The log-Sobolev inequality for spin systems of higher order interactions
Takis Konstantopoulos Ioannis Papageorgiou

 
 
The log-Sobolev  inequality for higher order interaction   systems.]The log-Sobolev  inequality for spin systems of higher order interactions.




 Address: ^1 Takis Konstantopoulos:
              Department of Mathematical Sciences,   University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL, United Kingdom.   
Email: T.Konstantopoulos@liverpool.ac.uk 
  ^2  Ioannis Papageorgiou:
               Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC), Avenida dos Estados, 5001 - Santo Andre - Sao Paulo, Brasil. 
Email: papyannis@yahoo.com ,  i.papageorgiou@ufabc.edu.br   

[2010] 60E15 26D10   















 

We study the infinite-dimensional log-Sobolev inequality for spin systems
on  ℤ^d with interactions of power higher than quadratic.
We assume that the one site measure without a boundary e^-ϕ(x)dx/Z    satisfies a log-Sobolev inequality and  we determine conditions  so that  the infinite-dimensional Gibbs measure also satisfies  the inequality.
As a concrete application, we prove that a
certain class of nontrivial Gibbs measures with non-quadratic
interaction potentials on an infinite product of Heisenberg groups satisfy the
log-Sobolev inequality.

 

[
    Takis Konstantopoulos^1 
 Ioannis Papageorgiou^2
    
====================================================










§ INTRODUCTION


 

 

Coercive inequalities, like the logarithmic Sobolev, play an important role in the study of ergodic properties of stochastic systems.  The inequality is  associated with strong properties about the type and speed of convergence of Markov semigroups to invariant measures. In particular, in the field  of infinite dimensional  interacting spin systems, they provide a powerful tool in the examination of  the infinite volume Gibbs measures. In the current paper we give a first explicit description of spin systems with interactions that are higher than quadratic that satisfy the log-sobolev inequality,  and thus provide a first example in the bibliography of spin systems  with high order interactions that converge exponentially fast to equilibrium.

Our focus is on the  typical logarithmic Sobolev (abbreviated as log-Sobolev or LS) 
inequality  for probability measures governing systems of unbounded spins on  
the d-dimensional lattice ℤ^d
with nearest neighbour interactions of order higher than 2.  
The aim of this paper is to investigate conditions on the local specification function
so that the inequality can be extended from the single-site interaction free  measure
to the infinite-dimensional Gibbs measure, assuming that the latter exists.
One crucial assumption  is that the single-site without interactions (consisting only of the phase) measure satisfies a  log-Sobolev inequality.  In addition, we assume that the power of the interaction is 
dominated by that of the phase. 
As an application, we show that the log-Sobolev inequality holds for the infinite Gibbs measure on spin systems 
with values in  the  Heisenberg group _1.

 

The single-site space will be denoted by  
(colloquially, “spins take values in ”)
and Ω:=^^d.
For Λ a finite subset of ^d,
denote by ^Λ,ω a probability measure on ^Λ
that depends on the boundary conditions ω∈^∂Λ.
These probability measures (known as local specifications)
satisfy the usual spatial Markov property which imposes
sever restrictions on them, namely, they must, under natural
assumptions, be of  Gibbs type with a Hamiltonian that
can be split into two parts: the phases (depending on single
sites) and the interaction (depending on neighboring sites).
Denote by ^Λ,ω integration with respect 
to ^Λ,ω;
and use the  convention that the former symbol be used in place of the
latter; see, e.g., Guionnet and Zegarlinski <cit.>.
Critiria for a measure (also a   local specification with quadratic interactions) to
satisfy a log-Sobolev inequality uniformly has been investigated by
Zegarlinski <cit.>,
Bakry and Emery <cit.>, 
Yoshida <cit.>, Ané et al. <cit.>,  
Bodineau and Helfer <cit.>, Ledoux <cit.>, Helfer <cit.> and  Baudoin and Bonnefont <cit.>.
Furthermore, in Gentil and Roberto <cit.> the spectral gap inequality 
is proved, while     Gentil,  Guillin and  Miclo,  in 
 <cit.> and   <cit.>,    Gozlan, Roberto and  Samson in  <cit.> and Papageorgiou in <cit.>  studied  the modified log-sobolev inequality.  p-logarithmic Sololev inequalities have been studied by
	  Balogh,    Engulatov,  Hunziker and   Maasalo  in <cit.>.
 
For the single-site measure  on the real line
with or without  boundary conditions 
necessary and sufficient  conditions in order that
the log-Sobolev inequality be satisfied uniformly over the boundary
conditions ω are presented in  Bobkov and Götze <cit.>, 
Bobkov and Zegarlinski <cit.> and Roberto and Zegarlinski <cit.>.


The log-Sobolev inequality for the infinite-dimensional Gibbs measure on the
lattice is examined in Guionnet and Zegarlinski <cit.>, 
and Zegarlinski <cit.>, <cit.>. 
The problem of passing from single-site to infinite-dimensional measure,
in presence of quadratic interactions,
is addressed by Marton <cit.>, Inglis and Papageorgiou <cit.>, 
Otto and Reznikoff <cit.> and Papageorgiou <cit.>.


Working beyond the case of quadratic interactions is the scope of
this paper.
Non-quadratic interactions have been considered in <cit.>, but for the
case of the one-dimensional lattice and the stronger log-Sobolev q-inequality.
In that paper, the inequality for the infinite-dimensional Gibbs measure 
was related to the inequality for the
finite projection of the Gibbs measure. In <cit.> conditions have been investigated so that the infinite dimensional Gibbs measure satisfies the inequality under the main assumption that the single-site measure satisfies a log-Sobolev inequality uniformly on the boundary conditions. Under the same framework, concentration properties have been studied in <cit.>.

The scope of the current paper is to  prove the log-Sobolev inequality for the Gibbs measure without setting conditions  neither on the local specification {^Λ,ω} nor on the one site measure 𝔼^{i},ω. What we actually show is that under appropriate conditions on the interactions,  the Gibbs measure  satisfies a log-Sobolev inequality whenever the boundary free one site measure μ(dx)=e^-ϕ(x)dx /(∫ e^-ϕ(x) dx)  satisfies a log-Sobolev inequality.
 In that way we improve the previous results since the log-Sobolev inequality is determined alone by the phase  ϕ of the simple without interactions measure μ on M, for which a plethora of criteria and examples of good measure that satisfy the inequality exist.

To explain the applicability of our general infinite-dimensional framework the
specific case of the Heisenberg group is presented. This will serve as a
specific example (see Theorem <ref>) derived from the more general
result of Theorem <ref>.






 §.§ General framework

Consider the d-dimensional integer lattice ^d equipped with 
the standard neighborhood
structure: two lattice points (sites) i, j ∈^d are neighbors (write i ∼ j)
if ∑_1≤ k ≤ d |i_k-j_k|=1. We shall be working with the configuration
space Ω = ^^d where  is an appropriate “spin space”. We consider the spin space  to be a group, and we denote · the group operation and  x^-1 the inverse of x∈ in respect to the group operation.
The coordinate ω_i of a configuration ω∈Ω is referred to
as the spin at site i; ω_i takes values in ^i ≡.
When Λ⊂^d we identify ^Λ
with the Cartesian product of the ^i when i ranges over Λ.
We assume that  comes with a natural measure; for example,
when  is a group then the measure is one which is invariant under
the group operation; we write dx_i for this measure on the copy
^i of  corresponding to site i ∈^d; 
and we use the symbol dx_Λ for a product measure, that is,
the product of the dx_i, i ∈Λ.    
It is assumed that ^{i},ω is absolutely continuous
with respect to dx_i. The Markov property implies then that,
for finite subsets Λ of ^d, the probability measures
^Λ, ω should be of a very special form (see <cit.>):

    ^Λ, ω (dx_Λ) = 1/Z^Λ,ω 
    e^-H^Λ,ω(x_Λ)  dx_Λ,

where Z^Λ,ω is a normalization constant
and where the function H^Λ, ω (the Hamiltonian) is of 
the form

    H^Λ,ω(x_Λ) 
    := ∑_i ∈Λϕ(x_i) 
    + ∑_i,j ∈Λ,  j ∼ i J_ij V(x_i, x_j)
    + ∑_i∈Λ,j ∈∂Λ,  j ∼ i J_ij V(x_i, ω_j),

the sum of the phase and the interactions.


It is implicitly assumed that the normalization constants are finite.
Several conventions are tacitly used in this business. 
When f is a function from ^^d into ,
we let ^Λ, ω f for the function
on ^^d obtained by integrating f with respect
to dx_Λ and by substituting x_∂Λ by ω,
while leaving all other coordinates the same.
When we simply write ^Λ f we shall understand this as 
above with ω = x_∂Λ. 
Thus, ^Λ can be thought of as a linear operator  that
takes functions on the whole of ^^d to functions
that do not depend on the variables x_i, i ∈Λ.
Similarly, we will write H^Λ for the Hamiltonian  H^Λ,ω.  If Λ is an  infinite subset of ^d with the property
that any two points in Λ are at lattice distance strictly
greater than 1 from one another then ^Λ, ω
is the product of ^{i}, ω_∂{i}.
Using these conventions, the spatial Markov property can then 
be expressed as

    ^Λ^K = ^Λ,    K ⊂Λ.

The Markov property written in this way, following the conventions above,
carries a lot of weight: in particular, it entails  that the law
of x_K given x_∂Λ is the
law of x_K given x_∂ K integrated over
x_∂ K when the later has the law obtained from ^Λ.
This Markov property can, naturally,   be seen to be 
equivalent to the usual Markov property for Markov
processes indexed by the one-dimensional lattice  (which
is often interpreted as “time” in view of the natural
total order of .)

We say that the probability measure ν on Ω = ^^d
is an infinite volume Gibbs measure for  the local
specifications {^Λ,ω} if
the Dobrushin-Lanford-Ruelle equations are satisfied:

    ν𝔼^Λ,∙=ν,   Λ⋐^d,

that is, if ν is an invariant measure for the Markov
random field.
We  refer to Preston <cit.>, Dobrushin <cit.> and  Bellisard and   Hoegn-Krohn <cit.> for details.
Throughout the paper we shall assume that we are in the case
where ν exists and is unique (although uniqueness can be
deduced from our main results).





We next make some assumptions about the nature of 
the spin space . 

We shall assume that  is a nilpotent  Lie group on
^d with Hörmander system X^1, …, X^n, n ≤ d,
satisfying the following relation:
if X^k = ∑_j=1^d a_kj∂/∂ x_j,
k=1,…, n, then a_kj is a function of x ∈^d
not depending on the j-th coordinate x_j; that is,
if x, y ∈^d have x_j=y_j then a_kj(x)=a_kj(y).
The gradient ∇ with respect to 
this system is the vector operator ∇ f = (X^1 f, …, X^n f),
whereas Δ = (X^1)^2 + ⋯ + (X^n)^2 is the
sublaplacian, where (X^k)^2 f = X^k(X^k f). 
We let ∇ f^2 := (X^1 f)^2 + ⋯ + (X^n f)^2 (for general hypocoercive-typeoperators type generators see Kontis, Ottobre   and  Zegarlinski  <cit.> and  <cit.>).
When these operators act
on functions on the spin space ^i at site i ∈^d 
they will be denoted by ∇_i and Δ_i, respectively. 
If Λ is a finite subset of  ^d we shall let 
∇_Λ := (∇_i, i ∈Λ) and 
∇_Λ f^2 := ∑_i ∈Λ∇_i f^2.
We shall assume that  comes equipped with a metric-like
function (x,y), x,y ∈.
For example, if  is a Euclidean space then  is the Euclidean
metric. If  is the Heisenberg group, 
then  is the Carnot-Carathéodory metric. 
More generally, the role of  only appears through
the assumptions we make.

In each and every case, the notation (x), for x ∈, stands
for (x, 0), where 0 is a special point of , for
example the origin if  is ^m or the identity element
if  is a Lie group.

The main assumption of the paper is that  the single site measure without interactions  (consisting only of the phase) 

    μ(dx)=e^-ϕ(x)dx/∫ e^-ϕ(x) dx
  
satisfies the log-Sobolev inequality, that is,
that there exists c>0 such that

    μ( f^2 logf^2/μ f^2) ≤ c μ∇ f^2

for any smooth function f: → such  that
 both sides make sense.



When the last inequality holds for ^Λ,ω in the place of μ for the constant c uniformly on the boundary conditions ω,  we say  that the log-Sobolev inequality holds for
^Λ,ω uniformly (in ω.)

We point out that when two measures satisfy the log-Sobolev inequality then their product also satisfies the inequality.   Similar thing is also true for 
spectral gap inequalities (a measure μ satisfies spectral gap inequality
with constant C if
μ |f-μ f|^2 ≤ C μ |∇ f|^2).

Proofs of these assertions can be found in  
Gross <cit.>, Guionnet and Zegarlinski <cit.> and 
Bobkov and Zegarlinski <cit.>. 
In that way, if for every i∈Λ, ^{ i },ω satisfies the log-Sobolev (similarly the Spectral gap) inequality uniformly and  Λ is a subset (finite or infinite) of ^d such that
any two points of Λ are at lattice distance strictly greater than one from
one another, then the log-Sobolev (similarly spectral gap) inequality holds for
^Λ,ω, with the same constant c, uniformly
in ω∈∂Λ.






 §.§ The Heisenberg group

The Heisenberg group _1 can be identified with 
ℝ^3 equipped with the group operation

    x·x̃ = 
    (x_1 + x̃_1, x_2 + x̃_2, x_3 + x̃_3 
    + 1/2(x_1x̃_2 - x_2x̃_1)).

It is a Lie group with Lie algebra 
which can be identified with the space of left-invariant vector 
fields on _1 in the standard way.  
See, e.g., <cit.>.
By direct computation, the
vector fields

    X_1    =   ∂_x_1 - 1/2x_2∂_x_3
    
    X_2    =   ∂_x_2 + 1/2x_1∂_x_3
    
    X_3    =   ∂_x_3 = [X_1, X_2] ,

where ∂_x_i denoted derivation with   respect to x_i, form a Jacobian basis.
From this it is clear that X_1, X_2 satisfy the Hörmander  condition 
(i.e., X_1, X_2 and their commutator [X_1, X_2] span the tangent space 
 at  every point of _1).   It is also 
easy to check that the left-invariant Haar measure 
(being also right-invariant measure owing to the fact that 
the group is nilpotent) 
is the Lebesgue measure on ^3. 


The gradient  is given by
∇ := (X_1,X_2),
and  the sub-Laplacian  by 
Δ := X_1^2 + X_2^2. 
A probability measure μ on _1
satisfies a log-Sobolev inequality if there exists
a positive constant c such that


    μ(
    f^2logf^2/μ f^2) ≤ c μ∇ f^2= c μ((X_1f)^2+(X_2 f)^2),

for all smooth  functions f: _1 →. Here, μ(g), or,
simply, μ g
stands for ∫__1 g   dμ.
The quantity on the left-hand  side is the μ-entropy
of the function f^2 or, equivalently, the Kullback-Leibler
divergence between the measure f^2 dμ and μ.
For example, the family of measures

    μ_p (dx) 
    := e^-β(x,e)^p/∫__1 e^-β(x,e)^pdx dx,

where p≥2, β >0,
and (x,e) is the Carnot-Carathéodory distance
of the point x ∈_1 from the identity element e of _1,
all satisfy a log-Sobolev inequality; this was shown by Hebisch and Zegarlinski in <cit.>.  For the inequality on Lie groups see  also Feng and Li <cit.>, Gordina and Luo <cit.>,  Chatzakou,  Kassymov and  Ruzhansky  <cit.> and Bonnefont,    Chafaï and  Herry <cit.>. Entropy dissipation  in the Heisenberg group has been studied by  Feng and   Li in <cit.>.

We briefly recall the   notion of the Carnot-Carathéodory metric on .

A Lipschitz curve
γ:[0,1]→ is said to be admissible if γ'(s) =
a_1(s)X_1(γ(s)) + a_2(s)X_2(γ(s)), a.e., for 
given measurable functions a_1(s), a_2(s),
and has length
l(γ) = ∫_0^1(a_1^2(s) + a_2^2(s))^1/2ds.
The Carnot-Carathéodory metric is then defined by

    (x,y) := inf{l(γ):γ is an admissible path joining xand y}.


We also have that x=(x_1, x_2, x_3)↦(x,e) is smooth 
for (x_1, x_2) ≠ 0, but has singularities at points of the form
(0,0, x_3). 
 Thus, the unit ball in the metric above has singularities on the x_3-axis.
In our analysis, we will  use the following result about the
Carnot-Carathéodory distance (see, for example, <cit.>, <cit.>).



Let ∇ be the  gradient and Δ be the sub-Laplacian on _1.
Then ∇(x,e) =1 for all x=(x_1, x_2, x_3)∈ such that
(x_1, x_2)≠0. Also there exists a positive constant K such  that Δ(x,e)<K/(x,e) in the sense of distributions.

 






§ ASSUMPTIONS AND MAIN RESULTS 


In this section we present the  hypothesis and the statement of the main result.  Without loss of generality, assume the single-site space to be the origin 0 ∈^d. Let  be the corresponding spin space. To ease the notation,
we denote the Hamiltonian by

    H(x) = ϕ(x) + ∑_j=-d^dJ_j V_j(x),    x∈,

where e_j ∈^d is the vector with components
e_j,i = _i=j and V_± j(x) := V(x, ω_± e_j),
j=1,…,d. In other words, we freeze the boundary conditions
ω_-e_d, …, ω_e_d at the 2d neighbors 
± e_1, …, ± e_d of the origin.
Of course, we need to assume that the functions ϕ and V_j
are such that ∫_exp(-H(x)) dx < ∞ so that
the measure with density exp(-H(x)) be normalizable to
a probability measure which (again suppressing the ω) we
simply denote as :

    (dx) = Z^-1 e^-H(x) dx.

Before stating the main results, we introduce a number of natural
hypotheses. 



  §.§.§ The main assumption


 The single site measure without interactions  (consisting only of the phase) 

    μ(dx)=e^-ϕ(x)dx/∫ e^-ϕ(x) dx
  
 satisfies
the log-Sobolev inequality with a constant c.



  §.§.§ Assumptions on the  phase and the interaction potential


We also assume that   J_j>0 and  that ϕ and the V_j are non negative twice  continuously
differentiable satisfying the following “geometric” conditions:
there exists a nonnegative 
C^2 function ϕ_1 
such that

    ∇ϕ = ϕ_1 ∇ .

Similarly, for each V_j:

    ∇ V_j=  U_j ∇,

where U_j are nonnegative C^2 functions.
 The gradient vector ∇ is uniformly bounded in magnitude from above and below:
there exist constants τ and ξ such that, for all
x ∈,

    ξ≤∇≤τ .

Instead of speaking of a metric , we shall, for the purposes of
this section, speak of positive functions ,
such that there exists a constant θ with

    |Δ| ≤θ/,

for all j and all x.
Moreover, we require that there exists k_0 >0 and 
p ≥ 2 
such that

    k_0 ϕ≤ dϕ_1  and d^p≤ϕ

and

    k_0V_j≤ U_j,

for all j and x.
 Furthermore,  we assume 
 
    V_j → +∞ as    (ω_ e_j)→ +∞

and that  
 ∃  s ≤ p  and k>0 such that

    ∇ V_j ^2 ≤ k + k ^s + k ^s(ω_ e_j),


    V_j ≤ k + k ^s + k ^s(ω_ e_j),

Three last  assumptions follow.  These, as shown in section <ref>,  are natural assumptions that are easily verified for Hamiltonians that are given as functions of .  For any x,y∈ we assume that there exists a λ>1 such that

    H(x · y)  ≤λ H(x)+λ H (y)

where · the group operation,  while for x^-1 the inverse of x in respect to the group operation,

    H(x^-1)= H (x)
    .

If we consider γ:[0,t]→ a geodesic from 0 to x ∈ then

    H(γ (s))≤ H (x)
 for every s∈ [0,t].


We can now state the main theorem related to the general framework.


 
Let f^ℤ^d→ℝ. 
If assumptions (<ref>)-(<ref>) hold,
if the single-site measure μ satisfies
a log-Sobolev inequality, 
 then the Gibbs measure
ν satisfies a log-Sobolev inequality:

    ν f^2logf^2/ν
    f^2≤ℭ ν∇ f
    ^2,

for some positive constant ℭ. 
 
tocsubsubsectionThe main theorem

The main   assumption about the   phase ϕ is  that the  single site
measure μ satisfies the log-Sobolev
inequality, while the main
assumption about the interactions
is that the phase ϕ(x) dominates over
the interactions, in the sense that

    ∇ V(x_i,ω_j)^2≤ k+k(
    d^s(x_i)+d^s(ω_j))≤ k+k( ϕ(x_i
    )+ ϕ(ω_j ))

for s≤ p.  
 

   
We briefly mention some consequences of this result. 

Let ν be as in Theorem <ref>. Then ν satisfies the spectral gap
inequality

    ν ( f-ν f )^2 ≤ℭν∇ f
    ^2

where ℭ is as in Theorem <ref>.

The proofs of the  next two can be found in [B-Z].

Let ν be as in Theorem <ref> and suppose f:Ω→ is such
that ∇ f  ^2 ∞ < 1. Then

    ν( e^λ f) ≤exp{λν(f) + ℭλ^2}

for all λ>0 where ℭ is as in Theorem <ref>.
Moreover, the following 'decay of tails' estimate holds true

    ν{| f - ∫ fdν|≥ h}≤
    2exp{-1/ℭh^2}

for all h>0.

Suppose that our configuration space is actually finite dimensional, so that we
replace ^d by some finite graph G, and Ω = ()^G. Then
Theorem <ref> still holds, and implies that if ℒ is a
Dirichlet operator satisfying

    ν(fℒ f) = -ν( |∇ f| ^2),

then the associated semigroup P_t = e^tℒ is ultracontractive.

For further properties of Markov semigroups on the lattice see Won <cit.>.
Next, we present an example of a measure that satisfies the
hypothesis of  Theorem <ref>.



 §.§ The Case of Heisenberg Group




 As an  example of a measure  ^i,ω that satisfies
 the conditions of Theorem   <ref>
 one can consider the following  measure on the Heisenberg group 

    𝔼^Λ,ω(dX_Λ) =
    e^-H^Λ,ωdX_Λ/Z^Λ,ω
 
  where for any Λ⋐^d,ω∈Ω the  Hamiltonian is defined as 

    H^Λ,ω(x_Λ) =
    ∑_i∈Λ^p(x_i) +
    δ∑_i∈Λ ,j∼ i((x_i)+(ω_j))^r

for δ>0 and p,r∈ℕ s.t. p+2/2≥ r>2, where  the
Carnot-Carathéodory  distance. Then the main result related to the infinite
volume Gibbs measure associated with this local specification follows:
 
Consider  the Heisenberg group and let
f^ℤ^d→ℝ. If
{𝔼^Λ,ω}_Λ⋐^d,ω∈Ω as in (<ref>). Then the infinite-dimensional Gibbs measure
ν for the local specification
{𝔼^Λ,ω}_Λ⋐^d,ω∈Ω satisfies the log-Sobolev  inequality

    ν f^2logf^2/ν f^2≤ℭ ν|∇
    f
    |^2
                              
for some positive constant ℭ. 


A few words about the structure of the paper. In section <ref> we show a coersive inequality as well as the Poincare inequality for the one site measure ^i. In the next section we present the first  sweeping out inequalities and show convergence to equilibrium, while in section <ref> a weak logarithmic Sobolev inequality is obtained for the  one site measure ^i.  Further sweeping out inequalities are obtained in section <ref> together with a log-Sobolev inequality for the product measure. In the next section <ref> we gather all the previous bits together to prove  the main result of Theorem <ref>. Finally, in section <ref> we present the proof of Theorem <ref>.






§ A COERCIVE INEQUALITY FOR THE SINGLE-SITE SPACE
 
In this section we present a single-site coercive inequality that will provide the main tool in order to control the higher interactions. This  coercive inequality is on the line of the U-bound inequalities presented in <cit.> in order to prove log-Sobolev inequalities on a typical analytic framework (see also  <cit.>, <cit.>  and  <cit.>).  Furthermore, as we show in Lemma <ref> this coercive inequality  will imply  the spectral gap inequality for ^i,ω uniformly on ω. In a recent work, weak U-bound inequalities have been used to prove Spectral Gap inequalities (see <cit.>).
 
Under   assumptions (<ref>)-(<ref>), there exists C_0>0 such that,
for all r ≤ p,

    ^r f^2 ≤ C_0|∇ f|^2 + C_0 f^2,

and

    Hf^2≤ C_0  |∇ f|^2+C_0 f^2

for any smooth function f with compact support.


It is clear that it suffices to prove the inequality for r=2(p-1).
Indeed, if ^2(p-1) f^2 ≤ C  |∇ f|^2 + C  f^2
holds then for all r ≤ 2(p-1) we have
^r f^2 =  [^r f^2; d ≤ 1] +  [^r f^2; d > 1]
≤ f^2 + ^2(p-1) f^2 ≤ C  |∇ f|^2 + (C+1)  f^2.
By homogeneity, in all calculations, we will forget the normalizing constant
Z and think of (dx) as being equal to e^-H(x) dx.
In other words, we may, without loss of generality, assume that Z=1.
Let f be a smooth function with compact support and write

    |∇ f|^2 = ∫ |∇ f|^2 e^-H dx.

Since 

    ∇ (f e^-H) = (∇ f) e^-H 
    -  (∇ H) e^-H f,

upon taking the inner product   with ∇ on both sides we get

    ⟨∇,∇ H⟩ e^-H  f=⟨∇,∇ f⟩ e^-H-⟨∇, ∇ (f e^-H)⟩ ,

Hence,

    ⟨∇,∇ H⟩ f__1=   ⟨∇,∇ f⟩-∫⟨∇, ∇ (f e^-H)⟩ dx  
    ≤   |∇||∇ f|-∫⟨∇, ∇ (f e^-H)⟩ dx
     
    ≤   τ |∇ f|- ∫⟨∇, ∇ (f e^-H)⟩ dx__2

where above we used  (<ref>). Let X be any of the Hörmander generators of . 
Then, by the structural assumption,
we have the integration-by-parts
formula

    ∫ F (XG) dx = - ∫ (XF) G dx

for smooth functions F and G with  compact support. As a consequence,
the integration-by-parts formula

    ∫ f ⟨∇Φ, ∇Ψ⟩ dx
    = - ∫⟨∇Φ, ∇ f ⟩Ψ dx
    - ∫ (ΔΦ) Ψ f dx,

holds, and so

    _2 
    =∫⟨∇, ∇ (f e^-H)⟩ dx=-∫|∇| f e^-H dx-∫(Δ)f e^-H dx
    ≥- τ f-θ f

because of  (<ref>) and (<ref>). Since H = ϕ+∑_j J_jV_j, the  first term is

    _1 
        = ⟨∇,∇ H⟩ f=⟨∇,∇ϕ⟩ f+∑_jJ_j⟨∇,∇ V_j⟩ f= 
    
       = ϕ_1 |∇| f+∑_jJ_j U_j|∇| f ≥ξ k_0ϕ f+ξ k_0∑_jJ_j V_j  f

where above we used at first (<ref>)-(<ref>) and in the last inequality (<ref>), (<ref>) and (<ref>). Combining all that  we arrive at

    ϕ f+∑_jJ_j  V_j  f≤ 1/ξ k_0
      (τ |∇ f|+τ f+θ f)

 If we replace f by f^2  and use   Cauchy-Schwarz  inequality we obtain

    Hf^2≤    1/ξ k_0(2τ f |∇ f|+ τ f^2+θ f^2 )
      ≤1/ξ k_0(τ  |∇ f|^2+τ^2f ^2+τ f^2+θ f^2)  
      =    1/ξ k_0{τ  |∇ f|^2+τ(I_{4τ/ξ k_0≤^p-2} +I__{4τ/ξ k_0>^p-2})^2f ^2+
        +τ(I_{4τ/ξ k_0≤^p-1}+I_{4τ/ξ k_0>^p-1}) f^2+ θ f^2}
    ≤   1/ξ k_0{τ  |∇ f|^2+ξ k_0/2^pf ^2+ (4τ(4τ/ξ k_0)^2/p-2+4τ(4τ/ξ k_0)^1/p-1+ θ)  f^2}
    ≤   1/2 Hf ^2+
      1/ξ k_0{τ  |∇ f|^2+ (4τ(4τ/ξ k_0)^2/p-2+4τ(4τ/ξ k_0)^1/p-1+ θ)  f^2}

since  d^p≤ϕ≤ H, because of
   (<ref>) and the non negativity of ϕ and V_j. Again, for the same reason we obtain

    ^p  f^2≤ Hf^2≤2/ξ k_0{τ  |∇ f|^2+ (4τ(4τ/ξ k_0)^2/p-2+4τ(4τ/ξ k_0)^1/p-1+ θ)  f^2}

which proves  the  inequality.
 We will now prove the  Poincare inequality for the one site measure ^i for a constant uniformly on the boundary conditions. For the Poincare inequality on Carnot Group see  Chatzakou, Federico and Zegarlinski (<cit.> and <cit.>) and  Dagher, Qiu, Zegarlinski and Mengchun Zhang  <cit.>. The proof follows closely  the proof of   the local Poincaré inequalities from <cit.> and <cit.>.
Under   assumptions (<ref>)-(<ref>),  ^i,ω satisfies the spectral gap inequality

    ^i(f-^if)^2≤ c_p^i ∇_if^2

for some constant c_p>0 uniformly on the boundary conditions.


We denote  set V(R)={x_i:H ^i≤ R}. Then,  if we  define  a(f)=1/| V_R|∫_V_Rf(x_i)dx_i, where | V_R|= ∫_V_Rdx_i,  we can  compute  

    ^i(f-a(f))^2=^i(f-a(f))^2I_V(R)__1+^i(f-a(f))^2I_V(R)^c__2

where V(R)^c the complement of V(R). Since ϕ, V and J_ij are all no negative, H^i≥0 and so the first term is 

    _1≤1/Z^i∫_V(R)( f(x_i)-a(f))^2dx_i.

If we now  use the invariance  of the  measure dx_i with  respect to the group operation we can write a(f)=1/| V(R) |∫ f(x_iz_i)I_V(R)(x_iz_i)dz_i. If we substitute this expression on the last inequality and use Cauchy-Schwarz inequality we obtain 


    _1≤1/Z^i1/| V(R) |∫( f(x_i)-f(x_iz_i))^2I_V(R)(x_iz_i)I_V(R)(x_i )dz_idx_i

where above we also considered R large enough so that | V(R)|>1, i.e. 1/| V(R) |^2≤1/| V(R) |. 
Consider  a geodesic  γ:[0,t]→ H  from 0 to z_i, such that |γ̇(t) |≤ 1. Then      we can write 

    ( f(x_i)- f(x_iz_i))^2 =   (∫_0^(z_i)d/ds f(x_iγ(s))ds)^2=(∫_0^(z_i)∇_i f(x_iγ(s))·γ̇(s)ds)^2  
    ≤   (z_i)∫_0^(z_i)∇_i f(x_iγ(s))^2 ds .

From  the last inequality, we can  bound

    _1≤1/Z^i1/| V(R)|∫(z_i)∫_0^(z_i)∇_i f(x_iγ(s))^2 dsI_V(R)(x_iz_i)I_V(R)(x_i )dz_idx_i.

We observe  that for   any x_i  ∈ V(R) and x_i z_i∈ V(R) we obtain 

    H^i(z_i)=H^i(x_i^-1x_iz_i)≤λ H^i(x_i^-1 )+λ H^i( x_iz_i) ≤λ +λ H^i(x_i  )+H^i( x_iz_i)≤2Rλ_:=r_1
 
because of (<ref>) and (<ref>). Furthermore, using (<ref>) and (<ref>)   we can calculate

    H^i( x_i γ(s))≤λ H^i(x_i)+λ H^i(γ(s))≤λ H^i(x_i)+λ H^i(z_i)< (2λ^2 +λ )R_:=r_2.

From  (<ref>), since (z_i)≤ϕ(z_i)^1/p   we can also bound

    (z_i)≤ϕ(z_i)^1/p≤ H^i(z_i)^1/p≤ r_1^1/p.

So, we get

    _1   ≤  r_1^1/p/Z^i| V(R) |∫∫∫_0^(z_i)∇_i f(x_iγ(s))^2  ℐ_V(r_2)( x_i γ(s))ℐ_V(r_1)(z_i )dsdz_idx_i.

Using again the  invariance   of  the measure we can write

    _1   ≤ r_1^1/p/Z^i| V(R)|∫∫∫_0^(z_i)∇_i f(x_i)^2   ℐ_V(r_2)(x_i)ℐ_V(r_1)(z_i )dsdx_idz_i
         = r_1^1/p/Z^i| V(R)|∫∫(z_i)∇_i f(x_i)^2  ℐ_V(r_2)(x_i)ℐ_V(r_1)(z_i )dx_idz_i .

Notice that for    z_i  ∈ V(r_1)  one can bound as before  (z_i)≤ϕ(z_i)^1/p≤ H^i(z_i)^1/p≤ r_1^1/p, and so

    _1   ≤ r_1^2/p/Z^i| V(R)|∫∫(∇_i f(x_i)^2  ℐ_V(r_2)(x_i))dx_iℐ_V(r_1)(z_i )dz_i 
       ≤ r_1^2/p| V(r_1)|/| V(R)| Z^i,ω∫∇_i f(x_i)^2   ℐ_V(r_2)(x_i)dx_i .

Since, for x_i ∈ V(r_2), we    have e^-H^i,ω≥ e^-r_2, the last quantity can be   bounded by 

    _1 ≤ e^r_2r_1^2/p| V(r_1)|/| V(R)|𝔼^i∇_i f^2   .

If now we take under account that | V(r_1)|/| V(R)|=| V(2Rλ)|/| V(R)|≥ 1, as well as that because of (<ref>), the limit | V(2λ R)|/| V(R)|→1 as ∑_j∼ i(ω_j)→∞,
we then  observe that  |  V( r_1)|/| V(R)| is bounded from above uniformly on ω from a constant. Thus, we finally obtain that

    _1   ≤  C(R) 𝔼^i∇_i f(x_i)^2

for some positive  constant C(R).


We will now compute _2. We have 

    _2≤^i(f-a(f))^2H^i/R≤C_0/R^i |∇ f|^2+C_0/R ^i(f-a(f))^2

where above we used  Lemma <ref>.
 Combining  all the above we obtain
 
    ^i(f-a(f))^2≤(C(R)+C_0/R)^i ∇ f^2+C_0/R ^i(f-a(f))^2

For R large   enough so that C_0/R<1 we get

    ^i(f-a(f))^2≤C(R)+C_0/R/1-C_0/R^i ∇ f^2.

Since ^i(f-^i(f))^2≤4 ^i(f-k)^2 for any real number k, the result follows.



§ SWEEPING OUT INEQUALITIES AND  CONVERGENCE TO THE GIBBS MEASURE
 

Recall  the  definition of the  operator ^Λ
and the definition of ∇_j f as being the gradient of a 
function f:^^d→ with respect to the coordinate
ω_j ∈^{j}.
Also, recall the assumption that there is a unique Gibbs measure ν.
By our notational conventions, for i ∈^d, the quantity ^i f
is a function on ^d that depends only on the 2d variables
x_j, with j∈^d ranging over the neighbors of i and the x_j's that comprise the input of   f excluding x_i.
Fixing a neighbor j, the 
gradient ∇_j ^i f is then gradient with respect
to x_j. Denoting by X_j^1, …, X_j^n the Hörmander
system for ^{j}, we have
∇_j ( f) = (X_j^1 f, …, X_j^n f),
so that ∇_j (^i f)^2 = ∑_α=1^n (X^α_j ^i)^2.
We have


Suppose that (<ref>)-(<ref>) hold.  Let i,j ∈^d be neighbors.
Then there are constants D_1>0 and 0<D_2< 1 such that

    ν∇_j (^i f)^2 ≤ D_1 ν∇_j f^2
    + D_2 ν∇_i f^2 .



Fix i ∈^d and let j be one of its neighbors.
We compute (X^α_j (^i f))^2. Letting ρ_i be the 
density of ^i with respect to dx_i, we have,
using Leibniz' rule and (a+b)^2 ≤ 2 a^2+2 b^2,

    (X^α_j (^i f))^2 
       =( 
    ∫ρ_i (X^α_j f) dx_i
    + ∫ (X^α_j ρ_i) f dx_i  
    )^2
    ≤ 
    2 ^i (X^α_j f)^2
    
    + 2( ∫ (X^α_j ρ_i) f dx_i)^2,

where we used Jensen's inequality to pass in the square inside the
expectation in the first term.
If we sum over α and integrate over ν, the first term
on the right becomes ν∇_j f^2, which is what we need.
For the second term, we need to take into account the specific form of
the density ρ_i = e^-H^i/Z^i. Note that H^i depends on x_i
and the variables x_ℓ, where ℓ ranges over the neighbors
of i, including j, but Z^i does not depend on x_i. 
Taking this into account and using Leibniz' rule again, 
we easily arrive at[
The computation is as follows:
X_j ρ_i = (X_j e^-H^i)/Z^i - e^-H^i(X_j Z^i)/ (Z^i)^2.
But X_j e^-H^i = - e^-H^i (X_j H^i), and
X_j Z^i = X_j ∫ e^-H^i dx_i = - ∫ e^-H^i(X_j H^i) dx_i.
So X_j ρ_i = -(e^-H^i/Z^i) (X_j H^i)
+ (e^-H^i/Z^i) ∫ (e^-H^i/Z^i) (X_j H^i) dx_i 
= -ρ_i (X_j H^i) + ρ_i ∫ρ_i (X_j H^i) dx_i.
]

    ∫ (X^α_j ρ_i) f dx_i
    = - ^i[ f(X^α_j H^i - ^i (X^α_j H^i))]
    = - ^i[(f-^i f)  (X^α_j H^i)]
 
At this point, we use Jensen's inequality again,

    ( ∫ (X^α_j ρ_i) f dx_i)^2
    ≤^i[(f-^i f)^2  (X^α_j H^i)^2],

and then take into account the specific form of H^i.
Since the differential operator X_j^α acts on x_j,
only the one of the interactions terms survives, giving

    X^α_j H^i = J_ij X^α_j V(x_i,x_j).

Therefore, using (<ref>)

    ∑_α=1^n ( ∫ (X^α_j ρ_i) f dx_i)^2
       ≤ J_ij^2  ^i [ (f-^i f)^2 ∇_j V(x_i,x_j)^2]
    
       ≤ k J_ij^2 ^i (f-^if)^2 
    +  k J_ij^2 ^i (f-^if)^2 (x_i)^s
    + k J_ij^2 ^i (f-^if)^2 (x_j)^s.

Summing up the first display of this proof over α
and integrating over ν we obtain 

    ν∇_j(^i f)^2 ≤ 2 ν∇_j f^2
    + 2k J_ij^2 ν[ (f-^if)^2 ]
    
    
    +  2k J_ij^2 ν[ (f-^if)^2 (x_i)^s]
    + 2k J_ij^2 ν[ (f-^if)^2 (x_j)^s].

From the single-site  coercive inequality of Lemma <ref>,

    ν[ (f-^if)^2 (x_i)^s] 
    = ν^i [ (f-^if)^2 (x_i)^s]
    ≤ C_0 ν∇_i f^2 + C_0 ν [(f-^if)^2],

and

    ν[ (f-^if)^2 (x_j)^s] 
    = ν^j [ (f-^if)^2 (x_j)^s]
    ≤ C_0 ν∇_j (f-^i f)^2 + C_0 ν [(f-^if)^2],
    
    ≤ 2C_0 ν∇_j f^2
    + 2C_0 ν∇_j(^i f)^2
    + C_0 ν [(f-^if)^2].

Substituting these last two into (<ref>) gives

    ν∇_j(^i f)^2  
    ≤ (2+4kJ_ij^2 C_0)ν∇_j f^2+  2kJ_ij^2 (1+2C_0) ν [(f-^if)^2] 
    +2kJ_ij^2 C_0 ν∇_i f^2
    
    
    + 4kJ_ij^2 C_0 ν∇_j(^i f)^2.

We can now use the Poincare inequality from Lemma <ref> to bound the variance

    ν∇_j(^i f)^2  
    ≤ (2+4kJ_ij^2 C_0)ν∇_j f^2+  2kJ_ij^2 (c_p+C_0+2c_pC_0) ν∇_i f^2+
    
    
    + 4kJ_ij^2 C_0 ν∇_j(^i f)^2.

Equivalently, we can write

    (1-4kC_0 J_ij^2)  ν∇_j(^i f)^2  
    ≤ (2+4kJ_ij^2 C_0)ν∇_j f^2+  2kJ_ij^2 (c_p+C_0+2c_pC_0) ν∇_i f^2.

We now need to make sure that 1-4kC_0 J^2 > 0, i.e., 
that J < (4kC_0)^-1/2 and that 2kJ_ij^2 (c+C_0+2cC_0)  /(1-4kC_0 J^2)<1,
that is, 2kJ_ij^2 (c_p+C_0+2c_pC_0) +4kC_0 J^2<1, or J<(2kc_p+4kc_pC_0+6kC_0 )^-1/2. 
But the latter inequality implies the former. So it is only the
latter that we need.
Therefore the inequality holds with
D_1 := (2+4kC_0 J^2)/(1-4kC_0 J^2)
and
D_2 := 2kJ_ij^2 (c_p+C_0+2c_pC_0),
provided that J<(2kc_p+4kc_pC_0+6kC_0 )^-1/2.


Assume (<ref>)-(<ref>). For some D_3 > 0, if i, j are neighbors in ^d, then

    ν[(f-^i f)^2 (x_j)^s] ≤  D_3 ν∇_j f^2
    + D_3 ν∇_i f^2.

and

    ν[(f-^i f)^2 (x_i)^s] ≤  D_3 ν∇_i f^2.


For the first  assertion,  replace ν∇_j(^i f)^2 in the right-hand side of (<ref>) 
by its upper bound from the inequality  in the statement of Lemma <ref>,
and bound the last term 
from the spectral gap inequality from Lemma <ref>. Similarly, the second assertion of the corollary follows from (<ref>) and Lemma <ref>, for a constant  D_3:=2C_0(4+c_p)

Next, let, for r =0,1, …, d-1, the set Γ_r
be defined by

    Γ_r := {i ∈^d:  i_1+⋯+i_d ≡ r  d}.

Note that the sets Γ_r, r =0,1, …, d-1,
form a partition of ^d and
inf{max_1≤ k≤ d |i_k-j_k|:  i ∈Γ_r, j ∈Γ_s}=1
if r ≠ s.


From now on, we shall work with the case d=2, for simplicity of notation.
The general case is analogous.



Assume (<ref>)-(<ref>).
There are constants R_1>0 and 0<R_2 < 1 such that
 
    ν∇_Γ_0 (^Γ_1 f)^2 
     ≤ R_1 ν∇_Γ_0 f^2 + R_2 ν∇_Γ_1 f^2

and

    ν∇_Γ_1 (^Γ_0 f)^2 
     ≤ R_1 ν∇_Γ_1 f^2 + R_2 ν∇_Γ_0 f^2.



Fix i ∈Γ_1. Denote by ∂{i} the set {i± e_1, i± e_2}
of the 2d=4 neighbors of i. Since ∂{i}⊂Γ_0,
we can write ^Γ_0 f = ^Γ_0∖∂{i}^∂{i}f.
Hence if X^α_i is one of the Hörmander generators
 of ^{i}, we have
X^α_i ^Γ_0 f = ^Γ_0∖∂{i} X^α_i(^∂{i}f).
By Jensen's inequality,
(X^α_i ^Γ_0 f)^2 
≤^Γ_0 ∖∂{i}[ (X^α_i(^∂{i}f))^2].
Summing over all α, we get
∇_i ^Γ_0 f^2 
≤^Γ_0 ∖∂{i}∇_i(^∂{i}f)^2.
Integrating over ν and using ν^Γ_0 ∖∂{i}=ν, we get
ν∇_i ^Γ_0 f^2
≤ν∇_i(^∂{i}f)^2.
Summing this over i ∈Γ_1 we have

    ν∇_Γ_1 (^Γ_0 f)^2
    ≤∑_i∈Γ_1ν∇_i(^∂{i}f)^2.

We estimate the term inside the sum using Lemma <ref>  as follows.
First let ∂{i} = {i+e_1, i+e_2, i-e_1,i-e_2}
={j_1, j_2, j_3, j_4}.
Then ∇_i (^∂{i}f) 
= ∇_i ^{j_1}^{j_2,j_3,j_4} f.
So

    ν∇_i (^∂{i}f)^2
    = ν∇_i ^{j_1}^{j_2,j_3,j_4} f^2
    ≤ D_1 ν∇_i ^{j_2,j_3,j_4} f^2
    + D_2 ν∇_j_1^{j_2,j_3,j_4} f^2

For the second term we have 
∇_j_1^{j_2,j_3,j_4} f
= ^{j_2,j_3,j_4}∇_j_1 f and so, by Jensen's inequality,

    ν∇_j_1^{j_2,j_3,j_4} f^2
    ≤ν^{j_2,j_3,j_4}∇_j_1 f^2
    = ν∇_j_1 f^2.

The first term is estimated using Lemma <ref> once more:

    ν∇_i ^{j_2,j_3,j_4} f^2
    = ν∇_i ^j_2^{j_3,j_4} f^2
    ≤ D_1 ν∇_i ^{j_3,j_4} f^2
    + D_2 ν∇_j_2^{j_3,j_4} f^2.

Continuing in this manner, we obtain (observe that D_1 > 1)

    ν∇_i (^∂{i}f)^2
    
    ≤ D_1^4 ν∇_i f^2
    + D_1^3 D_2 ν∇_j_4 f^2
    + D_1^2 D_2 ν∇_j_3 f^2
    + D_1 D_2 ν∇_j_2 f^2
    + D_2 ν∇_j_1 f^2
    
    ≤ D_1^4 ν∇_i f^2 + D_1^3 D_2 ∑_j ∈∂{i}∇_j f^2.

Summing up over all i ∈Γ_1,

    ν∇_Γ_1(^Γ_0 f)^2
    ≤ D_1^4 ν∇_Γ_0 f^2 
    + 4 D_1^3 D_2 ν∇_Γ_1 f^2.

We need to make sure that 4D_1^3 D_2 < 1. Substituting the
actual expressions for these constants we can see that this
inequality is satisfied for all sufficiently small positive J.
In particular, the inequality is true
for all J < (80 k(c+2cC_0+2C_0))^-1/2.
We have thus proved the second inequality with R_1 :=D_1^4
and R_2 := 4 D_1^3 D_2, provided that
J < (192( k^2+k)(c+2cC_0+2C_0+C_0^3))^-1/2.
Define now the symbol ^n to be ^0 f=f and ^n := ^Γ_0^n-1 when n is odd and  ^n := ^Γ_1^n-1 when n is even, 
with the understanding that ^n when n is even takes a functional g on ^^d,
integrates with respect to ^Γ_1, x_∂Γ_0(dx_Γ_1)
= ^Γ_1, x_Γ_0(dx_Γ_1)
 so that ^n g is a functional not depending on x_Γ_1. Analogously,   ^n g for n odd is a functional not depending on x_Γ_0. We used the fact that ∂Γ_0 = Γ_1 and
∂Γ_1 = Γ_0.


Under hypotheses (<ref>)-(<ref>),  we  have
that lim_n →∞^n f = ν f, ν-a.e.



We will estimate the ^2(ν) norm of the differences of ^n f.
From the spectral gap inequality for ^Γ_k,k=0,1 (which follows
from the product property of the spectral gap and the spectral gap for the one node from  Lemma <ref>) we have

    ^Γ_k(^n f-  ^n+1 f)^2=^Γ_k(^n f-^Γ_k^n f)^2 ≤ c_p ^Γ_0∇_Γ_k^n f^2.

Integrating with respect to ν we have

    ν(^n f-^Γ_k^n f)^2 ≤ c_p ν∇_Γ_k^n f^2.

The last term is estimated from Lemma <ref>, for n ≥ 2,

    ν[(^n f-^n+1 f)^2] ≤ c_p (R_1 +R_2)R_2^n-1ν∇_   f^2 ≤ R^n,

for some R (depending on f), with 0<R<1.
Let ϵ>0 be so small so that R(1+ϵ)<1.
Then

    ν{x∈Ω:  |^n f-^n+1f| > (R(1+ϵ))^n/2}≤ν[(^n f-^n+1 f)^2]/(R(1+ϵ))^n ≤ (1+ϵ)^-n.

Hence

    ν{x∈Ω:  |^n f - ^n+1 f| ≤ (R(1+ϵ))^n/2 for almost all  n} = 1.

By the triangle inequality,

    ν{x∈Ω:  |^n f - ^m f| ≤ (R(1+ϵ))^n/2/
    (R(1+ϵ))^1/2 for all large n and m}=1.

Hence ^n f converges ν-a.e. say to,  ξ (f).      At first we will show that ξ(f)  is a constant  that  does not depend on variables neither on  Γ_0 nor on Γ_1. We first observe that  ^n(f) is a function on Γ_1 or Γ_0 when n  is odd or even respectively. As a consequence the limits of the subsequences   lim_n  odd, n→∞^nf  and lim_n  even, n→∞^nf do not depend on  variables on Γ_0 and Γ_1 respectively.    However, since  the two  subsequences {^nf}_n even and {^nf}_n odd  converge to the same limit ξ(f) ν-a.e.  we conclude that 
    lim_n  odd, n→∞^nf=ξ(f)=lim_n  even, n→∞^nf

 from which we derive that ξ(f) is a constant. Furthermore, this implies that 

    ν(ξ (f) )=ξ(f)

To finish the proof, it remains to show that  ξ(f)=ν (f). One notices that since the sequence {^n f}_n ∈ℕ 
 converges ν-a.e, the same holds for  the sequence {^n f-ν^n f}_n ∈ℕ. 

At first assume  positive bounded functions  f. In this case we have 
 
    lim_n→∞(^n f-ν^n f)=ξ (f)-ν(ξ(f) ) =ξ (f)-ξ (f)=0

 by the dominated     convergence theorem and the fact that ξ(f) is constant. On the other hand, we also have 
 
    lim_n→∞(^n f-ν^nf)=lim_n→∞(^n f-ν f)=ξ (f)-ν (f)
 
by the definition of the Gibbs measure ν. From the last two we obtain ξ (f)=ν (f)
for bounded positive functions f. We will extend this to   no bounded positive functions f. For this we consider f_k(x):=max{f(x),k} for any k∈ℕ. Then  
    ξ(f_k)=lim_n→∞^nf_k= ν f_k
 ν a.e, since f_k(x) is bounded by k. 
 But since f^k is increasing on k, by the monotone convergence theorem we   get 

    ξ(f)=lim_k→∞ξ(f_k)=lim_k→∞ν(f_k)=ν(lim_k→∞f_k)=ν(f)    ν   a.e.

The assertions   can be  extended to no  positive functions f just by writing f=max{f,0}-min{f,0}.


  
 

§ LOG-SOBOLEV INEQUALITY FOR ONE SITE MEASURE.
 

In this section we show a weak version of the  log-Sobolev type inequality for the one site measure 𝔼^i,ω. 
 


  Assume (<ref>)-(<ref>) and  that the  measure μ satisfies the log-Sobolev inequality with a constant c. Then, for J sufficiently small,   the one site measure  𝔼^i,ω satisfies the following weak version of  a log-Sobolev  inequality 
    ν𝔼^i,ω(  f^2logf^2/𝔼^i,ωf^2)≤ c_1ν∇_i f^2 +c_2∑_ j∼ iν∇_j f^2

for some    positive constants c_1 and c_2<1.
          

 
  We begin with the main assumption about the measure μ(dx_i)=e^ϕ(x_i)dx_i/∫ϕ(x_i)dx_i, that it  satisfies a  log-Sobolev inequality with a constant c   

    μ  (f^2logf^2/μ f^2)≤ cμ∇_i f ^2

We will interpolate the phase ϕ by the interactions W^i:=∑_j ∼ iJ_ijV(x_i,ω_j) in order to form the Hamiltonian     of the   one site measure 𝔼^ i,ω. To achieve  this,   replace f by      e^-W ^i/2 f, 
 
    ∫ e^-H^i   f^2loge^-W^i f^2/∫ (e^-H^i f^2)dx_i /  ∫ e^-ϕ(x_i)
     dx_i)dx_i≤ c  ∫ e^- ϕ(x_i)∇_i (e^-W^i/2 f)
    ^2 dx_i.

 We denote  by  D_l and D_r  the  left and right hand side of (<ref>) respectively.
Use the Leibnitz rule
for the gradient on D_r, to bound ∇_i (e^-W^i/2 f) ^2 ≤ 2  e^-W^i∇_i  f ^2+ 1/2  e^-W^if^2∇_i  W^i ^2, so that  
 
    D_r≤( ∫ e^-H^idx_i) ( 2c𝔼^ i,ω∇_i f
    ^2+ c/2𝔼^ i,ω(f^2∇_iW^i^2)).

On the left hand side of (<ref>) we form the Hamiltonian H^i=ϕ(x_i)+W^i to  obtain the entropy for the measure 𝔼^ i,ω

    D_l =    ∫ e^-H^i f^2logf^2/∫ e^-H^if^2dx_i / ∫ e^-H^i
     dx_i dx_i +∫ e^-H^i f^2log(∫ e^-ϕ(x_i)
     dx_i)  e^-W^i/∫ e^-H^i
     dx_i dx_i
    =    ( ∫ e^-H^i
     dx_i) (𝔼^ i,ω(f^2logf^2/𝔼^ i,ωf^2) - 𝔼^i,ω( f^2W^i))+∫ e^-H^i f^2log∫ e^-ϕ(x_i)
     dx_i/∫ e^-H^i
     dx_i dx_i.

Since  W^i is no negative,  the  last  gives 
 
    D_l ≥( ∫ e^-H^i
     dx_i)(𝔼^ i,ω(f^2logf^2/𝔼^ i,ωf^2) - 𝔼^ i,ω( f^2W^i)).

Combining (<ref>) together    with (<ref>)  and (<ref>) we obtain

    𝔼^ i,ω(f^2logf^2/𝔼^ i,ωf^2) ≤   2c 𝔼^ i,ω∇_i f
    ^2+𝔼^ i,ω(f^2( c/2∇_iW^i
    ^2+W^i)) .

 We now consider the following bound for the entropy, shown in   <cit.> and <cit.>   
 
    𝔼^ i,ω(f^2logf^2/𝔼^ i,ωf^2) ≤   A𝔼^ i,ω( f-𝔼^ i,ωf )^2 +
     𝔼^ i,ω(( f-𝔼^ i,ωf )^2log( f-𝔼^ i,ωf )^2/𝔼^ i,ω( f-𝔼^ i,ωf )^2)

  for some positive constant A.  
 Use (<ref>) to bound the entropy appearing on the second term on the right hand side,  
 
    𝔼^ i,ω(f^2logf^2/𝔼^ i,ωf^2) ≤     A𝔼^ i,ω( f-𝔼^ i,ωf )^2 +
     2c 𝔼^ i,ω∇_i f
    ^2+
        +𝔼^ i,ω(( f-𝔼^ i,ωf )^2( c/2∇_iW^i
    ^2+W^i)).

If we take expectations with respect to  the Gibbs measure we have 

    ν(f^2logf^2/𝔼^ i,ωf^2) ≤     Aν( f-𝔼^ i,ωf )^2 +
     2c ν∇_i f
    ^2+
        +J∑_j∼ iν(( f-𝔼^ i,ωf )^2{2c ∇_iV(x_i,ω_j)
    ^2+V(x_i,ω_j)})
where above we use that J_i,j^2≤ J_i,j≤ J.  And so, from the bounds (<ref>) and  (<ref>) 
 
    ν(f^2   log f^2/𝔼^ i,ωf^2) ≤  (A+4Jk(1+2c))ν( f -𝔼^ i,ωf)^2  +2c ν∇_i f
    ^2+
       4(2c+1)kJν(( f-𝔼^i,ωf)^2^s(x_i) ) +(2c+1)kJ∑_ j∼ iν(( f-𝔼^i,ωf)^2^s(ω_j) ).

We bound the variance in the first  term by the spectral gap of Lemma <ref> 
and the third and the  fourth term   by Corollary <ref>
 
    ν(f^2logf^2/𝔼^ i,ωf^2) ≤   (  (A+4Jk(1+2c))c_p    +2c+(16c+8)kJ  D_3  )ν∇_i f
    ^2+
       +(2c+1)kJ D_3∑_ j∼ iν∇_j f^2

which finishes the proof of  the proposition for c_1=   (A+4Jk(1+2c))c_p    +2c+(16c+8)kJ  D_3 and c_2=(2c+1)kJ 2C_0(4+c_p)<1 for J<((2c_0+1)k2C_0(4+c_p))^-1.
   
 
 
 
 
 
   


§ FURTHER SWEEPING-OUT INEQUALITIES

In this section we prove the second set of sweeping-out inequalities.

Assume (<ref>)-(<ref>) and the log-Sobolev inequality for μ. If i ∼ j then for some G_1>0 and 0<G_2<1,

    ν∇_i √(^j f^2)^2 
    ≤ G_1 ν∇_i f^2 + G_2 ν∇_j f^2



Fix neighboring sites i, j.
Start with the left-hand side, 

    ν∇_i √(^j f^2)^2 = ∑_α=1^n
    (X_i^α√(^j f^2))^2,
 where 

    (X_i^α√(^j f^2))^2 
    = (X_i^α^j f^2)^2/4^j f^2,
 
estimate the numerator as in (<ref>): 

    (X_i^α^j f^2)^2
    ≤ 2 (^j(X_i f^2))^2 + 2(∫(X_i^αρ_j) f^2 dx_j)^2.

Use Leibnitz' rule, Cauchy-Schwarz and Jensen 
for the first summand and estimate
the second using (<ref>) and (<ref>):

    (X_i^α^j f^2)^2   ≤ 4 (^j f^2) ^j(X_i^α f)^2
    + 2 J_ji^2 _^j[f^2,  X_i^α V(x_j, x_i)] ^2,

where _μ(f, g) = μ(fg)-μ(f)μ(g) = μ(f(g-μ g)), 
for a probability measure μ.
Substituting into (<ref>) and summing over α, we get

    ∇_i √(^j f^2)^2  ≤^j∇_i f^2 + J^2/2∑_α_^j[f^2,  X_i^α V(x_j, x_i)] ^2/^j f^2.

Instead of using Jensen,  as we did in (<ref>), we use
the following inequality (see <cit.>):
 For a probability measure μ

    (_μ(f^2, g))^2 ≤ 8  (μ f^2) μ [(f-μ f)^2 (g^2 + μ g^2)].
 

We get

    ∇_i √(^j f^2)^2  ≤^j∇_i f^2 + 4J^2  
    ^j{(f-^j f)^2 (∇_i V(x_j,x_i)^2 + ^j ∇_i V(x_j,x_i)^2) }.

If we now use  condition  (<ref>)  to bound the interactions, and then take   expectations with  respect to ν  we obtain 

    ν∇_i √(^j f^2)^2  ≤   ν∇_i f^2 +8k J^2  
    ν[^j (f-^j f)^2 ]+ 4 k J^2  
    ν[^j[(f-^j f)^2 ^j (x_j)^s]]+ 
        +8k J^2  
    ν [(f-^j f)^2(x_i)^s] +  4 k J^2  
    ν[ (f-^j f)^2   (x_j)^s ].

At first notice that from Lemma <ref> we can bound ^j[(x_j)^r] ≤ C_0. So the sum of the second and third term can be bounded from the variance with respect to the one site measure ^i. Then the variance can be bounded by the spectral gap inequality obtained in Lemma <ref>. 

    8λ J^2  
    ν[^j (f-^j f)^2 ]+ 4 k J^2  
    ν[^j[(f-^j f)^2 ^j (x_j)^s]] ≤    4kJ^2  (2+C_0)ν [^j (f-^j f)^2]
    ≤    4k J^2  (2+C_0) c_pν∇_j f^2.

For the  remaining two last  terms in the right hand side of (<ref>), we can use  the two bounds presented in  Corollary <ref>. 
If we put all these bounds together we get

    ν∇_i √(^j f^2)^2  ≤   (1+8k J^2  D_3)ν∇_i f^2 +4k J^2(3
     D_3+  (2+C_0) c_p)ν∇_j f^2+

This proves the lemma with constants
G_1 = 1+8k J^2  D_3
and G_2 = 4k J^2(3
 D_3+  (2+C_0) c_p)<1, provided that 
 J < (4k(3
 D_3+  (2+C_0) c_p))^-1/2.


Assume (<ref>)-(<ref>) and the log-Sobolev  inequality for ^i,ω. There are constants C_1>0 and 0<C_2 < 1 such that
 
    ν∇_Γ_0√(^Γ_1 h^2)^2
       ≤ C_1 ν∇_Γ_0 h^2 + C_2 ν∇_Γ_1 h^2

and

    ν∇_Γ_1√(^Γ_0 h^2)^2
       ≤ C_1 ν∇_Γ_1 h^2 + C_2 ν∇_Γ_0 h^2



We will make frequent use of the following inequality. Let A, B
be subsets of ^2 at lattice distance at least 2 
and i ∈^2 such that 
∂{i}∩ A = ∅.
Then

    ν∇_i √(^A∪ B f^2)^2
    ≤ν∇_i √(^B f^2)^2.

To see this, let ∇_i = (X^α_i, α=1,…,n)
and write

    X^α_i √(^A∪ B f) 
    = X^α_i ^A∪ B f/2√(^A∪ Bf)
    = ^A X^α_i ^B f/2√(^A∪ Bf)
    = 2 ^A [ √(^B f) X^α_i √(^B f)]/2√(^A∪ Bf),

where the first and last inequalities are due to Leibnitz' rule,
while the middle one follows from the assumptions on A, B and i.
By Cauchy-Schwarz, (^A [ √(^B f) X^α_i √(^B f)])^2
≤ (^A ^B f)  ^A(X^α_i √(^B f))^2. 
Squaring the last display and  replacing by this inequality we obtain
(X^α_i √(^A∪ B f))^2 ≤^A(X^α_i √(^B f))^2.
Summing over α and integrating over ν proves the claim.

To save some space below, for F : ^^d→^n
we shall write F^2 instead of ∫F(x)^2 dν(x).
We shall also write E f instead of √( f). Thus the inequality
we showed is written as 

    ∇_i ^A∪ B f^2 ^2
    ≤∇_i ^B f^2^2
    QS

Using this we upper bound ν∇_Γ_1√(^Γ_0f^2)^2:

    ∇_Γ_1^Γ_0 f^2^2
    = ∑_i∈Γ_1∇_i ^Γ_0f^2^2
    ≤∑_i∈Γ_1∇_i ^∂{i}f^2^2_:=T_1(i),

Fix i ∈Γ_1 and denote its neighbors by i_1, i_2,i_3,i_4. Let also I := {i_2,i_3,i_4} = ∂{i}∖{i_1}.
Using Lemma <ref> we write

    T_1(i)
    =∇_i  ^i_1^I f^2^2
    ≤ G_1 ∇_i ^I f^2^2_:=T_2(i)
    + G_2 ∇_i_1^I f^2^2.

Using (QS) three times in the second term, we obtain 

    ∇_i_1^I f^2^2 ≤∇_i_1 f^2.

And so,

    T_1(i)≤ G_1 T_2(i)+   G_2  ∑_ℓ∼ i∇_ℓ f^2  
    .
  
   Now we sum  over i ∈Γ_1. Note that 
∑_i ∈Γ_1∑_ℓ∼ i∇_ℓ f^2
=4 ∑_j∈Γ_0∇_j f^2.

    ∑_i ∈Γ_1 T_1(i) 
    ≤ G_1 ∑_i ∈Γ_1 T_2(i)
    +   4G_2∇_Γ_0 f^2.

We proceed in the same manner to estimate T_2(i).
Let J={i_3,i_4},

    T_2(i) 
    := ∇_i ^i_2^J f^2^2
    ≤ 
    G_1 ∇_i ^J f^2^2_:=T_3(i)
    + G_2 ∇_i_2^J f^2^2.

Use (QS) for the second term,

    ∇_i_2^J f^2^2 
    ≤∇_i_2f^2.

Substituting  into (<ref>) 

    T_2(i) 
    ≤ 
    G_1 T_3(i)+  G_2  ∑_ℓ∼ i∇_ℓ f^2

and summing up over i ∈Γ_1,

    ∑_i∈Γ_1 T_2(i)
    ≤ G_1 ∑_i∈Γ_1 T_3(i) 
    +  4  G_2  ∇_Γ_0 f^2

The next term is similar:

    T_3(i) 
    = ∇_i ^i_3^i_4 f^2^2
    
    ≤ G_1 ∇_i ^i_4f^2^2
    + G_2 ∇_i_3^i_4f^2^2,

with the  terms  estimated as

    ∇_i ^i_4f^2^2    ≤
    G_1 ∇_i f^2
    + G_2 ∇_i_4 f^2
     
    ∇_i_3^i_4f^2^2    ≤∇_i_3 f^2,

so that

    T_3(i) 
    
    ≤  G^2_1 ∇_i f^2
    + (1+G_1 )G_2  ∑_ℓ∼ i∇_ℓ f^2

and summing over i∈Γ_1 
    ∑_i∈Γ_1 T_3(i)
    
    ≤ G^2_1 ∇_Γ_1 f^2
    + 4(1+G_1 )G_2  ∇_Γ_0 f^2


Substituting the terms involving the sums to one another and then
back to (<ref>) yields the second inequality in
the statement with 
C_1 =G_1^4 and  
C_2 = 4G_2 (1+4G_1+G^2_1++G^3_1). 
Since G_2 = 4k J^2(3 D_3+  (2+C_0) c_p)<1 we can choose J sufficiently small such that G_2 is small enough so that C_2<1.


In the next proposition we prove a weak log-Sobolev inequality for the product measures ^Γ_i, i=0,1.  
 
  Assume (<ref>)-(<ref>) and the log-Sobolev inequality for μ. Then the following log-Sobolev type inequality for the measure  𝔼^i,ω holds  
 
    ν𝔼^Γ_k(f^2logf^2/𝔼^Γ_k f^2) ≤C̃ν|∇_Γ_0
    f
    |^2+C̃ν|∇_Γ_1 f
    |^2
for k=0,1, and  some positive constant C̃.    

     Consider a node  i∈ℤ^2 with  four neighbours denoted as  {∼ i }=i_1,i_2,i_3,i_4 . We start by considering the following two quantities:  

    Φ(i):=   ν∇_i( 𝔼^{i_1 ,i_2,i_3,i_4}f^2)^1/2^2 +ν∇_i( 𝔼^{ i_2,i_3,i_4}f^2)^1/2^2+ν∇_i( 𝔼^{ i_3,i_4}f^2)^1/2^2+
       +ν∇_i( 𝔼^{ i_4}f^2)^1/2^2

and

    Θ(i):=    ν∇_i  f^2+ ∑_s ∼ i ν∇_s  f^2
 From the estimates (<ref>), (<ref>) and (<ref>)  about the components of  the sum of Φ(i) in the proof of   Lemma  <ref> together with   Lemma <ref> we   surmise that there exists a constant  R_3>0   such that   

    Φ(i) ≤  R_3Θ(i)
  Starting from the neighbourhood of  (0,0) we form a spiral  enumeration of all  nodes in Γ_1 as described below (see also depiction in figure <ref>).

 We start by denoting   a_1,a_2,a_3,a_4 the neighbours of (0,0). Obviously, since (0,0) ∈Γ_0, the nodes a_i∈Γ_1 for i=1,...,4. After choosing  a_1 from any of the four neighbours,  the rest are named clockwise. Then, we choose  a_5 to be any of the nodes in Γ_1 of distance two from a_4 and distance three from (0,0).  We continue in the same manner   clockwise the enumeration of the rest of the nodes in Γ_1 that have distance  three from (0,0), then distance four, and so on. In this way we   construct a spiral    comprising of the  nodes in Γ_1 always  moving clockwise while we move away  from (0,0).    We can then write 𝔼^Γ_1=⊓_i=1^+∞ 𝔼^ a_i.
Since we have obtain in Proposition <ref> a log-Sobolev inequality for the one node measure, we will express the entropy of the product measure ^Γ_1 in terms of the individual entropies as seen  below

    ν𝔼^Γ_1(f^2logf^2/𝔼^Γ_1f^2) = ∑_k=1^+∞ν𝔼^a_k(𝔼^ a_k-1...𝔼^a_1f^2log𝔼^a_k-1...𝔼^a_1f^2/𝔼^a_k...𝔼^a_1f^2)

so that we can  upper bound the one site entropies from the log-Sobolev inequalities,   
 
    ν𝔼^  a_k(𝔼^ a_k-1...𝔼^ a_1f^2log𝔼^a_k-1...𝔼^a_1f^2/𝔼^a_k...𝔼^a_1f^2) ≤ c_1ν∇_a_k f^2 +c_2∑_ j∼ a_kν∇_j (𝔼^ a_k-1...𝔼^ a_1f^2)^1/2^2

 where above in the computation of the first term  we used that a_i's have distance bigger than one from each other, and so ν∇_a_k (𝔼^ a_k-1...𝔼^ a_1f^2)^1/2^2 ≤ν∇_a_k f^2.  
 For the second summand in (<ref>) notice that the neighbours of  a_k can be distinguished into two categories. Those that have distance bigger than one from a_k-1,a_k-2,...,a_1 and those that neighbour with at least one of    a_k-1,a_k-2,...,a_1.  For j∼ a_k that belong to the first category, since they do not neighbour any of the nodes  a_k-1,a_k-2,...,a_1 we clearly get
 
    ν∇_j (𝔼^ a_k-1...𝔼^ a_1f^2)^1/2^2 ≤ν∇_j f^2

 For those   neighbours of a_k, that neighbour  with at least one of the a_k-1,a_k-2,...,a_1 we can write 

    ν∇_j (𝔼^ a_k-1...𝔼^ a_1f^2)^1/2^2 ≤Φ(j)

If we bound this by (<ref>) 

    ν∇_j (𝔼^ a_k-1...𝔼^ a_1f^2)^1/2^2 ≤ R_3Θ(j)
   
Gathering together    (<ref>), (<ref>) and  (<ref>) we have
   
    ν𝔼^  a_k(𝔼^ a_k-1...𝔼^ a_1f^2log𝔼^a_k-1...𝔼^a_1f^2 /𝔼^a_k...𝔼^a_1f^2) ≤    c_1ν∇_a_k f^2+c_2∑_ j∼ a_kν∇_j f^2+
         +c_2R_3∑_ j∼ a_kΘ(j)

 Then, if we combine this bound  together with (<ref>) we obtain 
  
    ν𝔼^Γ_1(f^2logf^2/𝔼^Γ_1f^2) ≤    c_1∑_k=1^+∞ν∇_a_k f^2+c_2∑_k=1^+∞∑_ j∼ a_kν∇_j f^2+
         +c_2R_3∑_k=1^+∞∑_ j∼ a_k∑_n=0^2∑_r:dist(s,j)=n ν∇_s  f^2

If we notice that for every node there are four nodes at distance one and eight at distance two, after rearranging the sums above we finally obtain  
 
    ν𝔼^Γ_1(f^2logf^2/𝔼^Γ_1f^2) ≤( c_1+13R_3c_2)ν∇_Γ_1 f^2+( 4c_2+13R_3c_2)ν∇_Γ_0 f^2

 



 



§ THE LOG-SOBOLEV INEQUALITY FOR THE GIBBS MEASURE


In this section we prove the main result stated in Theorem <ref>.
 We recall that  ^n is defined as ^0 f=f and ^n := ^Γ_0^n-1 when n is odd and  ^n := ^Γ_1^n-1 when n is even.
If Λ is a subset of ^d, we write
_^Λ for the entropy of the probability measure
^Λ,ω on ^Λ, that is, 
_^Λ(g) =  ^Λ[ g logg/^Λ g].
From this, with λ(x) := x log x, we have

    ^Λ[λ(g)] = _^Λ(g) + λ(^Λ g),

where we used the fact that ^Λ g  does
not depend on x_Λ. 

We claim that, for all n ≥ 1,

    ^n[λ(g)] 
    =    ∑_m=0, m odd^n-1^n-m-1^Γ_1[_^Γ_0(^m g)]
    + ∑_m=0,  m even^n-1^n-m-1 [_^Γ_1( ^m g)]
    + 
       + λ(^n g).

To see this, notice first that  the statement is trivial for n=1. Assuming it
true for some n ≥ 1, we prove the same thing with n+1 in place of n.
Apply (<ref>) with Λ = Γ_0 and ^n g for n  odd in place of g:

    ^Γ_0[λ(^n g)] = _^Γ_0(^n g) + λ(^Γ_0^n g),

and, again from (<ref>) with Λ = Γ_1 and
^n g for n  even in place of g,

    ^Γ_1[λ(^n g)] = _^Γ_1(^n g) + λ(^Γ_1^n g).

From the last two displays, for odd n we get  

    ^Γ_1[λ(^n g) ] = ^Γ_1[_^Γ_0(^n g)]
    + _^Γ_1(^Γ_0^n g) + λ(^n+1 g)

while for n even  

    ^Γ_0[λ(^n g)] = ^Γ_0[_^Γ_1(^n g)]
    + _^Γ_0(^Γ_1^n g) + λ(^n+1 g)

Using these, and  applying ^Γ_0 or ^Γ_1 to (<ref>) when n is even or odd respectively, we readily obtain
(<ref>) with n+1 in place of n. This shows the
veracity of (<ref>).
Using Lemma <ref>, we have ^n[λ(g)] →ν[λ(g)]
and λ^n(^n g) →ν[g], ν-a.e. From this
and Fatou's lemma, (<ref>) gives

    _ν(g)    ≤lim inf_n →∞{ν[
    ∑_m=0, m odd^n-1^n-m-1  [_^Γ_0(^m g)]
    + ∑_m=0, m even^n-1^n-m-1 [_^Γ_1(^Γ_0^m g)]
    ]
    },        
       = lim inf_n →∞{∑_^m=0, m odd^n-1ν[_^Γ_0(^m g)]
    + ∑_m=0, m even^n-1ν[_^Γ_1(^Γ_0^m g)]
    }

where we used the fact that ν is a Gibbs  measure to obtain
the last equality.
Let g=f^2 and   apply Proposition <ref> to bound the entropy
 
    ν[_^Γ_0(^m f^2)]
       ≤C̃ν∇_Γ_0√(^m f^2)^2
    
    ≤C̃ [C_1 C_2^m-1ν∇_Γ_1 f^2 
    + C̃C_2^mν∇_Γ_0 f^2 ]
    
    ν[_^Γ_1(^Γ_0^m f^2)]
       ≤C̃ν∇_Γ_1√(^Γ_0^m f^2)^2
    
    ≤C̃ [C_1 C_2^m-1ν∇_Γ_0 f^2 
    + C̃C_2^mν∇_Γ_1 f^2 ],

for m odd and even respectively,  where,  for the last inequalities we used Lemma <ref> and induction.
Substituting in (<ref>), we obtain (recall that 0< C_2 < 1)

    _ν(f^2) ≤C̃ (C_1C_2^-1+C_2)/1-C_2ν∇_Γ_1 f^2 
    + C̃ (C_1C_2^-1+C_2)/1-C_2ν∇_Γ_0 f^2 
    ≤C ν∇ f^2,

where C is the largest of the two coefficients.
This  is the log-Sobolev inequality for ν. 
 


§ EXAMPLE

We consider the Hamiltonian for a measure on the Heisenberg group defined as in (<ref>). Theorem <ref> follows from the main result presented in Theorem <ref>. Thus, we need to verify that the conditions of Theorem <ref> are satisfied for a local specification with a Hamiltonian as in (<ref>).

At first, we need to verify that the main hypothesis, that  the  single site measure without interactions  (consisting only of the phase) 
μ(dx)=e^-ϕ(x)dx/∫ e^-ϕ(x) dx  satisfies
the log-Sobolev inequality.  In our example where ϕ(x)=(x)^p, p>2,
as explained in the introduction in section <ref>, this is true, since the family of measures (<ref>) satisfies the log-Sobolev inequality, a result that has been  proven in <cit.>. Furthermore, hypothesis (<ref>) and (<ref>) about  the  Carnot-Carathéodory distance on the Heisenberg group _1 are true (see <cit.> and <cit.>).    

At first one notices, that   for   convenience    the interaction potential can be written in the  following  equivalent form: 

    V(x,ω)=  δ^r (x) +∑ _k=1^r-1a_k^r-k(x) ^k(ω)
where a_k= [ r; k;   ] the  binomial    coefficients. 

 
    

 

 For conditions (<ref>) - (<ref>),  the first one easily follows from ^r(xz)≤ 2^r-1(x)+2^r-1( z) for every r∈ℕ and the specific form of ϕ and V. The second and third plausibly from   (x^-1)= (x)  and (γ (s))≤ (z) for any geodesic from 0 to z, both by the definition of the Carnot-Carathéodory  distance.


  Finally, conditions (<ref>)-(<ref>) and (<ref>)-(<ref>) can easily be verified for any s=2p-2 and r≤p+2/2, if one writes     the interaction potential in the   form (<ref>).

alpha
Dillo 83




[A-B-C]A-B-CC. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer,
 Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses. Soc. Math,  10, France, Paris  (2000).  


 

[B-E]B-ED. Bakry and M. Emery,
Difusions hypercontractives ,Seminaire de Probabilites XIX, Springer Lecture Notes in Math. 1123, 177-206 (1985). 


[Ba-En-Hu-Ma]Ba-En-Hu-MaZ.M. Balogh,  A. Engulatov,  L. Hunziker and O. E. Maasalo,
	Functional Inequalities and Hamilton–Jacobi Equations in Geodesic Spaces, Potential Anal, 36, 401-432 (2012)



[BA-Bo12]BA-Bo12F. Baudoin and M. Bonnefont,
	Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality, J of Funct Analysis, 262, 2646-2676 (2012)

 


 
   [B-HK]B-HK J. Bellisard and R. Hoegn-Krohn,
Compactness and the maximal Gibbs state for random fields on the Lattice, Commun.
Math. Phys., 84, 297-327 (1982).

 [B-G]B-GS.G.  Bobkov and F. Gotze,
Exponential integrability and transportation cost
          related to logarithmic sobolev inequalities
 , J of Funct Analysis 163  1-28 (1999).
  
 

[B-L-U]B-L-UA. Bonfiglioni, E. Lanconelli and F. Uguzzoni,
Stratified Lie groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics. Springer, New York  (2007).

 
 

 [Bo-Ch-He]Bo-Ch-HeM. Bonnefont,  D. Chafaï and R. Herry,
On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group Ann. Fac. Sci. Toulouse Math.,  29,  335-355 (2020).
 
  

 [B-Z]B-ZS.G. Bobkov and B. Zegarlinski,
Entropy Bounds and Isoperimetry. Memoirs of the American Mathematical Society,  Vol: 176,  1 - 69 (2005).
 
  
 
 
[B-H] B-HT. Bodineau and B. Helffer,
Log-Sobolev inequality for unbounded spin systems , J of Funct Analysis 166, 168-178 (1999). 




[Ch-Ka-Ru] Ch-Ka-RuM. Chatzakou and A. Kassymov and M. Ruzhansky,
Logarithmic Sobolev inequalities on Lie groups. , arXiv:2106.15652 (2021). 




 

 
 
 [Da-Ze]Da-ZeE.B.  Dagher and  B. Zegarlinski,
Coercive Inequalities and U-Bounds on Step-Two Carnot Groups.,  Potential Anal 59,  589–612 (2021). 


 [Da22]Da22E.B.  Dagher,
Note on the q-logarithmic Sobolev and p-Talagrand inequalities on Carnot groups.  Commun. Contemp. Math. (2022). 


 



[Da-Qi-Ze] Da-Qi-ZeE.B.  Dagher and  Y. Qiu and B. Zegarlinski and  M. Zhang,
Spectral Gap Inequalities on Nilpotent Lie Groups in Infinite Dimensions. , arXiv:2211.14188 (2022). 


 
 
 [D]DR. L. Dobrushin,
The problem of uniqueness of a Gibbs random field and the problem of
phase transition,  Funct.
Anal. Apll. 2, 302-312 (1968).
 
 

[Fe-Li]Fe-LiQ.  Feng and W. Li,
Entropy  dissipation for Degenerate Stochastic
Differential Equations via
Sub-Riemannian Density Manifold, Entropy,
25  (2023).

 


[Fe-Li1]Fe-Li1Q.  Feng and W. Li,
Sub-Riemannian Ricci curvature via generalized Gamma z calculus, arXiv:2004.01863 (2020).


 


 
[Ge-Gu-M-05]Ge-Gu-M-05I. Gentil, A. Guillin and L.  Miclo,
Modified logarithmic Sobolev inequalities and transportation inequalities,Probab. Theory Relat. Fields,
133, 409-436 (2005).

 

[Ge-Gu-M-07]Ge-Gu-M-07I. Gentil, A. Guillin and L.  Miclo,
Modified logarithmic Sobolev inequalities in null curvature, Rev. Mat. Iberoamericana,
23, 235-258 (2007).








 

[G-R]G-RI. Gentil and C. Roberto,
Spectral Gaps for Spin Systems: Some
Non-convex Phase Examples, J. Func.
Anal.,
180, 66-84 (2001).
 




[Gor-Luo]Gor-LuoM. Gordina and  L. Luo,
Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups, J. Func.
Anal.,
283, 66-84 (2022).
 
 


[Go-Ro-Sa]Go-Ro-SaN. Gozlan and C. Roberto and P.-M. Samson,
Characterization of Talagrand's Transport-Entropy inequalities in metric spaces. .Ann. Probab,
5, 3112–3139 (2013).

 





[G]GL. Gross,
Logarithmic Sobolev inequalities, Am. J. Math. 97, 1061-1083 (1976).
 
 

[G-Z]G-ZA.Guionnet  and B.Zegarlinski,
Lectures on Logarithmic Sobolev Inequalities. IHP Course 98,  Seminare de Probabilite XXVI, Lecture Notes in
         Mathematics 1801, 1-134, Springer (2003).

 
 
 
  [H-Z]H-ZW.Hebisch  and B.Zegarlinski,
Coercive inequalities on metric measure spaces. J. Func. Anal., 258, 814-851 (2010).

 


  [H]HB. Helffer, Semiclassical Analysis, Witten Laplacians and Statistical Mechanics, World Scientific, Singapore (2002).
 


 
  [I-K-Z11]I-K-Z11J. Inglis and V. Kontis and B. Zegarliński,
From U-bounds to isoperimetry with applications to H-type groups. J. Func. Anal., 260, 76-116 (2011).

 
 




 [I-P1]I-P1J. Inglis and I. Papageorgiou,
Log-Sobolev inequalities for infinite-dimensional Gibbs measures with non-quadratic interactions. Markov Proc. Related Fields, 25, 879-898 (2019).

[I-P]I-PJ. Inglis and I. Papageorgiou,
Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group. Potential Anal., 31, 79-102 (2009).




[Ch-Fe-Ze1]Ch-Fe-Ze1M. Chatzakou, S. Federico and B. Zegarlinski,
Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators.arXiv:2211.09471 (2023).


 	
 

 


[Ch-Fe-Ze2]Ch-Fe-Ze2M. Chatzakou, S. Federico and B. Zegarlinski,
q-Poincare inequalities on Carnot Groups with filiform type Lie algebra. Potential Anal. (2023).


 	


 



 



 [KOZ2]KOZ2V. Kontis and M. Ottobre and B. Zegarlinski,
Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing. J. Func.
Anal.,
270, 3173-3223 (2016).        
 


 [KOZ1]KOZ1V. Kontis and M. Ottobre and B. Zegarlinski,
Long- and short-time behaviour of hypocoercive-type operators in infinite dimensions: An analytic approach. Infin. Dimens. Anal. Quantum Probab.,
20 (2017).        
 

 
 






  [Led]LedM. Ledoux,
Logarithmic Sobolev inequalities for unbounded spin systems revisited. Seminaire de Probabilites XXXV. Lecture
Notes in Math. 1755, 1167-194. Springer (2001).



 [M1]M1K. Marton,
An inequality for relative entropy and logarithmic Sobolev inequalities in Eucledian spaces. J. Func.
Anal.,
264, 34-61 (2013).        
 




 [Mo]MoR. Monto,
Some properties of Carnot-Carathéodory  balls in the Heisenberg group. , Rend. Mat. Acc. Lincei, 11, 155-167 (2007). 
 
 [O-R]O-RF. Otto and M. Reznikoff,
A  new criterion for the Logarithmic Sobolev Inequality and two  Applications , J. Func. Anal.,
243, 121-157 (2007). 

[Pa3]Pa3I. Papageorgiou,
The log-Sobolev inequality with quadratic interactions.,
J. Math. Phys., 59 (2018).


[Pa2]Pa2I. Papageorgiou, The logarithmic Sobolev inequality for Gibbs measures on infinite product of Heisenberg groups. Markov Proc. Related Fields, 20, 705-749 (2014).

 
[Pa1]Pa1I. Papageorgiou,
The  Logarithmic Sobolev Inequality in Infinite dimensions for Unbounded Spin Systems on the Lattice with non Quadratic Interactions,
Markov Proc. Related Fields, 16, 447-484 (2010).


[Pa4]Pa4I. Papageorgiou, Concentration inequalities for Gibbs measures,
Infin. Dimens. Anal. Quantum Probab. Relat., 14, 79-104 (2011).

 
[Pa5]Pa5I. Papageorgiou, A note on the modified Log-Sobolev inequality.,
 J. Potential Anal. 35, 275-286 (2011).

 
 
 

 [Pr]PrC.J.Preston,Random Fields, LNM 534, Springer (1976).

 [R-Z]R-ZC. Roberto and B. Zegarlinski,
Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups ,  J. Func.
Anal.,
243 (1), 28-66 (2007).

 [R]RO. Rothaus,
Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Func. Anal., 64, 296-313 (1985).

 [SC]SCL.Saloff-Coste,
Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series, 289. Cambridge University Press, Cambridge, (2002).


[V-SC-C]V-SC-CN. Th. Varopoulos, L. Saloff-Coste and  T. Coulhon,
Analysis and Geometry on Groups, Tracts in Mathematics, 100, Cambridge University Press, Cambridge, (1992).




[Won]WonY.S. Won,
An L2-approximation method for construction andsmoothing estimates of Markov semigroups forinteracting diffusion processes on a lattice, Infin. Dimens. Anal. Quantum Probab. Relat. 24  (2021).


 




[Y]YN.Yoshida,
The log-Sobolev  inequality for weakly coupled lattice field, Probab.Theor. Relat. Fields  115, 1-40 (1999).
 
 
[Z1]Z1B. Zegarlinski,
On log-Sobolev Inequalities for Infinite Lattice Systems, Lett. Math. Phys. 20, 173-182 (1990).
 
 
 
[Z2]Z2B. Zegarlinski,
The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice, Comm. Math. Phys. 175, 401-432 (1996).
 

