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Random Lie Group Actions on Compact Manifolds

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Random Lie Group Actions on Compact Manifolds The Annals of Probability 2010, Vol. 38, No. 6, 2224–2257 DOI: 10.1214/10-AOP544 c Institute of Mathematical Statistics, 2010 RANDOM LIE GROUP ACTIONS ON COMPACT MANIFOLDS: A PERTURBATIVE ANALYSIS1 By Christian Sadel and Hermann Schulz-Baldes Universit¨at Erlangen–N¨urnberg A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but suf- ficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg’s irreducibility hypothesis in related prob- lems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applica- tions to the theory of products of random matrices and a model of a disordered quantum wire are presented. 1. Main results, discussion and applications. This work provides a per- turbative calculation of invariant measures for a class of Markov chains on continuous state spaces and shows that these perturbative measures are unique and smooth. Let us state the main result right away in detail, and then place it into context with other work towards the end of this section and explain our motivation to study this problem. Suppose given a Lie group GL(L, C), a compact, connected, smooth Riemannian manifold withoutG⊂ boundary and a smooth, transitive group action : .M Thus, is a homogeneous space. Furthermore, let · G×M→M M λ,σ be a family of group elements depending on a coupling constant λT 0∈ and G a parameter σ varying in some probability space (Σ, p), which is of≥ the following form: arXiv:0802.2909v3 [math-ph] 19 Nov 2010 ∞ n (1) λ,σ = exp λ n,σ , T R P ! nX=1 Received February 2008; revised August 2009. 1Supported by the DFG. AMS 2000 subject classifications. 60J05, 37H05, 37H15. Key words and phrases. Group action, invariant measure, Birkhoff sum. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2010, Vol. 38, No. 6, 2224–2257. This reprint differs from the original in pagination and typographic detail. 1 2 C. SADEL AND H. SCHULZ-BALDES where and n,σ are measurable maps on Σ with compact image in the LieR algebra ∈ G g ofP such that G 1/n (2) lim sup sup( n,σ ) < n σ Σ kP k ∞ →∞ ∈ for some norm on g. This implies that λ,σ is well defined and analytic in λ for λ sufficiently small. The expectationT value of the first-order term P1,σ will be denoted by = p(dσ) 1,σ . Let us consider theP product probabilityP space (Ω, P)=(ΣN, pN). Associ- R ated to ω = (σn)n N Ω, there is a sequence ( λ,σn )n N of group elements. ∈ ∈ T ∈ An -valued Markov process xn(λ, ω) with starting point x0 is defined iterativelyM by ∈ M (3) xn(λ, ω)= λ,σn xn 1(λ, ω). T · − The averaged Birkhoff sum of a complex function f on is M N 1 N 1 1 − 1 − (4) I (f)= E f(x (λ, ω)) = (T nf)(x ), λ,N ω N n N λ 0 n=0 n=0 X X where in the second expression we used the Markov transition operator (T f)(x)= E (f( x)). Here and below, expectation values w.r.t. P (or λ σ Tλ,σ · p) will be denoted by E (or Eω and Eσ). Next, recall that an invariant mea- sure νλ on is defined by the property νλ(dx)f(x)= νλ(dx)(Tλf)(x). The operatorM ergodic theorem [16], Theorem 19.2, then states that I (f) R R λ,N converges almost surely (in x0) w.r.t. any invariant measure νλ and for any integrable function f. In the case that is a projective space and the action is matrix multiplication, one is in the worldM of products of random matrices. If then the group generated by λ,σ, with σ varying in the support of p, is noncompact and strongly irreducible,T Furstenberg, Guivarch and Raugi have proved [2, 9, 11] that there is a unique invariant measure νλ which is, moreover, H¨older continuous [2]. To our best knowledge, little seems to be known in more general situations and also concerning the absolute conti- nuity of νλ (except if p is absolutely continuous [18], for some and under supplementary hypothesis [4, 28]). Let p1 be the distribution of the random variable 1,σ on the Lie algebra g, that is, for any measurable b g one has p (b)=Pp( b ). We are ⊂ 1 {P1,σ ∈ } interested in a perturbative calculation of Iλ,N (f) in λ for smooth functions f with rigorous control on the error terms. This can be achieved if the support of p1 is large enough in the following sense. First, let us focus on the special case = 1 and = 0. R P Theorem 1. Let λ,σ be of the form (1) and assume = 1, = E( 1,σ)= 0. Let x be the associatedT Markov process on as givenR byP (3) andP let n M RANDOM LIE GROUP ACTIONS 3 v = Lie(supp(p1)) be the smallest Lie subalgebra of g that contains the sup- port of p1. Recall that µ(dx) denotes the Riemannian volume measure on . MCoupling hypothesis: Suppose that the smallest subgroup of containing exp(λ ), v, λ [0, 1] acts transitively on . (This isV a LieG subgroup with{ LieP algebraP ∈ v,∈ but it} may not be a submanifold.)M Then there is a sequence of smooth functions ρm with dµρm = δm,0 M and ρ0 > 0 µ-almost surely, such that for any M N and any function f C ( ), one obtains ∈ R ∈ ∞ M M 1 (5) I (f)= λm µ(dx)ρ (x)f(x)+ , λM+1 . λ,N m O Nλ2 m=0 X ZM 1 M+1 Here, the expression ( Nλ2 , λ ) means that there are two error terms, O 1 M+1 one of which is bounded by C1 Nλ2 and the other by C2λ with C1,C2 depending on f and M. Especially, C2 may grow in M so that we cannot deduce uniqueness of the invariant measure for small λ this way (cf. Remark 1 below). When = 1 or = 0 further assumptions are needed in order to control the BirkhoffR 6 sums.P We 6 assume that and generate commuting compact 1 R P groups, that is, − = Ad ( )= and the closed Abelian Lie groups RPR R P P = k : k Z and = exp(λ ) : λ R) are compact. While is alwayshRi {R connected,∈ } canhPi possibly{ beP disconnected.∈ } However, therehPi exists K N such that hRiK is connected. By considering the suspended Markov ∈ n hR i process (y )n N with yn = xKn corresponding to the family ∈ = Tλ,σ1,...,σK Tλ,σK ···Tλ,σ1 K K for (σ1,...,σK) (Σ , p ), one can always assume that is connected and we shall do∈ so from now on. Note that the product hRi is also a compact, connected, Abelian subgroup of which will be denotedhRihPi by , . G hR Pi All these groups are tori in and their dimensions are L , L and L , . TLR TLGP TLR,P RTL PRL RZPL Hence, ∼= , ∼= and , ∼= , where = /(2π ) is the LhRi-dimensionalhPi torus. The (chosen)hR Pi isomorphisms shall be denoted by R ,R and R , , respectively, for example, R (θ) GL(L, C) for R P R P TLR R ∈ hRi ⊂ θ = (θ1,...,θLR ) . The isomorphism∈ R directly leads to the Fourier decomposition of the function θ TLR f(R (θ) x), notably ∈ 7→ R · ıj θ (6) f(R (θ) x)= fj(x)e · , R · j ZLR ∈X where LR dθ ıj θ fj(x)= e− · f(R (θ) x), j θ = jlθl. LR TLR (2π) R · · Z Xl=1 4 C. SADEL AND H. SCHULZ-BALDES LP LR,P Similarly, the maps θ T f(R (θ) x) and θ T f(R , (θ) x) ∈ 7→ P · ∈ 7→ R P · lead to Fourier series. Definition 1. A function f C ( ) is said to consist of only low ∈ ∞ M frequencies w.r.t. if the Fourier coefficients f C ( ) vanish for j hRi j ∈ ∞ M with norm j = LR j larger than some fixed integer J> 0. Similarly, f k k l=1 | l| is defined to consist of only low frequencies w.r.t. or , . P hPi hR Pi The following definitions are standard (see [17] for references). Definition 2. Let us define θˆ TLR by R (θˆ )= and θˆ RLP R ∈ R R R P ∈ by R (λθˆ ) = exp(λ ). Then is said to be a Diophantine rotation or simplyP DiophantineP ifP there is someR s> 1 and some constant C such that for any nonzero multi-index j ZLR 0 one has ∈ \ { } ıj θˆR s e · 1 C j − . | − |≥ k k Similar, is said to be Diophantine, or a Diophantine generator of a rota- P tion, if there is some s> 1 and some constant C, such that for any nonzero multi-index j ZLP 0 one has ∈ \ { } s j θˆ C j − . | · P |≥ k k As final preparation before stating the result, let us introduce the measure p on the Lie algebra g obtained from averaging the distribution p1 of the lowest-order terms 1,σ w.r.t. the Haar measure dR on the compact group , , namely for anyP measurable set b g, hR Pi ⊂ 1 p(b)= dR p( σ Σ : R 1,σR− b ).
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