Convergence of Renormalization Group Transformations of Gibbs Random Field

Corresponding Author: Farida Kachapova Auckland University of Technology, Auckland, New Zealand Email: farida.kachapova@aut.ac.nz Abstract: Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group transformation. Here we consider a thermodynamic limit of a lattice model with weak interaction and we use semi-invariants to prove that random fields transformed by renormalization group converge in distribution to an independent field with Gaussian distribution as the distance scale infinitely increases; it is a generalization of the central limit theorem to weakly dependent fields on a lattice.


Introduction
The classical Central Limit Theorem (CLT) considers a sequence of independent random variables and their normalized sums. Here we consider a sequence of weakly dependent random fields on a multi-dimensional integer lattice. We are interested in the limiting distribution of normalized sums of these variables, similar to the sums in the classical CLT. Such problems arise in the research of Renormalization Group (RG) in statistical mechanics.
The concept of RG as a scale transformation was introduced and studied in works of Kadanoff (Kadanoff, 1966;Kadanoff, 2013), Wilson and Kogut (1974). Originally RG was defined in terms of Hamiltonian (interpreted as the interaction potential). A rigorous formula of the renormalized Hamiltonian was derived by Kashapov (1980). Bertini et al. (1999) and Lorinczi et al. (1998) studied Gibbs property of the renormalized Hamiltonian.
Other research on RG are based on limit theorems of probability theory. Sinai (1976) studied distributions invariant under the RG transformation and showed that Gaussian distribution is one of them. Newman (1980) proved the CLT on an integer lattice under Fortuin-Kasteleyn-Ginibre (FKG) conditions. Bolthausen (1982) proved the CLT on an integer lattice under some strong conditions.
In this paper we study the limiting distribution of Gibbs random field under the RG transformations and we improve our results from (Kachapova and Kachapov, 2015). We show that under the condition |λ| < C the limiting distribution in a high-temperature region is an independent Gaussian distribution. The novelty of our result is in finding a broad condition for the interaction parameter λ, for which the CLT on a lattice holds; this condition is |λ| < C for a constant C depending only on the lattice dimension. This is a simple condition and is easy to check and it is stated in a form preferable for physicists, without tedious technical details.
The FKG conditions in the Newman's version of CLT (Newman, 1980) do not always hold; for example, they hold for the ferromagnetic Ising model but not for the anti-ferromagnetic one; while our theorem covers both models and more.
The conditions in the Bolthausen's theorem (1982) involve supremum of probabilities and supremum of covariances, which are difficult to estimate. Also Bolthausen proved his theorem under the assumption of absolute convergence of three series and positivity of a fourth series, while in our paper we prove convergence of all necessary series. In his theorem Bolthausen did not consider RG transformations but only normalized sums of a random variable on finite sets and he proved convergence in distribution of these sums to a single random variable. In our paper we prove convergence in distribution of RG transformations of a random field to another random field.
Mathematicians doing research in statistical mechanics try to create the mathematical structures that make foundation of physical theories. They appreciate rigorous, non-contradictory and transparent theories.
They also make effort to obtain simplest possible proofs for existing theorems. In our paper we use a new approach in proving the CLT for weakly dependent random fields; this approach is based on estimation of semi-invariants.
We apply the techniques of Malyshev and Minlos (1991;Malyshev, 1980) to estimate semi-invariants of a random field and we use these estimations to prove a generalization of the CLT to weakly dependent random fields on a lattice.
Semi-invariants are synonyms for cumulants and Ursell functions. We give the definition of semiinvariants and briefly describe their properties in section 2. In section 2 we also introduce other necessary concepts from probability theory and statistical mechanics and briefly prove some relevant lemmas.
In section 3 we state the main result of this paper: the central limit theorem for Gibbs random field transformed by RG, with a brief discussion of its meaning.
The rest of the paper develops techniques for proving the main theorem. In particular, in section 4 we prove an inequality about the number of links in a set with a symmetric binary relation and apply it to estimate semiinvariants of a random field (Estimation Theorem).
In section 5 we prove the main theorem. In subsections 5.1 and 5.2 we prove a series of lemmas, which lead to the direct proof of the main theorem in subsections 5.3 and 5.4. In particular, we find an expression for the limiting variance and show the equality to 0 of all other limiting semi-invariants of the random field transformed by RG. We complete the proof of the main theorem by applying Carleman theorem to the limiting distribution.

Semi-Invariants
Denote E(X) the expectation of a random variable X. Semi-invariant is a generalization of the concepts of expectation and covariance. The following is a slight modification of the definition in (Malyshev and Minlos, 1991), pg. 27-33.

Definition 2.1
Suppose X 1 ,..., X m are random variables on the same probability space and M = {1, 2,..., m} is the set of their indices. For any S ⊆ M, we denote S . We assume that the expectation of every such product is finite. A semi-invariant of random variables X 1 ,..., X m is: Semi-invariants characterize the distribution and dependence of random variables. Other terms for a semiinvariant are cumulant and Ursell function.

Example 2.1
Suppose X, X 1 , X 2 and X 3 are random variables.
Denote µ the expectation of X and σ the standard deviation of X. Then the following hold: 4) 〈X, X〉 = σ 2 , the variance of X. 5) 〈X, X, X〉/σ 3 equals the skewness of X.
Proof can be found in (Malyshev and Minlos, 1991).
The following is a well-known lemma about semiinvariants of normal distribution.

Definition 2.2
We say that random variables Y 1 ,..., Y m satisfy the mvariate Carleman condition if: ( For example, if X is bounded, then it satisfies Carleman condition. The logarithmic normal distribution does not satisfy Carleman condition and is not defined uniquely by its moments. In order to prove that random variables X 1 , X 2 ,..., X m with identical distribution satisfy the m-variate Carleman condition, it is sufficient to check the 1-variate Carleman condition for only one of the random variables.
We will use the following version of Carleman theorem.

Lemma 2.3
If a random variable Z has the standard normal distribution, then it satisfies the 1-variate Carleman condition.

Lemma 2.4
Suppose random variables X 1 , X 2 , ..., X m are uncorrelated, identically distributed and satisfy the following condition: Then X 1 , X 2 , ..., X m are independent and have a multivariate normal distribution.

Proof
Denote µ = 〈X i 〉, σ 2 = 〈X i , X i 〉. Consider independent random variables Z 1 , Z 2 , ..., Z m , where each Z i has the standard normal distribution and denote Y i = σZ i . Then Y 1 ,..., Y m are independent and have a multivariate normal distribution.
By Lemma 2.3 each Z i satisfies the 1-variate Carleman condition and so does each Y i . Since Y 1 ,..., Y m are identically distributed, they satisfy the m-variate Carleman condition.

Interaction Model in Statistical Mechanics
For the rest of the paper we fix a natural number v ≥1 and consider a v-dimensional integer lattice: with the distance between any two points s and t defined by: , the origin. Fix a set D⊆ℝ with at least 2 elements and denote Ω = {ω | ω: ℤ ν →D}. An element ω of Ω is called a configuration and is interpreted as a state of a physical system in statistical mechanics.
For each t∈ℤ ν a function X t : Ω→D is defined by the following: We define Σ as the σ-algebra generated by sets of the form {ω∈Ω | ω(t) ≤ a} for all t∈ℤ ν and a∈D. We fix a probability measure P 0 on (Ω, Σ) such that: for any a∈D, P 0 (ω(t) < a) does not depend on t (2) and for any a 1 ,…, a n ∈D and distinct t 1 ,…, t n ∈ℤ ν : P 0 (ω(t 1 ) < a 1 ,…, ω(t n ) < a n ) = P 0 (ω(t 1 ) < a 1 ) ⋅… ⋅ P 0 (ω(t n ) < a n ). ( Then {X t | t ∈ℤ ν } is an independent random field on the probability space (Ω, Σ, P 0 ) and this field is translation invariant. Clearly, the random variables X t , t ∈ℤ ν , are identically distributed with respect to the measure P 0 .
We also assume that the following conditions are satisfied: 1 2 3 th t each X has a finite moment of m order,m = , , ,…; (4) each X t satisfies the 1-variate Carleman condition: We denote 〈⋅〉 0 the expectation with respect to the measure P 0 .

Note 1
There always exists a probability measure P 0 satisfying (2) -(3) and for which {X t | t ∈ℤ ν } satisfies the conditions (4) -(5). Here is an example. Let F(x) be a probability distribution function satisfying Carleman condition, that is: As mentioned before, the normal and exponential distributions are some of the distributions satisfying Carleman condition.
Probability measure P 0 is defined by: Then the conditions (2) -(5) are satisfied.
We fix an increasing sequence Λ N of finite subsets of ℤ ν such that Λ N ⊂ Λ N+1 for any N∈ℕ and Denote R = {{s, t}| s, t∈ ℤ ν and ||s−t|| = 1}. R is the set of all pairs of neighbouring nodes in the lattice ℤ ν . Denote RB = {u∈ℤ ν | one coordinate of u is 1 and the others are 0}. RB is the standard basis in ℝ ν .

Definition 2.3
Interaction model is defined by a triple of objects The interaction model includes a set Λ N (as defined before), potential Φ and interaction energy U N defined as follows.
1) For any B∈R we define a random variable Φ B on the probability space (Ω, Σ, P 0 ). Any B∈R has the form B = {r, r + u}, where u∈RB, so we define: Such Φ B represents interaction between neighbours r and r + u.
2) Function U N : Ω→R is defined by the following: This completes the definition of the interaction model.

Note 2
In the interaction model a union of the random fields {X t | t ∈ℤ ν } and {Φ B | B∈R} is translation invariant. This means: for any t 1 ,..., t m , r ∈ℤ ν and any B 1 ,..., B n ∈R the random vectors The interaction model describes a physical system with many particles represented by points of the set Λ N in the integer lattice. The random field X t describes some property of the physical system. The function U N characterizes the interaction energy of the system and |λ| is proportional to the inverse temperature of the system.
The parameter λ also characterizes the strength of interaction between particles and we assume that only neighbouring particles interact.

Example 2.2
The statistical model with λ = 0 describes a physical system with no interaction between its elements, e.g., ideal gas.
Since D is finite, each X t satisfies the Carleman condition. The rest of the model is defined by: ,

Gibbs Modification
Gibbs modification was introduced in (Malyshev and Minlos, 1991). Since it is important for our paper, we also provide the definition.

Definition 2.4
For the interaction model given by (N, λ, ϕ) we define the associated probability space (Ω, Σ N , P λ, N ) as follows: 1. The sample space is the set Ω as defined before. 2. Σ N is the sigma-algebra generated by X t , t∈Λ N . 3. For any event A∈Σ N the probability is defined by: where I A denotes the indicator of event A. The probability measure P λ,N is called (finite) Gibbs modification on Λ N . This completes the definition.

Example 2.4
Ising model with parameters N, λ and h is usually defined with probabilities: Σ N is defined in the same way as in the interaction model and probability measure P G is defined by: where the expectations are with respect to the probability measure P.
We can define the Ising model as a particular case of interaction model. We take D = {-1, 1}; P 0 is defined by:  (3); Then we show that the probability space (Ω, Σ N , P λ,N ) is the same as (Ω, Σ N , P G ) in the usual definition of the Ising model. Proof
2. {X t | t∈Λ N } is an independent random field with respect to P 0,N .
3. The distribution of the random field {X t | t∈Λ N } with respect to Gibbs modification P 0,N does not depend on N. 2. This part holds because the random variables X t , t∈Λ N , are independent with respect to the measure P 0 .

Gibbs Measure and Thermodynamic Limit
The definition of Gibbs measure is given in (Dobrushin, 1968); it is a probability measure on (Ω, Σ). For our results it is sufficient to consider Gibbs measure P λ as the limit of Gibbs modifications P λ,N as N→∞: Malyshev and Minlos (1991) established necessary and sufficient conditions when the equality (7) holds; the results in (Kashapov, 1980) imply that the equality (7) holds for all λ with |λ| < C, where C is the constant from our main theorem (Theorem 3.1).
Let us see what happens to the interaction model and associated probability space when N→∞. Clearly, the finite set Λ N transforms into the lattice ℤ ν , Σ N transforms into Σ and the Gibbs modification P λ,N transforms into the Gibbs measure P λ .

Definition 2.5
The thermodynamic or macroscopic limit of interaction model is the lattice ℤ ν together with the limiting probability space (Ω, Σ, P λ ).
Clearly, {X t | t ∈ℤ ν } is a random field on the limiting probability space. We denote 〈⋅,…,⋅〉 λ semiinvariants with respect to the Gibbs measure P λ .

Renormalization Group
The following concept was introduced by Kadanoff (1966).

Definition 2.6
Fix a natural number k >1 and a real number α ≥ v.
For each r = (r 1 , r 2 ,..., r v )∈ℤ ν consider a cube k r C of side length k with vertex kr: A renormalization group (RG) with parameters k and α is a transformation that assigns to each random field {Z t RG is a scaling transformation. It allows to study a physical system at different distance scales, such as atomic and molecular levels. Details of its physical interpretation can be found in (Kadanoff, 2013).
We are interested in the distribution of the result

Theorem 3.1. (Main Theorem)
Consider the thermodynamic limit of interaction model with parameter λ. Suppose a renormalization group with parameters k and α transforms the random There exists a positive constant C such that for any |λ| < C the following hold.
1. Suppose α > v. Then the field ( ) k r Y → 0 in mean square as k →∞.
2. Suppose α = v. Then as k→∞, the random field { ( ) k i Y | r∈ℤ ν } converges in distribution to an independent random field with Gaussian distribution (i.e., any finite subset of the field has a multivariate normal distribution). Each of the variables of the limiting field has 0 expectation and the positive variance given by: where each coefficient V n is a finite sum of semiinvariants of X t and Φ B with respect to P 0 (t∈ℤ ν , B∈R). Exact formula for coefficients V n is formula (27) in subsection 5.4.
Proof is given in section 5. This theorem can be considered as a generalization of the classical Central Limit Theorem (CLT). Instead of a sequence of independent random variables we have a weakly dependent random field {X t | t ∈ℤ ν }. It is weakly dependent because |λ| is small and λ characterizes the strength of interaction.
The classical CLT considers a sequence of independent identically distributed random variables with finite variances and states that their normalized sums converge in distribution to a normal random variable. Theorem 3.1.2) also states convergence in distribution and that the limiting distribution is normal but in this case it is the distribution of an independent normal field.
In other words, Theorem 3.1 states: in systems with weak interaction the distribution of the normalized sums over big regions is approximately independent and normal.

Estimation Theorem
The proof of the main theorem in Section 5 is based on estimations of semi-invariants. In this section we prove an inequality (Theorem 4.1), which will be applied to estimating semi-invariants. This was inspired by Estimates of Intersection Number in (Malyshev and Minlos, 1991). Here we have improved our estimate from (Kachapova and Kachapov, 2015) and simplified the proof.
In this section we consider a countable set with a reflexive, symmetric binary relation. If elements a, b of are in this relation, we say that a and b are linked. Thus, any element of is always linked to itself (reflexivity). If a is linked to b, then b is linked to a (symmetry).
Denote l(a) the number of elements in that are linked to a. In this section we assume that there is a constant L such that l(a) ≤ L for all a∈ .

Theorem 4.1. (Estimation Theorem)
For any sequence α of elements of : where |α| = m is the length of the sequence α, n j = n j (α) and υ j = υ j (α).
This estimate cannot be improved.
Denote J = J(α) for brevity. We define a link matrix d ij as follows: Then for any j = 1,..., m: (9) Next we use Jensen inequality, which states for a concave function f and numbers x 1 ,..., x n in its domain: We apply the Jensen inequality to the concave function logarithm and The following example shows that the estimate cannot be improved. is an arbitrary countable set and m is any positive integer. We take α = (a 1 ,..., a m ), where all a i are distinct. We assume that any a i and a j are linked

Application of the Estimation Theorem to Semi-Invariants
First we introduce some notations. We take = R∪{{t}| t∈ℤ ν }. Two subsets T and S of ℤ ν are said to be linked if T∩ S ≠ ∅. By Lemma 2.5.4), if sets T and S are not linked, then they correspond to independent random vectors. So links correspond to possible dependencies of random vectors.
Any element of the form {t} is linked to itself and to 2v elements of the form {t, r}, so l({t}) = 2v +1. Any element of the form {r, s} is linked to elements {r}, {s}, 2v elements of the form {r, t} and 2v elements of the form {s, t}, so the total is 4v +1 (because the element {r, s} is counted twice). Then L = 4v +1 (each element of is linked to at most 4v +1 elements). 2. The number n i is called the multiplicity of element T i in the family .
3. We denote the length of the family as | |= n 1 + n 2 +… + n m and ! = n 1 ! ⋅ n 2 ! ⋅…⋅n m ! We use letters , Ψ ,... for families. The same elements T 1 ,..., T n ∈ can be represented as a sequence or a family. where the maximum is taken over all sequences of numbers k 1 ,…, k l with k 1 + … + k l = m.

Semi-Invariants with Respect to Gibbs Measure
In this subsection we prove a series of lemmas about estimates and semi-invariants and later we use these lemmas to prove the main theorem.
1. We define its associated graph G( Ψ ) as follows. 2. We say that the family Ψ connects a sequence τ of elements of ℤ ν if the associated graph G( Ψ ) is connected and the set of its vertices contains all elements of the sequence τ.
Thus, the associated graph has | Ψ | = n 1 +...+ n k edges. The mapping ֏ is a one-to-one mapping of families of elements of R to this type of graphs on ℤ ν .
A semi-invariant is a symmetrical functional, the order of random variables is not important. If a sequence β of elements of R reduces to a family Ψ , we denote 0 0

Lemma 5.1
If a family Ψ of elements of R does not connect a sequence τ of elements of ℤ ν , then . Fix a sequence β that reduces to the family Ψ . Suppose Ψ does not connect τ. There are two cases.
Case 2. All elements of τ are vertices of G but G is not connected.
Then G = G 1 ∪ G 2 , where G 1 and G 2 are disjoint graphs. Without loss of generality we can assume: τ = (t 1 ,..., t q , s 1 ,..., s l ) and β = (A 1 ,..., A m , B 1 ,...,B  The following lemma is mentioned by several authors without a proof or with a complicated proof. Here we provide a short, simple proof giving an explicit value for the estimation constant.

Lemma 5.2
Denote C 2 = 4v 2 . Fix a sequence τ of points in ℤ ν and a natural number n ≥ 1. The number of families Ψ such that | Ψ | = n and Ψ connects τ, is not greater than (C 2 ) n .

Proof
For a family Ψ = {(B 1 , n 1 ),..., (B k , n k )} consider the associated graph G = G( Ψ ). A new graph G′ is obtained from G by adding for every edge another edge with the same ends. So G′ has 2n edges. Each vertex of G′ has an even degree and G′ is connected, hence G′ has an Eulerian trail, that is a closed path which includes every edge of the graph exactly once; the length of such a path is 2n.
Therefore the number of the families with | Ψ | = n that connect τ, is not greater than the number of paths with 2n steps through τ going along edges of the lattice ℤ ν . There are at most 2v directions at each vertex. Therefore the number of such paths is not greater than (2v) 2n = (C 2 ) n for C 2 = (2v) 2 .
For a sequence τ = (t 1 ,..., t m ) denote , the semi-invariant with respect to P λ . In the Definition 2.4 we expressed the measure P λ,N in terms of measure P 0 . The following lemma describes a connection between semi-invariants with respect to these measures.

Lemma 5.3
Denote C 3 = (2C 1 C 2 ) −1 , where C 1 is the constant from Lemma 4.1 and C 2 is the constant from Lemma 5.2. Fix N > 1 and a sequence τ of points in Λ N . The following equality holds: where the finite inner sum is taken over all families Ψ = {(B 1 , n 1 ),…, (B k , n k )} such that | Ψ | = n, Ψ connects τ and each B i ⊂ Λ N . The series (13) converges absolutely and uniformly for λ∈[−C 3 , C 3 ].

Proof
The semi-invariant with respect to Gibbs measure can be expanded in Taylor series:  (14) can be found in (Malyshev and Minlos, 1991), pg. 34. Expanding (14) we get: The last sum is finite and is taken over all sequences β = (B 1 , B 2 ,…,B n ) of elements of R such that each B i ⊂ Λ N . In this sum we can take only the sequences β that connect τ because for others the corresponding addends equal 0 by Lemma 5.1. If n = 0, then a N,0 = 0 \ X τ .
For each family Ψ of length n there are ! ! n Ψ sequences that reduce to Ψ . Therefore where the sum is taken over all families Ψ = {(B 1 , n 1 ),…, (B k , n k )} such that | Ψ | = n, Ψ connects τ and each B i ⊂ Λ N . So we have proven the equality (13).

Lemma 5.4
Fix a sequence τ of points in ℤ ν . For any |τ | < C 3 (where C 3 is the constant from Lemma 5.3) the following equality holds: where the finite inner sum is taken over all families Ψ with | Ψ | = n that connect τ. The series (17) converges absolutely and uniformly for λ∈[−C 3 , C 3 ].

Proof
It is proven in (Malyshev and Minlos, 1991) that By taking the limit of both sides of (13) as N → ∞ we get the equality (17). Similarly to Lemma 5.3 we estimate the common term of the series (17), which proves its absolute and uniform convergence.

Lemma 5.5
If |λ| < C 3 (where C 3 is the constant from Lemma 5.3), then any finite set of the random variables X t , t∈ℤ ν , satisfy the Carleman condition with respect to measure P λ .
Similarly to Lemma 5.4, we can show that for any natural l: where the finite inner sum is taken over all families Ψ with | Ψ | = n that connect τ = (t).
Similarly to the proof of Lemma 5.3, we can estimate the coefficient for λ n :

Proof
If we shift a sequence (r 1 , r 2 ,…, r m ) by vector r, then we get a new sequence (r 1 + r, r 2 + r,…, r m + r).
This shifts all sequences τ = (t 1 , t 2 ,…, t m ) and families Ψ by vector kr in (21)  1 where τ = (t 1 , t 2 ,…, t m ) and the second sum is taken over all families Ψ with | Ψ | = n that connect τ. Then: where K(m) = C(m, X)(C 1 ) m m! and C(m, X) is defined in Lemma 4.1.

Proof
For a non-zero addend in the sum, | Ψ | = n and Ψ connects (t 1 , t 2 ,…, t m ), hence each of t 2 ,…, t m is a vertex in the associated graph G( Ψ ). So there are at most n +1 choices for each of them. By Lemma 5.2, there are at most (C 2 ) n families Ψ with | Ψ | = n that connect τ.
Using also Lemma 4.1, we get:

Proof
Case m = 1 has been considered in Lemma 5.6.1). So we assume m > 1. To estimate the semi-invariant we use the series (21) from Lemma 5.6 and consider the corresponding series of absolute values: Here the sum is taken over all sequences τ and families Ψ as in (21).
and it is sufficient to show that lim 0 k k A →∞ = .

Finding the limiting covariances
From here till the end of this section we consider only the case when α = v and m = 2. Other cases are investigated earlier.

Proof
We consider four cases.
Case 1: 1 0 r = and the first coordinate of r 2 is negative. Denote r = r 2 . Clearly, for any t = (t 1 ,..., t v ) ∈ 0 k C , we have: 0 ≤ t 1 ≤ k−1. Similarly, for any So the distance between such t and s is at least l + 1.
Next we prove: Proof of (32) Suppose k n s K ∈ . We will show that the sums (27) (27) there is an equal addend in (28). Similarly we can show that for every addend in (28) there is an equal addend in (27). The proof of (32) is completed.
Next we estimate the variance of ( )

Proof of the Main Theorem (Theorem 3.1)
Suppose |λ| < C, where C is the constant from Theorem 5.1.

Case α > v.
By Lemma 5.6.1) the limiting expectation of each ( ) k r Y is 0 and by Corollary 5.1 the limiting variance is 0. This proves part 1 of the theorem.