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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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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