada), Adv. Stud. Pure Math., 39 (2004), 325–343.
  • H. Osada, Tagged particle processes and their non-explosion criteria, J. Math. Soc. Japan, 62 (2010), 867–894.
  • H. Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probability Theory and Related Fields, 153 (2012), 471–509.
  • H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Ann. of Probab., 41 (2013), 1–49.
  • H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field, Stochastic Processes and their applications, 123 (2013), 813–838.
  • H. Osada and S. Osada, Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality, J. Stat. Phys., 170 (2018), 421–435.
  • H. Osada and H. Tanemura, Cores of Dirichlet forms related to Random Matrix Theory, Proc. Jpn. Acad., Ser. A, 90 (2014), 145–150.
  • H. Osada and H. Tanemura, Strong Markov property of determinantal processes with extended kernels, Stochastic Processes and their Applications, 126 (2016), 186–208.
  • H. Osada and H. Tanemura, Infinite-dimensional stochastic differential equations and tail $\sigma$-fields, arXiv:1412.8674.
  • H. Osada and H. Tanemura, Infinite-dimensional stochastic differential equations related to Airy random point fields, arXiv:1408.0632.
  • J. A. Ramírez, B. Rider and B. Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc., 24 (2011), 919–944.
  • D. Ruelle, Superstable interactions in classical statistical mechanics, Commun. Math. Phys., 18 (1970), 127–159.
  • A. Soshnikov, Determinantal random point fields, Russian Math. Surveys, 55 (2000), 923–975.
  • H. Tanemura, A system of infinitely many mutually reflecting Brownian balls in $\mathbb{R}^d$, Probab. Theory Relat. Fields, 104 (1996), 399–426.
  • L. C. Tsai, Infinite dimensional stochastic differential equations for Dyson's model, Probab. Theory Relat. Fields, 166 (2016), 801–850.