Dyson’s model in infinite dimensions is a system of Brownian particles that interact via a logarithmic potential with an inverse temperature of β = 2. The stochastic process can be represented by the solution to an infinite-dimensional stochastic differential equation. The associated unlabeled dynamics (diffusion process) are given by the Dirichlet form with the sine2 point process as a reference measure. In a previous study, we proved that Dyson’s model in infinite dimensions is irreducible, but left the ergodicity of the unlabeled dynamics as an open problem. In this paper, we prove that the unlabeled dynamics of Dyson’s model in infinite dimensions are ergodic.
REFERENCES
1.
F. J.
Dyson
, “A Brownian-motion model for the eigenvalues of a random matrix
,” J. Math. Phys.
3
, 1191
–1198
(1962
).2.
3.
H.
Spohn
, “Interacting Brownian particles: A study of Dyson’s model
,” in Hydrodynamic Behavior and Interacting Particle Systems
, IMA Volumes in Mathematics and its Applications Vol. 9, edited by G.
Papanicolaou
(Springer-Verlag
, Berlin
, 1987
), pp. 151
–179
.4.
H.
Spohn
, “Tracer dynamics in Dyson’s model of interacting Brownian particles
,” J. Stat. Phys.
47
, 669
–679
(1987
).5.
H.
Osada
, “Infinite-dimensional stochastic differential equations related to random matrices
,” Probab. Theory Relat. Fields
153
, 471
–509
(2012
).6.
H.
Osada
, “Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions
,” Commun. Math. Phys.
176
, 117
–131
(1996
).7.
H.
Osada
, “Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
,” Ann. Probab.
41
, 1
–49
(2013
).8.
B.
Valkó
and B.
Virág
, “Continuum limits of random matrices and the Brownian carousel
,” Invent. Math.
177
, 463
–508
(2009
).9.
H.
Osada
, “Tagged particle processes and their non-explosion criteria
,” J. Math. Soc. Jpn.
62
(3
), 867
–894
(2010
).10.
H.
Osada
, “Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field
,” Stochastic Process. Appl.
123
, 813
–838
(2013
).11.
N.
Ikeda
and S.
Watanabe
, Stochastic Differential Equations and Diffusion Processes
, 2nd ed.
(North-Holland
, 1989
).12.
H.
Osada
and H.
Tanemura
, “Infinite-dimensional stochastic differential equations and tail σ-fields
,” Probab. Theory Relat. Fields
177
, 1137
–1242
(2020
).13.
L.-C.
Tsai
, “Infinite dimensional stochastic differential equations for Dyson’s model
,” Probab. Theory Relat. Fields
166
, 801
–850
(2016
).14.
Y.
Kawamoto
, H.
Osada
, and H.
Tanemura
, “Uniqueness of Dirichlet forms related to infinite systems of interacting Brownian motions
,” Potential Anal.
55
, 639
–676
(2021
).15.
Y.
Kawamoto
, H.
Osada
, and H.
Tanemura
, “Infinite-dimensional stochastic differential equations and tail σ-fields II: The IFC condition
,” J. Math. Soc. Jpn.
74
, 79
–128
(2022
).16.
Y.
Kawamoto
and H.
Osada
, “Finite particle approximations of interacting Brownian particles with logarithmic potentials
,” J. Math. Soc. Jpn.
70
(3
), 921
–952
(2018
).17.
Y.
Kawamoto
and H.
Osada
, “Dynamical bulk scaling limit of Gaussian unitary ensembles and stochastic differential equation gaps
,” J. Theor. Probab.
32
, 907
–933
(2019
).18.
Y.
Kawamoto
and H.
Osada
, “Dynamical universality for random matrices
,” Partial Differ. Equations Appl.
3
, 27
(2022
).19.
H.
Osada
and R.
Tsuboi
, “Dyson’s model in infinite dimensions is irreducible
,” in Dirichlet Frms and Related Topics, (Springer
, Singapore
, 2022
), Vol. 394
, pp. 401
–419
.20.
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
, Advanced Studies in Pure Mathematics Vol. 39, edited by T.
Funaki
and H.
Osada
(Math. Soc. Japan
, Tokyo
, 2004
), pp. 325
–343
.21.
R.
Lang
, “Unendlich-dimensionale Wienerprozesse mit wechselwirkung I
,” Z. Wahrscheinlichkeitstheorie Verw. Geb.
38
, 55
–72
(1977
).22.
R.
Lang
, “Unendlich-dimensionale Wienerprocesse mit wechselwirkung II
,” Z. Wahrscheinlichkeitstheorie Verw. Geb.
39
, 277
–299
(1977
).23.
J.
Fritz
, “Gradient dynamics of infinite point systems
,” Ann. Probab.
15
, 478
–514
(1987
).24.
H.
Tanemura
, “A system of infinitely many mutually reflecting Brownian balls in
,” Probab. Theory Relat. Fields
104
, 399
–426
(1996
).25.
M.
Fradon
, S.
Roelly
, and H.
Tanemura
, “An infinite system of Brownian balls with infinite range interaction
,” Stochastic Process. Appl.
90
(1
), 43
–66
(2000
).26.
R.
Honda
and H.
Osada
, “Infinite-dimensional stochastic differential equations related to Bessel random point fields
,” Stochastic Process. Appl.
125
(10
), 3801
–3822
(2015
).27.
Y.
Kawamoto
, “Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices
,” Mod. Stochastics: Theory Appl.
9
, 89
–122
(2022
).28.
H.
Osada
and H.
Tanemura
, “Infinite-dimensional stochastic differential equations arising from Airy random point fields
,” arXiv:1408.0632 [math.PR].29.
S.
Albeverio
, Y. G.
Kondratiev
, and M.
Röckner
, “Analysis and geometry on configuration spaces: The Gibbsian case
,” J. Funct. Anal.
157
(1
), 242
–291
(1998
).30.
D.
Ruelle
, “Superstable interactions in classical statistical mechanics
,” Commun. Math. Phys.
18
, 127
–159
(1970
).31.
I.
Corwin
and X.
Sun
, “Ergodicity of the Airy line ensemble
,” Electron. Commun. Probab.
19
(49
), 1
–11
(2014
).32.
M.
Fukushima
, Y.
Oshima
, and M.
Takeda
, Dirichlet Forms and Symmetric Markov Processes
, 2nd ed.
(Walter de Gruyter
, 2011
).33.
A.
Soshnikov
, “Determinantal random point fields
,” Russ. Math. Surv.
55
, 923
–975
(2000
).34.
S.
Ghosh
, “Determinantal processes and completeness of random exponentials: The critical case
,” Probab. Theory Relat. Fields
163
(3–4
), 643
–665
(2015
).35.
A. I.
Bufetov
, “Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel
,” Bull. Math. Sci.
6
, 163
–172
(2016
).36.
A. I.
Bufetov
, Y.
Qiu
, and A.
Shamov
, “Kernels of conditional determinantal measures and the proof of the Lyons–Peres conjecture
,” J. Eur. Math. Soc.
23
, 1477
–1519
(2021
).37.
R.
Lyons
, “A note on tail triviality for determinantal point processes
,” Electron. Commun. Probab.
23
, 1
–3
(2018
).38.
H.
Osada
and S.
Osada
, “Discrete approximations of determinantal point processes on continuous spaces: Tree representations and tail triviality
,” J. Stat. Phys.
170
, 421
(2018
).39.
L. R.
Bellet
, “Ergodic properties of Markov processes
,” in Open Quantum Systems II
, Lecture Notes in Mathematics Vol. 1881 (Springer
, Berlin
, 2006
), pp. 1
–39
.© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.