NON-STATIONARY VERSION OF FURSTENBERG THEOREM ON RANDOM MATRIX PRODUCTS ANTON GORODETSKI AND VICTOR KLEPTSYN

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NON-STATIONARY VERSION OF FURSTENBERG THEOREM

ON RANDOM MATRIX PRODUCTS

ANTON GORODETSKI AND VICTOR KLEPTSYN

Abstract. We prove a non-stationary analog of the Furstenberg Theorem

on random matrix products (that can be considered as a matrix version of

the law of large numbers). Namely, we prove that under a suitable genericity

conditions the sequence of norms of random products of independent but not

necessarily identically distributed SL(d, R) matrices grow exponentially fast,

and there exists a non-random sequence that almost surely describes asymp-

totical behaviour of the norms of random products.

1. Introduction

The asymptotic behavior of sums of i.i.d. random variables is very well studied in

the classical probability theory. Analogous questions on random products of matrix-

valued i.i.d. random variables were initially formulated in the simplest case of 2 ×2

matrices with positive entries by Bellman [Bel]. Later these questions attracted

lots of attention due to the results by Furstenberg-Kesten [FurK] who showed that

exponential rate of growth of the norms of the random products (nowadays called

Lyapunov exponent) is well deﬁned almost surely, and Furstenberg [Fur1, Fur2],

where it was shown that under some non-degeneracy conditions Lyapunov expo-

nent must be positive. Since then enormous amount of literature on the subject

appeared, e.g. see [Ber, Fur3, FurKif, GM, GR, KS, Kif1, KifS, L, R, SVW, Vi]. Ap-

plications of random matrix products appear in a natural way in smooth dynamical

systems [V1, W1, W2], spectral theory and mathematical physics [CKM, D15, S],

geometric measure theory [HS, PT, Sh], and other ﬁelds. Far reaching general-

izations in terms of random walks on groups were developed, see [BQ], [Fu], and

references therein. Nonlinear one-dimensional analogues of Furstenberg Theorem

were also obtained in [A, DKN, KN, M]. Another series of generalizations (in

terms of positivity of Lyapunov exponents for a generic linear cocycle) was derived

in the dynamical systems community, e.g. see [ASV], [BGV], [Bo], [BoV1], [BoV2],

[BoV3], [V2], and the monograph [V1].

The most famous result is the following Furstenberg Theorem, that we recall

here in its classical form:

Theorem (Furstenberg [Fur1, Theorem 8.6]).Let {Xk, k ≥1}be independent

and identically distributed random variables, taking values in SL(d, R), the d×d

matrices with determinant one, let GXbe the smallest closed subgroup of SL(d, R)

Date: April 21, 2023.

A. G. was supported in part by Simons Fellowship and NSF grant DMS–1855541.

V.K. was supported in part by ANR Gromeov (ANR-19-CE40-0007) and by Centre Henri

Lebesgue (ANR-11-LABX-0020-01) .

arXiv:2210.03805v2 [math.DS] 20 Apr 2023

2 A. GORODETSKI AND V. KLEPTSYN

containing the support of the distribution of X1, and assume that

E[log kX1k]<∞.

Also, assume that GXis not compact, and there exists no GX-invariant ﬁnite union

of proper subspaces of Rd. Then there exists a positive constant λFsuch that with

probability one

lim

n→∞

nlog kXn. . . X2X1k=λF>0.

In this paper we generalize Furstenberg Theorem to the case when the random

variables {Xk, k ≥1}do not have to be identically distributed. Let us say a couple

of words about a motivation for our results before providing the formal statements.

First of all, the study of asymptotic behavior of sums Sn=xn+. . . +x1of

independent real valued random variables, in particular, the Law of Large Num-

bers, is a foundational statement in classical probability theory. As we mentioned

above, many of the limit theorems were proven for that non-commutative case. In

particular, Furstenberg Theorem can be considered as a non-commutative version

of the Law of Large Numbers. But in all of the obtained results on random matrix

products the matrices were assumed to be identically distributed, while most (if not

all) of the results on sums Sn=xn+. . . +x1of independent real valued random

variables do not actually require x1, . . . , xnto be identically distributed. It is a

very natural task to close this gap.

As an important application, we mention the non-stationary version of so called

Anderson Model. Classical Anderson Model on a one-dimensional lattice is given

by a discrete Schr¨odinger operator on l2(Z) with random potential:

(Hu)(n) = u(n−1) + u(n+ 1) + V(n)u(n),

where {V(n)}is an iid sequence of random variables. It is known that under suitable

assumptions this operator almost surely has pure point spectrum, with exponen-

tially decreasing eigenfunctions (this property is usually referred to as Anderson

Localization). Other properties of this operators (such as dynamical localization,

properties of the integrated density of states etc.) were also studied in details,

see [AW, D15, DKKKR] for some recent surveys. But from the physical point of

view it is very natural to consider the case when the potential {V(n)}is given

by independent but not identically distributed random variables. Indeed, suppose

we try to study the transport properties of an electron in the random media with

some ﬁxed non-random background. In this case it is natural to consider potential

V(n) = Vrandom(n) + Vbackground(n), where {Vbackground(n)}is a ﬁxed bounded se-

quence, and {Vrandom(n)}is a sequence of iid random variables. Since the spectral

properties of the 1D discrete Schr¨odinger operator are closely related to the prop-

erties of the products of the corresponding transfer matrices, this setting leads to

the question about properties of a random matrix products in the non-stationary

case. In particular, the results of this paper allow to prove spectral and dynamical

localization in non-stationary Anderson Model, as we plan to show in our next

paper [GK].

Up to now there were literally no results on random matrix products in non-

stationary case, since there were no suitable techniques available. Indeed, in most

cases the proofs in the stationary case are based on existence of a stationary mea-

sure, which restricts all the existing techniques either to the case of identically

distributed matrices (or, at least, with the distributions given by some stationary

NON-STATIONARY FURSTENBERG THEOREM 3

process, as in [Kif2]), or to the context of Oseledets Theorem [O] (see also [R1]),

with some exceptions that are usually focused on speciﬁc models, with the proofs

heavily based on the special features of the model. Our ﬁrst main result, Theo-

rem 1.1, describes the growth of the norms of random matrix products in the case of

independent but not identically distributed matrices. As an intermediate step, we

provide a general result, Theorem 1.14, that we called Atom Dissolving Theorem,

that can be used in non-stationary context as a tool to replace the statements on

regularity of stationary measures, but is also of independent interest, see Section

1.3.

Let us now provide the formal statements of our main results.

1.1. Non-stationary version of Furstenberg theorem. Let Kbe a compact

set of probability measures on SL(d, R); as a particular case, one can consider

K={µi}i=1,...,k being a ﬁnite set. For any A∈SL(d, R) we will denote by

fA:RPd−1→RPd−1the induced projective transformation.

For a given sequence (µi)i∈N,µi∈K, we let Ai∈SL(d, R) be chosen randomly

with respect to distribution µi, set

Tn=AnAn−1. . . A1,

and denote

(1) Ln=Elog kTnk,

where the expectation is taken over the distribution µ1×µ2×. . . ×µn. This

expectation exists once the log-moment of the norm Elog kAkis ﬁnite for all µ∈K,

and we will be always imposing (at least) this assumption.

Our ﬁrst main result is the following theorem:

Theorem 1.1. Assume that the following hold:

•(ﬁnite moment condition) There exists γ > 0,Csuch that

(2) ∀µ∈KZSL(d,R)

kAkγdµ(A)< C

•(measures condition) For any µ∈Kthere are no Borel probability

measures ν1,ν2on RPd−1such that (fA)∗ν1=ν2for µ-almost every

A∈SL(d, R)

•(spaces condition) For any µ∈Kthere are no two ﬁnite unions U,

U0of proper subspaces of Rdsuch that A(U) = U0for µ-almost every

A∈SL(d, R).

Then the sequence Ln=Elog kTnkgrows at least linearly, i.e. then there exists

λ > 0such that for any nand any µ1, . . . , µn∈Kwe have

Ln≥nλ,

and it predicts the growth of the norm of the random products in the following sense:

almost surely, we have

lim

n→∞

n(log kTnk − Ln) = 0.

The particular case of 2 ×2 matrices, and the existence of an exponentially

contracted random vector in that case, are important in the setting of 1D Ander-

son Localization (see [D15, His]). For this case, Theorem 1.1 has the following

addendum:

4 A. GORODETSKI AND V. KLEPTSYN

Proposition 1.2 (Contracted direction).Let d= 2, and assume that the ﬁnite

moment and measures conditions of Theorem 1.1 hold. Then almost surely there

exists a unit vector ¯v∈R2such that |Tn¯v| → 0as n→ ∞. Moreover,

lim

n→∞

n(log |Tn¯v|+Ln)=0

We will prove Theorem 1.1 (and thus Proposition 1.2) by actually establishing

a stronger conclusion, the Large Deviations Estimates Theorem:

Theorem 1.3 (Large Deviations for Nonstationary Products).Under the assump-

tions of Theorem 1.1, for any ε > 0there exists δ > 0such that for all suﬃciently

large n∈Nwe have

P{|log kTnk − Ln|> εn}< e−δn,

where P=µ1×µ2×. . . ×µn. Moreover, the same estimate holds for the lengths of

random images of any given initial unit vector v0:

∀v0∈Rd,|v0|= 1 P{|log kTnv0k − Ln|> εn}< e−δn.

Remark 1.4. The lower bound on nin Theorem 1.3 depends on ε > 0 and the

compact set of distributions K, but can be chosen uniformly over all the speciﬁc

choices of the sequences of measures µ1, µ2, . . . from Kand, in the second part, the

unit vector v0.

1.2. Exponential growth of the norms. The sequence {Ln}in Theorem 1.1

must grow at least linearly. This statement by itself holds under weaker assumption

than those in Theorem 1.1, and so we formulate the statement on growth of the

random matrix products separately:

Theorem 1.5. Assume that the following hold:

•(log-moment condition) For any µ∈Kone has

ZSL(d,R)

log kAkdµ(A)<∞

•(measures condition) For any µ∈Kthere are no Borel probability

measures ν1,ν2on RPd−1such that (fA)∗ν1=ν2for µ-almost every

A∈SL(d, R)

Then there exists λ > 0such that for any nand any µ1, . . . , µn∈Kwe have

Ln≥nλ.

In particular, for any ﬁxed sequence (µi)i∈N∈KNwe have

lim inf

n→∞

nLn≥λ > 0.

Questions regarding exponential growth of nonstationary random matrix prod-

ucts were discussed and popularized by I. Goldsheid for a long time. In his recent

paper [G], it was shown that under the same “measure condition” as in Theorem 1.5,

there exists a positive λsuch that almost surely

lim inf

n→∞

nlog kTnk ≥ λ > 0.

The proof was obtained by completely diﬀerent methods.

NON-STATIONARY FURSTENBERG THEOREM 5

Our methods allow to consider Theorem 1.5 as a particular case of a more

general result. Namely, consider any closed manifold Mand the set of its C1-

diﬀeomorphisms Diﬀ1(M). Assume that Mis equipped with a Riemannian metric,

so that one can consider the Lebesgue measure LebMon Mand the corresponding

Jacobian

Jac(f)|x=|det df|x|=dLebM

df∗LebMx

We will measure the maximum volume contraction rate of a diﬀeomorphism by the

following quantity:

N(f) := max

x∈MJac(f)|−1

x= max

x∈M

df∗LebM

dLebMx

Let KMbe a compact subset of the space of probability measures on Diﬀ1(M)

(equipped with the weak-∗convergence topology). We then have the following

theorem, providing a lower estimate for the (averaged) growth of the maximum

volume contraction speed:

Theorem 1.6. Let M,KMsatisfy the following assumptions:

•(log-moment condition) For any µ∈KMone has

ZDiﬀ1(M)

log N(f)dµ(f)<∞

•(measures condition) For any µ∈KMthere are no Borel probability

measures ν1,ν2on Msuch that f∗ν1=ν2for µ-almost every f∈Diﬀ1(M).

Then there exists h>0such that for any nand any µ1, . . . , µn∈KMwe have

Elog N(Fn)≥nh,

where Fn=fn◦ · · · ◦ f1, and every fiis chosen independently with respect to the

corresponding measure µi, so that the expectation is taken over the distribution

µ1×µ2×. . . ×µn.

In particular, for any given sequence (µi)i∈N,µi∈KMof measures on Diﬀ1(M)

we have

lim inf

n→∞

nElog N(Fn)≥h>0,

where Fn=fn◦ · · · ◦ f1, and the expectation is taken with respect to the inﬁnite

product measure Qiµi.

Remark 1.7. Theorem 1.6 could be considered as a generalization of the famous

Baxendale Theorem [Bax], claiming (in the stationary case) the existence of an

ergodic measure with a negative volume Lyapunov exponent in the case of absence

of a common invariant measure.

Remark 1.8. The conclusions of Theorems 1.5 and 1.1 in the case of products of

i.i.d. random matrices correspond to the classical Furstenberg Theorem. Propo-

sition 1.2 can be considered as a non-stationary analog of Proposition II.3.3 and

Corollary IV.1.7 from [BL], or of results from [R1].

Remark 1.9. One can replace the assumptions of Theorem 1.5 by a more general

one. Namely, instead of the measures condition, it is enough to assume that there

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NON-STATIONARYVERSIONOFFURSTENBERGTHEOREMONRANDOMMATRIXPRODUCTSANTONGORODETSKIANDVICTORKLEPTSYNAbstract.Weproveanon-stationaryanalogoftheFurstenbergTheoremonrandommatrixproducts(thatcanbeconsideredasamatrixversionofthelawoflargenumbers).Namely,weprovethatunderasuitablegenericityconditionsthesequence...

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