Introduction to Ergodic Theory

Transcription

1 Introduction to Ergodic Theory Marius Lemm May 20, 2010 Contents 1 Ergodic stochastic processes Canonical probability space Shift operators Basic results on ergodicity Birkhoff s ergodic theorem Ergodic operators Definitions Basic results for ergodic operators Non-randomness of the spectrum

2 This document is meant to supplement my Seminar talk on the notion of ergodicity, wherein the structure was mainly based on [2]. In particular, we are always concerned with applications to the discrete Anderson Model, in this case a random Hamiltonian H ω = H 0 + V ω on l 2 (Z d ), where H 0 is the discrete Laplacian and V ω is a multiplication operator, which at every i Z d multiplies with the corresponding member of the family of iid {v ω (i)} i Z d. 1 Ergodic stochastic processes In this section the notion of an ergodic stochastic process (e.s.p.) will be introduced and some basic properties of such processes are discussed. Finally, a complete proof of Birkhoff s ergodic theorem (Strong law of large numbers for e.s.p.) will be given. 1.1 Canonical probability space Let X : (Ω, F, P) (R, B(R)) be a random variable (r.v.). For our purposes, the detailed strucure of X is unimportant. Only the values of the random variable and their probabiliy, given by the image measure P X (A) := P({ω Ω X(ω) A}), influence H ω. We say P X is the distribution of X. From this point of view, we might as well consider the new r.v. ˆX := id (R,B(R),PX ), which obviously has distribution P X. This can be generalized to a stochastic process {X i } i I (for us: I = Z d ). If we consider the whole process as the identity on R I, then each X i becomes the projection on the i-th coordinate, namely: X i (ω) =ω(i) i I. (1) These {X i } are then called the canonical process on the underlying probability space. In our case this is (R Zd, B Zd (R), P), where P = P Zd X. For more on this, see [1]. 1.2 Shift operators Definition 1.1 (ergodic family). Let (Ω, F, P) be a probability space, T :Ω Ωa r.v. T is called measure-preserving iff P(T 1 A)=P(A), A F. Given a measure-preserving family of {T i } i I. Then an A F is called invariant iff T 1 i A = A, i I and the {T i } i I are called an ergodic family, (short: ergodic) iff all invariant A F are trivial, i.e. have probabiliy 0 or 1. Theorem 1.2. For iid {ω i } i Z d, the family of shift operators {T i } i Z d, defined by (T i ω)(j) := ω(j i) i, j Z d, is ergodic. Proof. Step 1: Show {T i } i Z d is a measure-preserving family. Let A F be a cylinder set, i.e. A = {ω R Zd ω(i 1 ) A 1,..., ω(i n ) A n } (2) for some n N,i 1,..., i n Z d,a 1,..., A n B(R). A corresponds to a spatially limited event in the Z d - lattice. We denote the set of all cylinder sets by F 0. Then for any 2

3 j Z d, we have P(T 1 j A) =P({ω R Zd ω(i 1 + j) A 1,..., ω(i n + j) A n }) =Π n l=1p(ω(i l + j) A l ) =Π n l=1p(ω(i l ) A l ) = P(A), (3) where we have used the fact that the v ω (i) =X i (ω) =ω(i) are iid, so for finitely many events P is just the product measure. We see that the set of all (now general) A F satisfying (3), denoted by F 1, form a σ-algebra: A = R Zd is an invariant event (under the {T i } i Z d), in particular (3) holds. Observe that T 1 j = T j. If we take A, B F 1 with B A and note that T i (A \ B) = T i A \ T i B (this is not true for general measure-preserving transformations), we get P(T j (A \ B)) = P(T j A) P(T j B)=P(A) P(B) =P(A \ B). So A \ B F 1. Take a pairwise disjoint family {A n } n N F 1. Then P (T i ( A n )) = P ( i A n )) = P (A n )=P( n N n N(T A n ), n=0 n N so that F is a Dynkin system. That F 1 is -closed can be seen from the explicit form of the {T i } i Z d. From (3) we know: F 0 F 1. Recall from measure theory, that B Zd (R) is constructed from B(R) by taking the σ-algebra generated by all cylinder sets, i.e. σ(f 0 )=B Zd (R). Therefore we can conclude B Zd (R) F 1. Step 2: We show P((T i A) B) P(A)P(B) ( i ), (4) for all A, B B Zd (R). Pick A, B F 0, and make i large enough s.t. the spatially limited sets T i A and B have none of their (finitely many) indices in common and are therefore independent. Then P becomes the product measure again, and (4) holds. We see that F 2, the set of all A, B that satisfy (4) is a σ-algebra. Firstly, P(T i R Zd B) =P(B) =P(R Zd )P(B) B F 2, showing R Zd F 2. Now take A, B, C F 2 s.t. B A, then P(T i (A \ B) C) =P((T i A \ T i B) C) = P(T i A C) P(T i B C) P(C)(P(A) P(B)), so A \ B F 2. As in Step 1, we can argue for countable, pairwise disjoint unions and finite intersections. So, F 2 is a σ-algebra containing F 0. As in Step 1, we then get that (4) holds for all A, B B Zd (R). Now, let M be an invariant set. Then by (4) P (M) =P ((T i M) M) i P (M) 2 P (M) {0, 1}. 3

4 Remark 1.3. The property shown in (3) for cylinder sets is called stationarity (or translational invariance) of the process {ω i } i Z d. In Step 1 we have thus shown that stationarity is equivalent to the fact that the shift operators are a measure preserving family (the backward direction is trivial). For more on stationarity, see [1] and [6]. Remark 1.4. (4) is called a mixing property of the stochastic process and it is stronger than ergodicity. In fact, ergodicity is equivalent to Cesaro-mixing, namely: 1 P((T (2L + 1) d i A) B) P(A)P(B) (L ), (5) i L where the average is taken over the d-dimensional centered cube (notation: Λ L (0)). The notion of mixing is treated in [4]. 1.3 Basic results on ergodicity Let I denote the σ-algebra of all invariant events. The following presentation is based on Ch.20 in [4], up to and including Theorem 1.9. Theorem 1.5. Let T be a measure-preserving transformation on (Ω, F, P). Then (i) A measurable f : (Ω, F) (R, B(R)) is I-measurable iff f T = f. (ii) T is ergodic iff all I-measurable f are P - a.s. constant. Proof. (i) We first show the claim for f = A. To complete the proof it suffices to approximate a general measurable function by simple functions and notice that both properties directly carry on to the limit. We have A(ω) = A (T (ω)) = T 1 A(w), which is equivalent to T 1 A = A. And by checking the possible f-preimages of sets B in the generator ɛ = {(a, ) a R} of B(R), we see that f 1 (B) {, A, Ω} These form a subset of I iff T 1 A = A. Thus the claim holds for f = A. (ii) Let T be ergodic and f I-measurable. For fixed a R, f 1 ((a, )) I P(f 1 ((a, ))) {0, 1}. Therefore f = inf{a >0 P(f 1 ((a, ))) = 0} P a.s. Take A I. Then A is I-measurable and therefore P-a.s. either 0 or 1, i.e. P(A) {0, 1}. Corollary 1.6 (Our case). Given an ergodic family of {T i } i Z d and a r.v. Y which is invariant under {T i } i Z d (meaning: Y T i = Y, i Z d ). Then Y is P-a.s. constant. The following Lemma 1.8 is merely preparatory for the Proof of Birkhoff s Theorem in the next section. The convention used is that we take only one measure-preserving transformation τ and a measurable f :Ω R. We then define the e.s.p. by X n (ω) := f τ n (ω), n N. Then (X n ) n N is a stationary stochastic process. We also fix some notation: n 1 S n := X i M n := max{0,s 1,..., S n } n N (6) i=0 Since we will prove a more general version of Birkhoff s Theorem than the one only for e.s.p., we remind the reader of the following 4

5 Definition 1.7 (conditional expectation value). Let X L 1 ((Ω, F, P)) and A F a sub-σ-algebra. Then a r.v. Y statisfying E[X A ]=E[Y A ] A A. (7) is called conditional expectation of X given A and we denote E[X A] := Y. Lemma 1.8 (Hopf s maximal ergodic lemma). Let X 0 L 1 (P). Then E[X 0 {Mn>0}] 0 n N Proof. Let k n. By definition, M n (τ(ω)) S k (τ(ω)) X 0 +M n τ X 0 +S k τ = S k+1. This gives X 0 S k+1 M n τ k {1,..., n} (8) Note that M n τ 0 always holds. Since S 1 = X 0, (8) holds also for k = 0, which yields X 0 max{s 1,..., S n } M n τ. Appliying this to the quantity of interest gives us E[X 0 {Mn>0}] E[(max{S 1,...S n } M n τ) {Mn>0}] = E[(M n M n τ) {Mn>0}], (9) which should be 0. Using again that M n τ 0 always holds, we see {M n > 0} c {M n =0} { M n τ 0} {M n M n τ 0}. Applied to (9), this yields (together with the fact that τ is measure-preserving) E[(M n M n τ) {Mn>0}] E[M n M n τ] =E[M n ] E[M n ]= Birkhoff s ergodic theorem Theorem 1.9 (Birkhoff s ergodic theorem, 1931). Let f = X 0 L 1 (P). Then 1 n 1 n X k E[X 0 I] n i=0 P a.s. Corollary 1.10 (Ergodic case). Let additionaly τ be ergodic, then 1 n 1 n X k E[X 0 ] n i=0 P a.s. Proof. Let A I. Because τ is ergodic, it follows P (A) {0, 1} A is P a.s.constant. Therefore E[X 0 A ]=E[X 0 ]P(A) =E[E[X 0 ] A], so by (7) E[X 0 I] =E[X 0 ]. Proof of 1.9. Clearly, E[X 0 I] is I-measurable. By Theorem 1.5 we get E[X 0 I] τ = E[X 0 I] P-a.s. So ˆX n := X n E[X 0 I] = ˆX n 1 τ and we can assume w.l.o.g. E[X 0 I] = 0 (10) Consider the r.v. Z := lim sup n 1 n S n 5

6 and for fixed ɛ> 0 the event F := {Z > ɛ}. We will show P(F ) = 0, which implies Z 0 a.s. By the same token (applied to X := X), we get that lim sup n ( 1 n S n)= lim inf n ( 1 n S n) 0 a.s. The upshot is that 1 n S n n 0 Since Z τ = Z, we know by Theorem 1.5 that Z is I-measurable, so F I. Set X ɛ n := (X n ɛ) F n 1 Sn ɛ := i=0 X ɛ i a.s. M ɛ n := max{0,s ɛ 1,..., S ɛ n} F n := {M ɛ n > 0}. Clearly, (F n ) n N is an increasing sequence and 1 1 (F n )={sup k Sɛ k > 0} = {sup k N k S k >ɛ} F = F, n N k N which implies F n F. By dominated convergence (recall: X 0 L 1 (P)), we get E[X0 ɛ E[X0], ɛ where the l.h.s. is nonnegative, due to Hopf s Lemma. Therefore F n ] n 0 E[X ɛ 0]=E[(X 0 ɛ) F ]=E[E[X 0 I] F ] ɛp(f )= ɛp(f ), where we have used the definition (7) in the second equality and (10) for the last step. We can thus conclude P(F ) = 0. 2 Ergodic operators In this section we first introduce the notion of an ergodic operator and then look at some properties for such operators. Finally, we will prove statements about the non-randomness of the spectrum for our random Hamiltonian. An overview can be obtained from Ch.4 of [2]. A much more thorough discussion of measurability is found in [1]. 2.1 Definitions Definition 2.1 (measurability of operators). (i) A family {A ω } ω Ω of bounded operators on H is called weakly measurable iff the map is measurable for all ψ, φ H. ω φ A ω ψ (ii) A family {A ω } ω Ω of self-adjoint (s.-a.) operators is called measurable iff for all bounded, measurable f : C C we have that f(a ω ) is weakly measurable. Remark 2.2. The above defined notion of measurability continues to hold for sums, products and (weak) limits of measurable functions in the usual way. 6

7 Remark 2.3. There exist nicer criteria to check for measurability of s.-a. operators, e.g. if z C \ R s.t. ω (A ω + z) 1 is weakly measurable, then A ω is measurable. Definition 2.4 (ergodic operators). Let (Ω, F, P) be a probability space with measurepreserving transformations {T i } i I. A family {H ω } ω Ω of s.-a. operators on a separable Hilbert space H is called ergodic iff {U i } i I unitary operators on H s.t. H Ti ω = U i H ω U i (11) Note 2.5. For the following, recall that in our case the Hamiltonian is given by H ω = H 0 + V ω on H = l 2 (Z d ), where H 0 is the discrete Laplacian and V ω is a multiplication operator, which at every i Z d multiplies with the corresponding member of the family of iid {v ω (i)} i Z d. Proposition 2.6. The family {H ω } ω Ω of Hamiltonians is ergodic. Proof. From Theorem 1.2 we know that the {T i } i Z d of shift operators (on Z d ) is ergodic, in particular it is measure-preserving. Define unitary shift operators (U i φ)(m) := φ(m i) φ l 2 (Z d ). From (U i V ω U i φ)(m) =v ω (m i)(u i φ)(m i) =v ω (m i)φ(m) =v Ti ω(m)φ(m) we see that V Ti ω = U i V ω U i, i.e. V ω is an ergodic operator. Since H 0 is non-random, this implies that H ω is ergodic. 2.2 Basic results for ergodic operators Since most proofs in this section could be completed in the Seminar talk, we give only the ideas of proof before finally proving Theorem Proposition 2.7. Let {H ω } be an ergodic family of s.-a. operators on a separable Hilbert space H. Let f be a measurable, bounded function. Then f(h ω ) is ergodic, i.e. f(h Ti ω)=u i f(h ω )U i (12) Idea of proof: Notice that (12) holds for monomials. By linearity, (12) then extends to Polynomials. Now one can re-do the construction of the continuous and then the Borel functional calculus, to see that (12) transports to the limit. Proposition 2.8. Let {P ω } ω Ω be an ergodic family of orthogonal projections. Then dimr(p ω ) is P-a.s. constant. Idea of proof: This is an application of Theorem 1.5. It is easy to show that dimr(p ω ) is invariant under the {T } Z d. To see that ω dimr(p ω ) is measurable, note that dimr(p ω )=trp ω, which is clearly measurable. 7

8 2.3 Non-randomness of the spectrum Theorem 2.9 (Pastur). Let {H ω } be an ergodic family of s.-a. operators on a sep. Hilbert space H. Then Σ R s.t. σ(h ω )=Σ P a.s. (13) Proof. We use the criterion: λ σ(h ω ) dimr(p ω (λ ɛ, λ + ɛ)) 0 ɛ > 0 dimr(p ω (a, b)) 0 a, b Q s.t. a < λ < b (14) Now we use Proposition 2.7 on the spectral resolution of H ω denoted by P ω to get: a, b Q c N s.t. P({dimR(P ω (a, b)) = c} }{{} =:Ω a,b ) = 1 (15) Now we define Ω := a,b Q Ω a,b and note that P (Ω) = 1 P ( a,b, Q Ωc a,b ) 1 a,b Q P (Ωc a,b ) = 1, so Ω has full measure. Fix ω 1,ω 2 Ω. Due to (15), dimr(p ω1 (a, b)) = dimr(p ω2 (a, b)) = c, a, b Q. Because of the equivalence in (14), this yields: σ(h ω1 )= σ(h ω2 ). Remark Similarly, there exist Σ c, Σ pp, Σ ac R s.t. Σ κ = σ κ (H ω ) P a.s. (κ {c, pp, ac}) The proof is similar to the one of Theorem 2.9, the only complication here is that the measurability of e.g. P ω P c (ω) (where P c (ω) the projection onto the continuous subspace, which also depends on ω) is non-trivial. Remark Note that σ pp (H ω ) denotes the closure of ɛ(h ω ), the set of all eigenvalues of H ω. In fact, with some additional assumption, one can show that λ R : dimr(p ω ({λ})) {0, }P a.s., which demonstrates nicely that the individual eigenvalues fluctuate very fast. Their closure however is non-random. Theorem 2.12 (Kunz-Souillard). Let {H ω } and Σ be as in Theorem 2.9. Denote the single-site distribution of the iid {v ω (i)} i Z d by P. Then Σ = [0, 4d]+supp(P), where the sum is to be understood as a summation of sets. Note Recall that σ(h 0 ) = [0, 4d] in our convention and σ(v ω )=supp(p), since V ω is a multiplication operator. For a more general version of this Theorem, see [5]. Proof. We use Weyl s criterion. Let E = λ + ν [0, 4d]+supp(P) and ɛ> 0. Step 1: We show that Ω ɛ,e ψ l 2 (Z d ):P(Ω ɛ,e )=1 ψ =1 ω Ω ɛ,e : (H ω E)ψ < ɛ. 8

9 Since ν supp(p) we have that P(ν ɛ,ν + ɛ ) > 0. We define the event 2 2 A n = A n (L, x n, v, ɛ) ={ω : v ω (y) ν < ɛ 2, y Λ L(x n )}, (16) where L N,x n Z d are fixed and Λ L (x n ) := {i Z d : i x n L} denotes the cube of side length L and Volume (2L + 1) d. Since the{v ω (i)} i Z d are iid, the probability measure factorises and P(A n )= (P(ν ɛ 2,ν + ɛ ) (2L+1) d 2 ) > 0. (17) For fixed L N, we can pick a sequence (x n ) n N Z d s.t. the (Λ L (x n )) n N are pairwise disjoint, which implies that the corresponding (A n ) n N are independent. Because of this and (17), we can apply the Borel-Cantelli-Lemma to get P(lim sup n A n ) = 1. By taking the (countable) intersection over all L, we get: P ( L N ) lim sup A n =1 n Since λ σ(h 0 ), we can apply the Weyl criterion to get a normalized ψ l 2 (Z d ) s.t. (H 0 λ)ψ < ɛ 2. (18) Since H 0 is bounded (by 4d), we can first consider H 0 to be just defined on the dense subspace l 2 0(Z d ) of compactly supported l 2 -functions and then extend later. Therefore, we can assume ψ l 2 0(Z d ). Since H 0 is stationary (it looks the same at each i Z d ), we can choose supp(ψ) freely in Z d, which allows us to maneuver said support in such a way that n, L N : supp(ψ) Λ L (x n ). Putting (17) and (18) together, we can conclude (H ω λ ν)ψ ( H 0 λ)ψ + ψ < ɛ 2 <ɛ +1 sup v ω (y) ν y Λ L (x n) sup y supp(ψ) v ω (y) ν Step 2: We eliminate the E-dependence by fixing a countable dense set {E k } k N [0, 4d]+ supp(p). We define Ω ɛ := k N Ω ɛ,e k P( Ω ɛ ) = 1. Let ω Ω ɛ. For any E [0, 4d]+ supp(p), we find k 0 N s.t. E E k is minimized for k = k 0. Let ψ k0 be the ψ we get from applying Step 1 to E k0 and ɛ. We have 2 (H ω E)ψ k0 (H ω E k0 )ψ k0 + E k0 E ψ k0, }{{} =1 where both terms can be made smaller than ɛ 2. Step 3: We eliminate the ɛ-dependence by defining Ω := n N Ω 1/n, 9

10 which clearly has full measure. Altogether, we have shown: ω Ω E [0, 4d]+supp(P) (ψn ) n N l 2 (Z d ) : lim n (H ω E)ψ n =0 and the (ψ n ) n N are normalized. This is just the Weyl criterion, so E Σ. Note that a statement of the form The spectrum of the sum of two s.-a. operators is contained in the sum of the spectra of the individual operators is false in general. This direction is a direct application of the follow property dist{σ(u + V ),σ(v )} U. (19) Define U := H 0 2dI, V := V ω +2dI, where I denotes the identity. Then we have U + V = H ω and from Note 2.13, we can conclude U =2d and σ(v )=supp(p)+2d. Therefore (19) provides us with the estimate dist{σ(h ω ), suppp +2d} 2d, from which we can conclude σ(h ω ) supp(p) + [0, 4d]. This concludes our presentation of ergodic operators. For further reading, we recommend [1], [3] and [5]. References [1] W. Kirsch, Random Schrödinger operators: a course, Schrödinger operators, Lecture Notes in Physics 345 (1989), [2], An invitation to random Schrödinger operators, Panoramas et Syntheses 25 (2008), [3] W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, Journ.Reine Angew. Math. 334 (1982), [4] A. Klenke, Wahrscheinlichkeitstheorie, Springer, Berlin, [5] H. Kunz and B. Souillard, Sur le spectre des operateurs aux differences finies aleatoires, Commun. Math. Phys. 78 (1980), [6] L.Pastur and A.Figotin, Spectra of random and almost-periodic operators, Springer, Berlin,