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Jan 26, 2017 - DS] 26 Jan 2017. A GENERAL RENORMALIZATION PROCEDURE ... continuous potentials U, such that U is constant. Date: January 27, 2017.
A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

arXiv:1701.07656v1 [math.DS] 26 Jan 2017

A. O. LOPES

Inst. de Matem´atica, UFRGS - Porto Alegre, Brasil

A BSTRACT. We present a general form of Renormalization operator R acting on potentials V : {0,1}N → R. We exhibit the analytical expression of the fixed point potential V for such operator R. This potential can be expressed in a naturally way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by A. Baraviera, R. Leplaideur and A. Lopes (see [2]) where the fixed point potential V was of Hofbauer type (see [16]). For the potentials of Hofbauer type (a well known case of phase transition) the decay is like n−γ , γ > 0 (see [22] [23], [14] or [10]) Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential V of our general renormalization procedure. In some cases we get polynomial √ decay like n−γ , γ > 0, and in others a decay faster than n e− n , when n → ∞. The potentials g we consider here are elements of the so called family of Walters potentials on {0,1}N (see [27]) which generalizes the potentials considered initially by F. Hofbauer in [16]. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice dn → 0 of real numbers varying with n ∈ N there exist a potential g on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

1. I NTRODUCTION It is a classical problem in Statistical Mechanics the study of phase transitions (see [12], [13], [1] , [15], [28] and [29]) for general potentials. In several examples it is known that on the point of phase transition one gets polynomial decay of correlation for the associated equilibrium probability. As far as we know, there is no unified way for understanding all kinds of phase transitions. One important technique on the analysis of such class of problems is the use of the renormalization operator (see for instance [5], [18], [4] and [8]). The potential which is fixed for the renormalization operator in most of the cases is quite relevant for the understanding of the problem. Phase transitions in the setting of Thermodynamic Formalism has been considered from different points of view (see for instance [2], [14], [19], [20], [21], [7] , [6], [22], [23], [9], [10], [11] and [17]). In [2] it was introduced a renormalization operator (acting on potentials) on the one-dimensional lattice {0, 1}N and the fixed point one gets is the well known Hofbauer potential. The equilibrium probability for such potential exhibit polynomial decay of correlation for a certain class of observables. Here we consider a more general procedure of renormalization of potentials defined on the lattice {0, 1}N . We want to exhibit a fixed point potential for such procedure. The fixed point potential will be on a class of functions previously considered by P. Walters in [27] (which includes the Hofbauer potential). We are particulary interested in the decay of correlation (for a certain natural function) associated to the equilibrium probability for such potential. In some cases (first type) the fixed point will produce polynomial decay of correlation and in others √ cases (second type) not of polynomial type (some of them faster than n e− n , when n → ∞). In order to give an idea to the reader of the kind of results we will get here we present an example: in a particular case the renormalization operator R acts on continuous potentials U, such that U is constant Date: January 27, 2017. 1

2

A. O. LOPES

on cylinder sets of the form |0 .{z . . 0} 1 and |1 .{z . . 1} 0, for all Q ∈ N. For each fixed Q the values are the same on Q

Q

both cylinder sets We define U2 = R(U1 ) in the following way: suppose x is of the form x = (0 . . 0}, 1, ...), then | .{z Q

U2 (x) = R(U1 ) (x) = U1 (0| .{z . . 0}, 1) + U1(0 . . 0}, 1). | .{z 3Q

3 Q−2

In this case the fixed point potential V for the renormalization operator R is the function V (x) = −

Z

K

1 d ν (t) (Q − t)α

when x = (0 . . 0}, 1, ...), Q ∈ N, and where ν is the Hausdorff probability on the classical Cantor set (a | .{z Q

2 subset of the interval [0, 1]) and which has Hausdorff dimension α = log log 3 . In several distinct classes of problems the fixed point potential for the renormalization operator can be expressed on an integral form associated to a fractal probability (see for instance [5], [3], [18], [4] and [24]). Here we will also get a similar kind of expression.

We will present now the main definitions giving more details on the specific problems we want to analyze. We will consider a subclass of potentials g : {0, 1}N → R which were described in [27]. Using the notation of that paper, consider a real sequence an , n ∈ N, and assume that an → 0, when n → ∞. In our model a = 0 = c. Given a2 , a3 , ..., an , ... < 0 define η j , j ≥ 1, in such way that (a2 + a3 + ... + a j+1) = log η j+1 , j ≥ 1. γ For the sequence 1 ≥ ηn > 0, n ∈ N, n ≥ 1, η1 = 1, we assume that, ηn → 0, and ∑∞ n=1 ηn = W (β ) > 1 for any β > 1. 1/2

Particular cases which we are interested are when ηn = e−n , ηn = e−n Consider the two infinite collections of cylinder sets given by Ln = |000...0 {z } 1 n

1−

log 2 log 5

or ηn = n−γ , γ > 2.

and Rn = |111...1 {z } 0, for all n ≥ 1. n

Using these parameters we can define a continuous potential g = gβ : Ω → R, which is simply denote by g, in the following way: for any x ∈ Ω and β ≥ 1  β an , if x ∈ Ln , for some n ≥ 2;       if x ∈ Rn , for some n ≥ 2;  β a n , g(x) =

− logW (β ),      − logW (β ),    0,

if x ∈ L1 ;

if x ∈ R1 ; if x ∈ {1∞ , 0∞ }.

In the above we are taking cn = an , for all n. We are mainly interested in potentials g which are not of H¨older type. Definition 1. Given A : Ω = {0, 1}N → R, the Ruelle operator LA acts on functions ψ : Ω → R in the following way

ϕ (x) = LA (ψ )(x) =

d

∑ eA(ax) ψ (ax).

a=1

By this we mean LA (ψ ) = ϕ . From the general result described in page 1341 in [27] applied to our particular case, we get that the eigenfunction ϕβ of the Ruelle operator for A = β g, for β ≤ 1, is such that, for any n ≥ 1

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

ϕβ (0 1...) = α n

1+

ϕβ (1 0...) = b(β ) n

ϕβ (0∞ ) = α

ηnβ

1+

and

β

ηn+ j ∑ λ (β ) j j=1 ∞

1

1

β

!

ηn+ j ∑ λ (β ) j j=1 ∞

3

,

(1)

!

(2)

, ηnβ ϕβ (1∞ ) = b(β ).

The eigenfunction is unique. We assume that for β = 1 we get λ (β ) = 1 = α = b(β ) When β = 1, it is easy to see that if it is well defined the probability ρ on the Borelians of Ω, such that, for any natural number q ≥ 1, we have that   ρ 0q 1 = ηq ρ 1q 0 = ηq . and (3)

then, ρ is an eigenprobability for the dual of the Ruelle operator for g. Therefore, from a general procedure (see [25]) one gets that the probability µ , such that, ∞ ∞   µ 0q 1 = ηq ϕ (0n 1...) = ∑ ηn+ j µ 1q 0 = ηq ϕ (1n 0...) = ∑ ηn+ j and j=0

(4)

j=0

is the equilibrium probability for g.

We will show in Theorem 6 that decay of correlation dq , as a function of q ∈ N, of the equilibrium probability µ for the potential g and for the indicator of the cylinder 0 is of order ∞



∑ ∑ η(k+q+s) .

s=1 k=0

Given a certain sequence dq → 0, there exists (under mild conditions) a choice of ηn such that dq = ∞ ∑∞ s=1 ∑k=0 η(k+q+s) . Indeed, first for the given sequence dq , q ∈ N, take inductively cn such that cn = dn − dn+1, n ∈ N. In this way ∑∞j=q c j = dq . Now, given such sequence cn , n ∈ N, take a new sequence ηr , such that, ηr = cr − cr+1 , r ∈ N. In this case ∑∞ i=p ηi = c p . ∞ It follows that for all q we get dq = ∑∞ s=1 ∑k=0 η(k+q+s) . Part of the results presented here were a consequence of fruitful discussions with A. Baraviera and R. Leplaideur which contributed in a non trivial way to our reasoning. Our thanks to them. In sections 2 and 3 we present results about the fixed point potential for a certain renormalization operator. On sections 4 and 5 we estimate the decay of correlations for the class o potentials of Walters type and we show that a large class of types of decay of correlations are attained by such family. On the appendix we use results of the previous sections to show that for some of these fixed point potentials the decay is not of purely polynomial type. 2. A

GENERAL RENORMALIZATION

-

FIRST TYPE

The potentials f which we are interested are on the class f (0n 1z) = an , f (01n 0z) = bn ,

f (0∞ ) = a, f (01∞ ) = b,

f (10n 1z) = dn , f (1n 0z) = cn ,

f (10∞ ) = d, f (1∞ ) = c,

and is assumed that an → a, bn → b, cn → c and dn → d. We will assume that bn = dn = b = d. The renormalization operator will act on such set of functions. Fix a natural number k ≥ 2. We will generalize the kind of renormalization described in [2] where k = 2.

Now take for n > 1

Hk (0n 1x) = 0k n −(k−2) 1x Hk (1n 0 y) = 1k n −(k−2) 0y

and Hk (01m x) = 0 1m x

4

A. O. LOPES

Hk (1 0m y) = 1 0m y and also Hk (0∞ ) = 0∞ , Hk (1∞ ) = 1∞ . Now define the renormalization operator as Rk (V )(x) = V (Hk (x)) + V (σ Hk (x)) + V (σ 2 Hk (x)) + ... + V(σ k−1 Hk (x)). We will show that it is possible to exhibit a fixed point V for the renormalization operator Rk . The class of functions we consider here in the section 1 is large enough to be able to get fixed points V . This means we will be able to find a sequence an , n ∈ N, which will define such V An interesting question is to estimate the decay of correlation of the equilibrium probability for the fixed point potential V . One can show (see section 5) that for the case of renormalization of first type the potential V is such that we get polynomial decay of correlation. If V is fixed for the renormalization operator Rk , then V (1∞ ) = V (0∞ ) = 0 = a = c. Moreover, V (01∞ ) = V (01∞ ) + V (1∞ ) + V (1∞ ) = b and ∞ ∞ ∞ a2 = V (00 1∞) = V (00...000 | {z } 1 ) + V (00...000 | {z } 1 ) + ... + V(0001 ) = a3 + a4 + ... + ak+2. k+2

k+1

∞ ∞ ∞ a3 = V (000 1∞) = V (00...000 | {z } 1 ) + V (00...000 | {z } 1 ) + ... + V(00...000 | {z } 1 ) = ak+3 + ak+4 + ... + a2 k+2. 2 k+2

2 k+1

k+3

As we will se once we fix the value of a2 ad b the potential V will be determined. The Walters potential V which is the fixed point for Rk should satisfy for n ≥ 2: an = ak (n−1)+2 + ak (n−1)+1 + ... + ak(n−2)+3 an = cn = ck (n−1)+2 + ck (n−1)+1 + ... + ck(n−2)+3

and dn = bn = b = d are constant. Now we solve α (2) as the solution of a2 = − log Now, α (n) is defined by induction for n ≥ 2,

2 + α (2) . 2 + α (2) − 1

αk (n−1)+2 = αk (n−1)+1 = ... = αk(n−2)+3 ,

and

αk (n−1)+2 = k α (n) + (k − 2). Taking an = dn , n ≥ 2, of the form an = − log

n + α (n) 2, one gets polynomial decay of correlation of the form n2−γ similar to the Hofbauer case. This kind of question was previously consider in [22] [23] [14] [10]. This can be also obtained from the general proof of section 5.

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

3. A

GENERAL RENORMALIZATION

- SECOND

5

TYPE

We consider in this section a far more general generalization of the renormalization operator and we exhibit the fixed point potential V . We point out that in many examples the fixed point potential of a renormalization operator is described by an expression obtained by the integral of a kernel with respect to a certain fractal probability (see for instance [5], [3], [18], [4] and [24]). Here we will get a similar kind of result. Let us denote Ln the cylinders [00 . . . 01] (with n zeros) and Rn the cylinders [11 . . . 10] (with n ones). Lets us define a potential that is an on Ln and bn on Rn . In order to define the second type renormalization operator, we first introduce the map H. Let k be an integer larger or equal than 2; take h(0) = 00 . . .0} and h(1) = 11 . . . 1}; hence we define | {z | {z k

k

H(x1 , x2 , x3 , . . .) = (h(x1 )h(x2 )h(x3 ) . . .).

It is easy to see that σ k H = H σ . Hence . . . 0} 1 . . . H(00 . . . 0} 1 . . .) = 00 | {z | {z n

and

say, if x ∈ Ln then H(x) ∈ Lkn

kn

H(11 . . . 1} 0 . . .) = 11 . . . 1} 0 . . . | {z | {z n

say, if x ∈ Rn then H(x) ∈ Rkn

kn

We now define the renormalization operator as the map on functions defined as follows: fix l such that 2 ≤ l ≤ k and the constants 0 ≤ c1 < c2 < . . . < cl ≤ k. Then RV (x) := V (σ c1 H(x)) + V (σ c2 H(x)) + · · · + V (σ cl H(x))

One can see that the Renormalization operator of last section (the case l = k) is a particular case of this new one . The equation above for the potential V can be rewritten, analogously to the expressions in the previous section, as Ran = akn−c1 + akn−c2 + · · · + akn−cl

(and a similar expression holds to cn ). We are interested on finding a fixed potential V for such renormalization operator. For a potential on the class of Walters the fixed point equation is an = akn−c1 + akn−c2 + · · · + akn−cl Then we get the following result: Lemma 2. Given N ≥ 1 then

R N V (x) = ∑ V (σ j H N (x)) j

where j is of the form j = b0 k0 + b1k1 + b2k2 + · · · + bN−1 kN−1

for bi chosen on the set {c1 , c2 , . . . , cl }.

Proof. Indeed, the expression it obviously valid for N = 1 (by definition of the operator R). Now, admit that it holds for N; we need to show that this implies the expression for R N+1V . But R N+1V = R(R N V ) = R N σ c1 H + R N σ c2 H + · · · + R N σ cl H =

∑ V (σ j H N σ c1 H) + . . . j

Now it is easy to see (also by induction) that H σ P H = σ Pk H 2 and

P

P

H P σ H = σ k H 2 (?) = σ k H P+1

6

A. O. LOPES

With this two expressions we get N

H N σ c H = σ ck H N+1 Then we can show that R N+1V is

∑ V (σ j H N σ c1 H) + . . . = j

∑ V (σ j σ c1

kN

j

where j = b0

k0 + b

1

k1 + b

2

k2 + · · · + b

H N+1 ) + . . . = ∑ V (σ j+c1 k H N+1 ) + . . . N

j

N−1

kN−1 ;

hence the expression above is indeed

∑ V (σ J H N+1 ) + . . . J

where J = d0 k0 + d1k1 + d2k2 + · · · + dN−1 kN−1 + dN kN , showing that R N+1 is as claimed.

Now we want to show that we have a fixed point of R that in the neighborhood of 0∞ behaves like 1 V (00 . . . 0} 1 . . .) ∼ − log l | {z n log k n



As we will see in a moment it will be natural to consider the (Cantor like) set K = K(l, k) which is the closure of the set {x = ∑ ai k−i for ai ∈ {c1 , . . . , cl }, n ∈ Z } n≥1

Note that when k = l the set K will be the interval [0, 1]. Example 3. Consider the following particular case. Define H3 : Ω = {0, 1}N → Ω by:

H3 ((0, ..., 0, 1, ..., 1 0, ..., 0, 1, ...)) = (0, ..., 0, 1, ..., 1 0, ..., 0, 1, . . .), | {z } | {z } | {z } | {z } | {z } | {z } c1

and

c2

c3

c2

3c1

c3

H3 ((1, . . . , 1, 0, . . . , 0 1, . . . , 1, 1, . . .)) = (1, . . . , 1, 0, . . . , 0 1, . . . , 1, 1, . . .). | {z } | {z } | {z } | {z } | {z } | {z } c1

c2

c3

3c1

c2

c3

In this case the renormalization operator is

R3 (V )(x) = V (H3 (x)) + V (σ H3 (x)) + V (σ 2 H3 (x))

1 If x = (0, ..., 0, 1, ..) or x = (1, ..., 1, 0, ..) denote V (x) = log c1c−1 . Note that for each c1 we get that | {z } | {z }

c1

c1

3 c1 − 1 3 c1 − 2 c1 3 c1 + log + log = log . 3 c1 − 1 3 c1 − 2 3 c1 − 3 c1 − 1 In this case V is a fixed point for the renormalization operator. We can choose the signal of V and then, we take V in the form log

V (1, ..., 1, 0, ..) = V (0, ..., 0, 1, ..) = − | {z } | {z } c1

c1

Z 1 0

c1 1 dt = − log c1 − t c1 − 1

In order to find a good guess (in the general case) for the fixed point potential consider 1 U(x) = α , for any Q Q when, x = 0, ..., 0, 1, ..), or, x = 1, ..., 1, 0, ..). | {z } | {z } Q

Q

Then, by iteration of U we get:

R N U(x) = ∑ j

1 1 1 = Nα (QkN − j)α ∑ k (Q − kNj )α j

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

7

for j = b0 k0 + b1k1 + b2k2 + · · · + bN−1 kN−1 (and bi ∈ {c1 , . . . , cl }). From the above seems natural to try to find a fixed point V of the form V (x) = −

Z

K

1 d ν (t), (Q − t)α

when (0| .{z . . 0}, 1, ..), for some probability ν on the interval. Q

Example 4. Let us fix k = 3, l = 2 and c1 = 0, c2 = 2. In this case the Cantor set K we generate is indeed the classical Cantor set. We can write K = K0 ∪ K2 where K0 ⊂ [0, 1/3] and K2 ⊂ [2/3, 1]. Note that the transformation T (x) = 3 x (mod 1) is two to one when restricted to K. Moreover, T [0, 1/3] = [0, 1] = T [2/3, 1]. Let ν be the measure supported on K and consider α = log 2/ log 3, that is the Hausdorff dimension of K. Note that the Radon-Nykodin derivative of ν in each injective branch of T restricted to [0, 1/3] and [2/3, 1] is equal to 2. We want to show that the function V (0 . . 0} 1) = − | .{z Q

Z

K

1 d ν (t) (Q − t)α

is a fixed point of the renormalization operator, i.e.,

V (x) = V (H(x)) + V (σ 2 H(x)) If x = |0 .{z . . 0} 1 this corresponds to show that Q

Z

K

1 d ν (t) = (Q − t)α

Z

K

1 d ν (t) + (3Q − t)α

Z

K

1 d ν (t) (3Q − 2 − t)α

Note that the second expression is  Z Z 1 1 1 d ν (t) + d ν (t) = α α 3α K (Q − 2/3 − t/3) K (Q − t/3) Z  Z 1 1 1 d ν (t) + d ν (t) α 2 K (Q − t/3)α K (Q − 2/3 − t/3) We remind that 32α = 1. In the first integral we can use the change of variables x = t/3, then we get Z

K

1 d ν (t) = (Q − t/3)α

Z

1 2d ν (x) (Q − x)α

K0

In the second we use x = 2/3 + t/3 and we get Z

K

1 d ν (t) = (Q − 2/3 − t/3)α

Z

K2

1 2d ν (x). (Q − x)α

Therefore, 1 3α

 Z 1 1 d ν (t) + d ν (t) = α α K (Q − 2/3 − t/3) K (Q − t/3) Z  Z 1 1 1 2 d ν (x) + d ν (x) = α α 3α K2 (Q − x) K0 (Q − x)

Z

Z

as claimed.

K0 ∪K2

1 d ν (x) = (Q − x)α

Z

K

1 d ν (x) (Q − x)α

8

A. O. LOPES

Theorem 5. Let x of the form: x = 00 . . . 0} 1 . . ., or x = 11 . . . 1} 0 . . ., then the operator | {z | {z Q

Q

RV (x) := V (σ H(x)) + V (σ H(x)) + · · · + V (σ cl H(x)) c1

c2

has a fixed point V of the form V (x) = −

Z

K

1 d ν (t) (Q − t)α

log l where α = log k , and where ν is the measure over K that maximizes the entropy for the transformation T (x) = k x (mod 1) acting on K . log l Proof. The reasoning is similar to the last example. Consider α = log k. The transformation T (x) = k x (mod 1) acting on K is l to one. The maximal entropy probability ν has entropy log l. There exists l sets K 1 , K 2 , ..., K l ⊂ [0, 1] such that T (K j ∩ K) = K, and by considering T restricted to j K = [ c j /k, (c j +1)/k ], j = 1, 2, ..., l, we get an injective map. Consider the induced probability ν j = T ∗ (ν ) on K j . The Radon-Nykodin derivative of ν j with respect to ν in each set K j is equal to l. If x = (0 . . 0}, 1, ...) (or, x = (1| .{z . . 1}, 0, ...)) we want to show that | .{z Q

Q

V (x) = −

Z

K

1 d ν (t) = − (Q − t)α

l



Z

j=1 K

1 d ν (t) = RV (x). ( (k Q − c j ) − t )α

For each j we have that Z Z Z 1 1 1 d ν (t) = d ν (t) = d ν (t). cj c +t t α j α α ( (k Q − c ) − t ) K k (Q − K l (Q − k − k )α K j k )

For each j we consider the change of coordinate t → t/k on the set K j . Then, we get that Z Z 1 1 l d ν (y). c j +t α d ν (t) = j (Q − y )α K K (Q − k ) As Z l Z 1 1 d ν (t) = d ν (t) ∑ α α j K (Q − t) j=1 K ( Q − t ) we get the claim.

 Note that −

Z

K

1 d ν (t) ∼ − Q−α (Q − t)α 1−α

for large Q. In this case ηn will be of order e− n √. When k = 5 and l = 2 we will get that the equilibrium state for fixed point V has a decay faster than n e− n (see Appendix). 4.

THE ASSOCIATED JACOBIAN

Now we are interested on the estimation of the decay of correlation associated to equilibrium probabilities associated to the fixed point potentials we get via the renormalization operator. We will show that it is not always of polynomial type. In fact, we will present the decay of correlation for a general potential of Walters type. Suppose g is of Walter type. If we add a constant to a potential its equilibrium state will not change. Then, we can assume without lost of generality that the potential g has pressure zero. The associated Jacobian J for the potential defined by g at the inverse temperature β = 1 is such that log J = g + log ϕ − log(ϕ ◦ σ ), where ϕ is the eigenfunction of the Ruelle operator for g (see (2). We will define r(q) for q ≥ 1. We split the expression of log J(x) in six cases:

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

a) for q ≥ 2 and x ∈ Lq or x ∈ Rq we have



log J(x) = β aq + log 1 + ηq

∑ η(n+q−1)

n=2

!

− log 1 + η(q−1) q

9



∑ η(n+q−2)

n=2

!

:= β ak + logr(q) − logr(q − 1).

z }| { b) for x = 0 111...1 0... ∈ L1 we have

log J(x) = − log 1 + ηq



∑ η(n+q−1)

n=2

!

= − log(r(q)).

!

= − log(r(q)).

q

z }| { c) for x = 1 000...0 1... ∈ R1 we have

log J(x) = − log 1 + ηq



∑ η(n+q−1)

n=2

d) for x = 0∞ or x = 1∞ we have J(x) = 1. e) for x = 10∞ or x = 01∞ we have J(x) = 0. Note that r(1)η1 = W. The equilibrium probability µ for g has Jacobian J and ∗ Llog J (µ ) = µ .

5.

DECAY OF CORRELATION

In this section we want to estimate the decay of correlation of the observable I[0] for the equilibrium probability µ at the critical inverse temperature T where β = 1 = T1 . We assume from now on that β = 1. That is, we will estimate the asymptotic behavior with k of the decay of correlation of the observable I0 Z



(I[0] ◦ σ q ) [ I[0] − µ1 [0] ] d µ ∼ α (q),

where µ is the equilibrium probability for g, when β = 1. We will show, as a particular case, that if ηq ∼ q−γ , with γ > 2, then, α (q) goes to zero like q2−γ . The next result is quite general. Theorem 6. The decay of correlation as a function of q for the equilibrium probability µ (for the potential g) for the indicator of the cylinder 0 is ∞



∑ ∑ η(k+q+s)

s=1 k=0

in the case is well defined. Proof: The technique is similar to the one employed in [14] and [10]. First we need to estimate the asymptotic limit q µ1 [0] − LlogJ (I[0] )(01..).

We claim that q µ1 [0] − LlogJ (I[0] )(01..) ∼

We will present the proof of this fact later.

q



∑ ∑

n=1 j=q+1

η j.

(5)

10

A. O. LOPES

Now, for a fixed q and any s ≥ 2 we need to evaluate the difference q µ1 [0] − LlogJ (I[0] )(00...0 | {z } 1..). s

For fixed q and variable s, let psn =

r(s+n)ηs+n r(s)ηs ,

Bsq

=

n = 1, .., q − 2, s ∈ N. η

s z }| { q LlogJ (I[0] )(00...0 1..),

t (I )(01...) and finally denote A(t) = Llog J [0]

r(s+q)

. We denote α (q, s) = (s+q) ηs r(s) One can show (see figure 3 page 1092 and expression in the bottom of page 1090 [14] where a similar case is described) that for any s, q ≥ 2 Bsq = Aq−1 + eas+1

r(s + 1) r(s + 2) r(s + q − 2) Aq−2 + eas+1 +as+2 Aq−3 + ... + eas+1+as+2 +...+aq+s−2 A1 + r(s) r(s) r(s)

eas+1 +as+2 +...+aq+s−1 +aq+s

Aq−1 +

r(s + q) = r(s)

r(s + 1) η(s+1) r(s + q − 2) η(s+q−2) r(s + q) η(s+q) r(s + 2)ηs+2 Aq−2 + Aq−3 + ... + A1 + = r(s) ηs r(s)ηs r(s) ηs r(s) ηs

Aq−1 +

r(s + q − 2) η(s+q−2) r(s + 1) η(s+1) r(s + 2)ηs+2 Aq−2 + Aq−3 + ... + A1 + α (n, s). r(s) ηs r(s)ηs r(s) ηs

(6)

This is not a kind of renewal equation. Let us now consider for n ≥ 1 fixed, and s ≥ 2 the sequence Vns = µ1 [0] − Bsn. We also introduce, for n ≥ 1, the sequences Vn = µ1 [0] − An, and Uns = −µ [0](

r(s + q − 2) η(s+q−2) r(s + 1) η(s+1) r(s + 2)ηs+2 + + ... + ) − α (n, s). r(s) ηs r(s)ηs r(s) ηs

From the equation (6) we deduce that for q fixed q s s s s s s µ1 [0] − LlogJ I[0] (00...0 | {z } 1..) = µ [0] − Bq = Vq = Vq−1 + Vq−2 p1 + Vq−3 p2 + ... + V1 pq−2 + Uq .

(7)

s

Remember that (4) claims

 µ 0s 1 = ηs ϕ1 (0s 1...) = ηs r(s).

(8)

For each fixed q we will have to estimate later for each s the value

 q    q µ ( 0s 1 ) Llog J I[0] (00...0 | {z } 1..) − µ [0] = ηs r(s) Llog J I[0] (00...0 | {z } 1..) − µ [0] = s

s

Vq−1 ηs r(s) + Vq−2 r(1 + s)η1+s + Vq−3 r(2 + s)η2+s + ... + V1 r(s + q − 2)ηs+q−2 − r(s + q)ηs+q.

(9)

As mentioned before the Ruelle operator Llog J is the dual (in the L 2 (Ω, B, µ1 ) sense) of the Koopman operator K (ϕ ) = ϕ ◦ σ , using this duality, expressions (9) and (15) we get that for fixed q that

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

Z



(I[0] ◦ σ q ) [ I[0] − µ1 [0] ] d µ = = ∞

=



Z

s=1 Ω

Z

11

q I[0] Llog J [ I[0] − µ [0] ] d µ

Z



 q  I[0] (x) Llog J I[0] (x) − µ [0] d µ (x)

  q I[0] (00...0 | {z } 1..) Llog J I[0] (00...0 | {z } 1..) − µ [0] d µ s

s



  q = ∑ µ [00...0 | {z } 1] Llog J I[0] (00...0 | {z } 1..) − µ [0] s=1

s



s

= − ∑ µ [0] ( r(s + 1) η(s+1) + r(s + 2)ηs+2 + ... + r(s + q − 2) η(s+q−2) ) ∞

s=1

+ ∑ Vq−1 ηs r(s) + ... + V1 r(s + q − 2)ηs+q−2 − r(s + q)ηs+q ∼ s=1



q−1



j=1

Vq− j

∑ η(k+s+ j−1) −

k=0

Note that in the case ηk = e− k zero when s → ∞. Moreover,

1/2









∑ η(k+q+s) ∼ − ∑

s=1 k=0

∑ η(k+q+s) .

(10)

s=1 k=0

we get that r(s)ηs is of order ∞





∑ ∑ η(k+q+s) ∼ q e−

√ q

√ − √s se (see Appendix) which goes to

.

s=1 k=0

In the case log ηn ∼ −γ log n, n ≥ 1, one easily get from the above that the decay of correlation is of polynomial type n2−γ . Now we prove that q µ [0] − LlogJ (I[0] )(01..) = Vq ∼

q



∑ ∑

η j.

n=1 j=q+1

q Using the scheme of smaller trees on the right side we get that the general term of Llog J (I[0] )(01...) for 1 ≤ j ≤ q − 1 is given by the following expression η(q−1) 1 q Llog J (I[0] )(01 . . .) = W (γ ) Llog J (I[0] )(01 . . .) + . . .

+ Denote pq =

η(q+1) r(q + 1) 1 η2 q−2 q−1 LlogJ LlogJ (I[0] )(01 . . .) + (I[0] )(01 . . .) + W (γ ) W (γ ) W (γ )

ηq W (β ) ,

q ≥ 1, and α (q) =

η(q+1) r(q+1) W (β )

and K = µ [0] = 1/2 =

(11)

∑q α (q) . ∑q q p q

q (I[0] )(01..), for q ≥ 1. Define Vq = µ [0] − A(q) = µ [0] − LlogJ For example, V2 = p1 V1 + µ1 [0] (p2 + p3 + ...) − α (3). We want to obtain the behavior of V (q), when q → ∞. From the renewal equation (11) we get another renewal equation: for q ≥ 3 " # q−1

Vq =

∑ V j pq− j +

µ [0]



∑ p j − α (q)

.

(12)

j=q

j=1

We denote by Kq the last term on the above equality, that is, Kq = µ [0]



∑ p j − α (q),

j=q

q ≥ 1. 1 (I[0] )(01..) = V1 . Note that K1 = µ [0] − α (1) = µ [0] − LlogJ Note also that

(13)

12

A. O. LOPES

∑∞j=q+1 η j W

α (q) = and



Kq = 1/2

ηj

∑ W (β ) −

j=q

∑∞j=q+1 η j ∼− W (β )





j=q+1

η j → 0,

when q → ∞. Now consider the following formal power series ∞





f (z) =

∑ p j z j−1 ,

V (z) =

∑ Vj z j,

and K(z) =

j=1

j=1

j=1

∑ K j z j−1 .

From the renewal equation (12) we get that V (z) f (z) + K(z) = V (z). Therefore V (z) =

K(z) K(z) 1 − z = . 1 − f (z) 1 − z 1 − f (z)

We assume that pn , n ∈ N, is such that f (z) is differentiable on z = 1 (this is an assumption on ηn , n ∈ N) and the derivative is not zero. Up to a bounded multiplicative constant we get K(z) V (z) ∼ . 1−z from where we obtain (asymptotically) ∞

V (z) ∼

∑ Kj z j

(1 + z + z2 + z3 + ...)

j=1

K1 + (K1 + K2 ) z + (K1 + K2 + K3 )z2 + (K1 + K2 + K3 + K4 )z3 .... In this way we get the following recurrence relation: Vn ∼ K1 + K2 + K3 + ... + Kn. This argument complete the proof of

q µ [0] − LlogJ (I[0] )(01..) = Vq ∼



q

∑ ∑

η j.

(14)

n=1 j=q+1

This goes to zero because ∑q q ηq < ∞. 1/2

When ηk = e− k , we get that Vk ∼ k2 e−k

In the same way

1/2

q µ [0] − LlogJ (I[0] )(10..) ∼

.

(15) ∞

q

∑ ∑

η j ..

(16)

n=1 j=q+1

 6. A PPENDIX − k1/2

and we will take the estimation (10). First we consider the case √ ηk = e ∞ − n+ j+k The estimate of ∑k=0 e , for n, j fixed, can be done via the integral Z ∞

e−

√ n+ j+x

dx. 1 √ Taking the change of coordinates y = n + j + x, we get the equivalent expression 2 As

Z ∞

e √ n+ j+1

−y

y dy.

d (e−y y) dy = e−y dy − √ dy n+ j+1 √ √ R √ we get that 1∞ e− n+ j+x dx is of order n + j + 1e− n+ j+1 . √ (e−y y) |∞ n+ j+1 =

Z

Z

Z

y e−y dy,

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE AND DECAY OF CORRELATIONS

− Now for n fixed, we want to estimate the expression ∑∞j=1 ∑∞ k=0 e In order to to that we consider the integral Z ∞√ √ n + x e− n+x dx. 1 √ Considering the change of coordinates y = n + x we get that √ = (e−y y2 ) |∞ n+1

and finally





d (e−y y2 ) dy = 2 √ dy n+1

Z



∑ ηn+ j+k ∼

j=1 k=0

In a similar way, if ηk = e− k

1−

log 2 log 5

Z

Z ∞√ 1

n + x e−



n+1

√ n+x

√ n+ j+k

e−y y dy −

dx ∼ n e−

13

= ∑∞j=1 ∑∞ k=0 ηn+ j+k .

Z

√ n

y2 e−y dy,

.

we get a similar result, that is, the decay is faster than polynomial. log 2 1− log 5

1/2

log 2 log 2 −k ≤ e− k . Therefore, Indeed, log 5 = 0.4306, and then 1 − log 5 > 0.5. From this follows that e for the case of the fixed point V for the renormalization√ operator when k = 5 and l = 2, we get that the associated equilibrium state has a decay faster than n e− n .

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