A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions, it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure, showing that with high probability, the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.
REFERENCES
1.
Y.
Benoist
and J.-F.
Quint
, Random Walks on Reductive Groups
(Springer
, Cham
, 2016
).2.
P.
Bougerol
and J.
Lacroix
, Products of Random Matrices with Applications to Schrödinger Operators
(Birkhäuser
, Boston
, 1985
).3.
F.
Dorsch
and H.
Schulz-Baldes
, “Random perturbations of hyperbolic dynamics
,” Electron. J. Probab.
24
(89
), 1
–23
(2019
).4.
Y.-C.
Wong
, “Differential geometry of Grassmann manifolds
,” Proc. Natl. Acad. Sci. U. S. A.
57
, 589
–594
(1967
).5.
R.
Carmona
and J. M.
Lacroix
, Spectral Theory of Random Schrödinger Operators
(Birkhäuser
, Basel
, 1990
).6.
R.
Römer
and H.
Schulz-Baldes
, “The random phase property and the Lyapunov spectrum for disordered multi-channel systems
,” J. Stat. Phys.
140
, 122
–153
(2010
).7.
H.
Schulz-Baldes
, “Perturbation theory for Lyapunov exponents of an Anderson model on a strip
,” Geom. Funct. Anal.
14
, 1089
–1117
(2004
).8.
J.
Bellissard
, “Random matrix theory and the Anderson model
,” J. Stat. Phys.
116
, 739
–754
(2004
).9.
M.
Aizenman
and S.
Warzel
, Random Operators
(AMS
, Providence
, 2015
).10.
C.
Sadel
and B.
Virág
, “A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
,” Commun. Math. Phys.
343
, 881
–919
(2016
).11.
S.
Bachmann
and W.
De Roeck
, “From the Anderson model on a strip to the DMPK equation and random matrix theory
,” J. Stat. Phys.
139
, 541
–564
(2010
).12.
B.
Valkó
and B.
Virág
, “Random Schrödinger operators on long boxes, noise explosion and the GOE
,” Trans. Am. Math. Soc.
366
, 3709
–3728
(2014
).13.
C.
Sadel
and H.
Schulz-Baldes
, “Random Lie group actions on compact manifolds: A perturbative analysis
,” Ann. Probab.
38
, 2224
–2257
(2010
).14.
Y.
Kato
, Perturbation Theory for Linear Operators
(Springer International
, New York
, 2013
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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