## The Annals of Probability

### Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

#### Abstract

We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in $\mathbb{R}^{d}$ and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures.

We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in $\mathbb{R}$, while the latter is in $\mathbb{R}^{2}$. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions $d=1,2$, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.

#### Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 1-49.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.aop/1358951980

Digital Object Identifier
doi:10.1214/11-AOP736

Mathematical Reviews number (MathSciNet)
MR3059192

Zentralblatt MATH identifier
1271.60105

#### Citation

Osada, Hirofumi. Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41 (2013), no. 1, 1--49. doi:10.1214/11-AOP736. https://projecteuclid.org/euclid.aop/1358951980

#### References

• [1] Albeverio, S., Kondratiev, Y. G. and Röckner, M. (1998). Analysis and geometry on configuration spaces: The Gibbsian case. J. Funct. Anal. 157 242–291.
• [2] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ.
• [3] Fritz, J. (1987). Gradient dynamics of infinite point systems. Ann. Probab. 15 478–514.
• [4] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin.
• [5] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
• [6] Katori, M., Nagao, T. and Tanemura, H. (2004). Infinite systems of non-colliding Brownian particles. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 283–306. Math. Soc. Japan, Tokyo.
• [7] Katori, M. and Tanemura, H. (2007). Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals. Probab. Theory Related Fields 138 113–156.
• [8] Katori, M. and Tanemura, H. (2007). Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. 129 1233–1277.
• [9] Katori, M. and Tanemura, H. (2011). Markov property of determinantal processes with extended sine, Airy, and Bessel kernels. Markov Process. Related Fields 17 541–580.
• [10] Lang, R. (1977). Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. I. Existenz. Z. Wahrsch. Verw. Gebiete 38 55–72.
• [11] Lang, R. (1977). Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. II. Die reversiblen Masse sind kanonische Gibbs-Masse. Z. Wahrsch. Verw. Gebiete 39 277–299.
• [12] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin.
• [13] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam.
• [14] Osada, H. (1996). Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Comm. Math. Phys. 176 117–131.
• [15] Osada, H. (1998). Interacting Brownian motions with measurable potentials. Proc. Japan Acad. Ser. A Math. Sci. 74 10–12.
• [16] Osada, H. (2004). Non-collision and collision properties of Dyson’s model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 325–343. Math. Soc. Japan, Tokyo.
• [17] Osada, H. (2012). Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Related Fields 153 471–509.
• [18] Osada, H. and Shirai, T. (2008). Variance of the linear statistics of the Ginibre random point field. In Proceedings of RIMS Workshop on Stochastic Analysis and Applications 193–200. Res. Inst. Math. Sci. (RIMS), Kyoto.
• [19] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071–1106.
• [20] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
• [21] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
• [22] Shiga, T. (1979). A remark on infinite-dimensional Wiener processes with interactions. Z. Wahrsch. Verw. Gebiete 47 299–304.
• [23] Shirai, T. (2006). Large deviations for the fermion point process associated with the exponential kernel. J. Stat. Phys. 123 615–629.
• [24] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160.
• [25] Spohn, H. (1987). Interacting Brownian particles: A study of Dyson’s model. In Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986). IMA Vol. Math. Appl. 9 151–179. Springer, New York.
• [26] Tanemura, H. (1996). A system of infinitely many mutually reflecting Brownian balls in $\mathbb{R}^{d}$. Probab. Theory Related Fields 104 399–426.
• [27] Tanemura, H. (1997). Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in $\mathbb{R}^{d}$. Probab. Theory Related Fields 109 275–299.
• [28] Yoo, H. J. (2005). Dirichlet forms and diffusion processes for fermion random point fields. J. Funct. Anal. 219 143–160.
• [29] Yoshida, M. W. (1996). Construction of infinite-dimensional interacting diffusion processes through Dirichlet forms. Probab. Theory Related Fields 106 265–297.