## Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials.(English)Zbl 1271.60105

Summary: 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 thermodynamic 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 thermodynamic limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J60 Diffusion processes 82C22 Interacting particle systems in time-dependent statistical mechanics 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
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### 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. · Zbl 0931.58019 [2] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34 . Princeton Univ. Press, Princeton, NJ. · Zbl 1217.82003 [3] Fritz, J. (1987). Gradient dynamics of infinite point systems. Ann. Probab. 15 478-514. · Zbl 0623.60119 [4] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19 . de Gruyter, Berlin. · Zbl 0838.31001 [5] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277-329. · Zbl 1031.60084 [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. · Zbl 1074.82020 [7] Katori, M. and Tanemura, H. (2007). Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals. Probab. Theory Related Fields 138 113-156. · Zbl 1116.60042 [8] Katori, M. and Tanemura, H. (2007). Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. 129 1233-1277. · Zbl 1136.82035 [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. · Zbl 1259.82065 [10] Lang, R. (1977). Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. I. Existenz. Z. Wahrsch. Verw. Gebiete 38 55-72. · Zbl 0349.60103 [11] Lang, R. (1977). Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. II. Die reversiblen Masse sind kanonische Gibbs-Masse. Z. Wahrsch. Verw. Gebiete 39 277-299. · Zbl 0342.60067 [12] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of ( Non-Symmetric ) Dirichlet Forms . Springer, Berlin. · Zbl 0826.31001 [13] Mehta, M. L. (2004). Random Matrices , 3rd ed. Pure and Applied Mathematics ( Amsterdam ) 142 . Elsevier/Academic Press, Amsterdam. · Zbl 1107.15019 [14] Osada, H. (1996). Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Comm. Math. Phys. 176 117-131. · Zbl 0837.60073 [15] Osada, H. (1998). Interacting Brownian motions with measurable potentials. Proc. Japan Acad. Ser. A Math. Sci. 74 10-12. · Zbl 0909.60066 [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. · Zbl 1061.60109 [17] Osada, H. (2012). Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Related Fields 153 471-509. · Zbl 1253.82061 [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. · Zbl 1143.62056 [19] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071-1106. · Zbl 1025.82010 [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. · Zbl 0633.60001 [21] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127-159. · Zbl 0198.31101 [22] Shiga, T. (1979). A remark on infinite-dimensional Wiener processes with interactions. Z. Wahrsch. Verw. Gebiete 47 299-304. · Zbl 0407.60098 [23] Shirai, T. (2006). Large deviations for the fermion point process associated with the exponential kernel. J. Stat. Phys. 123 615-629. · Zbl 1110.60051 [24] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107-160. · Zbl 0991.60038 [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. · Zbl 0674.60096 [26] Tanemura, H. (1996). A system of infinitely many mutually reflecting Brownian balls in $$\mathbb{R}^{d}$$. Probab. Theory Related Fields 104 399-426. · Zbl 0849.60087 [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. · Zbl 0888.60075 [28] Yoo, H. J. (2005). Dirichlet forms and diffusion processes for fermion random point fields. J. Funct. Anal. 219 143-160. · Zbl 1104.60065 [29] Yoshida, M. W. (1996). Construction of infinite-dimensional interacting diffusion processes through Dirichlet forms. Probab. Theory Related Fields 106 265-297. · Zbl 0859.60068
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