Log-Sobolev Inequalities for Infinite dimensional Gibbs measures of higher order Interactions

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by  Ioannis Papageorgiou

LOG-SOBOLEV INEQUALITIES FOR INFINITE DIMENSIONAL
GIBBS MEASURES OF HIGHER ORDER INTERACTIONS.
IOANNIS PAPAGEORGIOU
Abstract. WefocusontheinfinitedimensionalLog-Sobolevinequalityforspin
systems on the d-dimensional Lattice (d≥1) with interactions of higher power
than quadratic. We show that when the one dimensional single-site measure
4
with boundaries satisfies the Log-Sobolev inequality uniformly on the bound-
1
0 ary conditions then the infinite dimensional Gibbs measure also satisfies the
2 inequality if the phase dominates over the interactions.

n
a
J

4
1. Introduction
1

Ourfocusisonthethetypical LogarithmicSobolevInequality (LS)formeasures
]
A
related to systems of unbounded spins on a d-dimensional lattice, for d ≥ 1, with
F
nearest neighbour interactions oforder higherthantwo. Theaimofthispaperisto
.
h investigate appropriate conditions on the local specification so that the inequality
t
a can be extended from the one site to the infinite dimensional Gibbs measure.
m
The main assumption is that the Log-Sobolev Inequality is true for the single
[
site measure with a constant uniformly bound on the boundary conditions and

1 that the power of the interaction is dominated by that of the phase.
v RegardingtheLog-SobolevInequalityforthelocalspecification{EΛ,ω}
9 Λ⊂⊂ZdG,ω∈Ω
1 on a d-dimensional Lattice, criterions and examples of measures EΛ,ω with qua-
2
dratic interactions that satisfy the Log-Sobolev -with a constant uniformly on the
3
set Λ and the boundary conditions ω− are investigated in [Z2], [B], [B-E], [B-
.
1
L], [Y], [A-B-C] and [B-H]. Furthermore, in [G-R] the Spectral Gap Inequality is
0
4 proved. For the measure E{i},ω on the real line, necessary and sufficient conditions
1
are presented in [B-G], [B-Z] and [R-Z], so that the Log-Sobolev Inequality is sat-
:
v isfied uniformly on the boundary conditions ω. Furthermore, the problem of the
i
X Log-Sobolev inequality for the Infinite dimensional Gibbs measure on the Lattice
r is examined in [G-Z], [Z1] and [Z2]. Still in the case of bounded interactions, in
a
[M], [I-P] and [O-R], criterions are presented in order to pass from the Log-Sobolev
Inequality for the single-site measure E{i},ω to the LS for the Gibbs.
2000 Mathematics Subject Classification. 60E15,26D10.
Key words and phrases. Logarithmic Sobolev Inequality, Gibbs measure, Spin systems.
Address: Department of Mathematics, Uppsala University, P.O Box 480, Uppsala 751 06,
Sweden.
Email: [email protected],[email protected].
1
2 IOANNISPAPAGEORGIOU
Relatedtothecurrent caseofnonquadraticinteractionsin[Pa2]conditionswere
presented for the stronger Logarithmic Sobolev q inequality for q ≤ 2 for spins on
theonedimensional lattice. There theinequality fortheinfinitedimensional Gibbs
measurewasrelatedtotheinequalityforthefiniteprojectionoftheGibbsmeasure.
In the current paper we focus on the typical Logarithmic Sobolev inequality
(q = 2) for spin systems on the d-dimensional lattice, for d ≥ 1.
Consider the one dimensional measure
E{i},ω(dx ) = e−φ(xi)−Pj∼iJijV(xi,ωj)dXi with k∂ ∂ V(x,y)k = ∞
i Z{i},ω x y ∞
Assume that E{i},ω satisfies the (LS) inequality with a constant uniformly on ω.
Our aim is to set conditions, so that the infinite volume Gibbs measure ν for the
local specification {EΛ,ω} satisfies the LS inequality.
Λ⊂⊂Zd,ω∈Ω
Our general setting is as follows:
The Lattice. When we refer to the Lattice we mean the d-dimensional square
Lattice Zd for d ≥ 1.
The one site space M. We consider continuous unbounded random variables in
an n-dimensional space M, representing spins. The space M, is a n-dimensional
noncompact metric spaces. Wewill denote d the distance and∇ the (sub)gradient
for which we assume that
0 < ξ < |∇d| ≤ τ
for some τ,ξ ∈ (0,∞), and
|∆d| < θ
outside the unit ball {d(x) < 1} for some θ ∈ (0,+∞).
When we refer to the (sub)gradient ∇ or (sub)Laplacian ∆ of M related to a
specific node, say i ∈ Zd, we will indicate this by the use of indices, i.e. we will
write ∇ and ∆ .
i i
The Configuration space. Our configuration space is Ω = MZd. We consider
functions f : Ω → R. Accordingly we define ∇ f(ω) := ∇f (x|ω)| and
i i x=ωi
∆ f(ω) := ∆f (x|ω)| for suitable f, where ∇ and ∆ are the (sub)gradient
i i x=ωi
and the (sub)Laplacian on M respectively. For Λ ⊂ Zd, set ∇ f = (∇ f) and
Λ i i∈Λ
|∇ f|2 := |∇ f|2.
Λ i
i∈Λ
X
We will write ∇ = ∇, since it will not cause any confusion. For any ω ∈ Ω and
Zd
Λ ⊂ Zd we denote
ω = (ω ) ,ω = (ω ) ,ω = (ω ) and ω = ω ◦ω
i i∈Zd Λ i iǫΛ Λc i iǫΛc Λ Λc
where ω ∈ M. When Λ = {i} we will write ω = ω . Furthermore, we will
i i {i}
write i ∼ j when the nodes i and j are nearest neighbours, that means, they are
connected with a vertex, while we will denote the set of the neighbours of k as
{∼ k} = {r : r ∼ k}.
LS INEQUALITIES WITH NON QUADRATIC INTERACTIONS. 3
The functions of the configuration. Let f: Ω → R. We consider integrable
functions f that depend on a set of variables {x },i ∈ Σ for a finite subset
i f
Σ ⊂⊂ Zd. The symbol ⊂⊂ is used to denote a finite subset.
f
The Measure on Zd. For any subset Λ ⊂⊂ Z we define the probability measure
e−HΛ,ωdX
EΛ,ω(dX ) = Λ
Λ ZΛ,ω
where
• X = (x ) and dX = dx
Λ i iǫΛ Λ iǫΛ i
• ZΛ,ω = e−HΛ,ωdX
Λ Q
• HΛ,ω = φ(x )+ J V(x ,z )
R iǫΛ i iǫΛ,j∼i ij i j
and
P P
x ,iǫΛ
j
• z = x ◦ω =
j Λ Λc
ω ,i ∈/ Λ
( j
We call φ the phase and V the potential of the interaction. For convenience we
will frequently omit the boundary symbol from the measure and we will write
EΛ ≡ EΛ,ω.
For the phase φ we make the following assumptions
• (H1.1) there exists some p ≥ 3 and k > 0 such that
0
∂ φ(x ) ≥ k dp−1(x )
d(xi) i 0 i
• (H1.2) there exists some k > 0 such that
1
∂2 φ(x ) ≤ k +k ∂ φ(x )
d(xi) i 1 1 d(xi) i
Furthermore, for the in(cid:12)teractions(cid:12)V(xi,ωj) we make the following assumptions
(cid:12) (cid:12)
• (H1.3) V(x,y) is a function of the distance d such that
∂ V(x,y) ≥ 0
d(x)
• (H1.4) there exists some k > 0 such that
2
∂2 V(x ,ω ) 6 k +k ∂ V(x ,ω )
d(xi) i j 2 2 d(xi) i j
for any d(xi) >(cid:12) M∗. (cid:12)
(cid:12) (cid:12)
• (H1.5) there exist an 0 < s ≤ p−1 such that for some k > 0 and any
j ∼ i
(a) Ej,ω|∇ V(x ,ω )|2 ≤ k +k ds(ω )
j i j i
i∼j
X
(b) Ej,ωd(x ) ≤ k +k ds(ω )
j i
i∼j
X
(c) |∇ V(x ,ω )|2 ≤ k +k(ds(x )+ds(ω ))
j i j i j
4 IOANNISPAPAGEORGIOU
(d) |V(x ,ω )| ≤ k +k(ds(x )+ds(ω ))
i j i j
as well as
(e) Ej,ωeǫ|∇jV(xi,ωj)|2 ≤ ekekPi∼jds(ωi)
for some ǫ > 0 sufficiently small.
• (H1.6) The coefficients J are such that |J | ∈ [0,J] for some J < 1
i,j i,j
sufficiently small.
The Infinite Volume Gibbs Measure. The Gibbs measure ν for the local spec-
ification {EΛ,ω} is defined as the probability measure which solves the
Λ⊂G,ω∈Ω
Dobrushin-Lanford-Ruelle (DLR) equation
νEΛ,⋆ = ν
for finite sets Λ ⊂ Z (see [Pr]). For conditions on the existence and uniqueness
of the Gibbs measure see e.g. [B-H.K] and [D]. In this paper we consider local
specifications for which the Gibbs measure exists and it is unique. It should be
noted that {EΛ,ω} always satisfies the DLR equation, in the sense that
Λ⊂⊂Zd,ω∈Ω
EΛ,ωEM,∗ = EΛ,ω
for every M ⊂ Λ. [P]
We denote
EΛ,ωf = fdEΛ,ω(X )
Λ
Z
We can define the following inequalities
The Log-Sobolev Inequality (LS). We say that the measure EΛ,ω satisfies the
Log-Sobolev Inequality, if there exists a constant C such that for any function
LS
f, the following holds
|f|2
EΛ,ω|f|2log ≤ C EΛ,ω|∇ f|2
EΛ,ω|f|2 LS Λ
with a constant C ∈ (0,∞) uniformly on the set Λ and the boundary conditions
LS
ω.
The Spectral Gap Inequality. We say that the measure EΛ,ω satisfies the Spectral
Gap Inequality, if there exists a constant C such that for any function f, the
SG
following holds
EΛ,ω f −EΛ,ωf 2 ≤ C EΛ,ω|∇ f|2
SG Λ
with a constant C ∈ (0,∞) uniformly on the set Λ and the boundary conditions
SG (cid:12) (cid:12)
ω. (cid:12) (cid:12)
Remark 1.1. We will frequently use the following two well known properties about
the Log-Sobolev and the Spectral Gap Inequality. If the probability measure µ sat-
isfies the Log-Sobolev Inequality with constant c then it also satisfies the Spectral
Gap Inequality with a constant less or equal than c. Furthermore, if for a family I
LS INEQUALITIES WITH NON QUADRATIC INTERACTIONS. 5
of sets Λ ⊂ Zd, dist(Λ ,Λ ) > 1 ,i 6= j the measures EΛi,ω,i ∈ I satisfy the Log-
i i j
Sobolev Inequality with constants c ,i ∈ I, then the probability measure E{∪i∈IΛi},ω
i
also satisfies the (LS) Inequality with constant c = max c . The last result is also
i∈I i
true for the Spectral Gap Inequality. The proofs of these properties can be found
in [G], [G-Z] and [B-Z].
2. The Main Result
WewanttoextendtheLog-SobolevInequalityfromthesingle-sitemeasureE{i},ω
to the Gibbs measure for the local specification {EΛ,ω} on the entire d
Λ⊂⊂Zd,ω∈Ω
dimensional Lattice. In the remaining of this paper we will refer to the hypothesis
about thephase andtheinteractions (H1.0)-(H1.4)collectively as(H1). Themain
hypothesis about the one site measure will be denoted as (H0):
(H0): The one dimensional measures Ei,ω satisfy the Log-Sobolev Inequality with
a constant c uniformly with respect to the boundary conditions ω.
Now we can state the main theorem.
Theorem 2.1. Let f: MZd → R. If hypothesis(H0) and(H1) for{EΛ,ω}
Λ⊂⊂Zd,ω∈Ω
are satisfied, then the infinite dimensional Gibbs measure ν for the local specifica-
tion {EΛ,ω} satisfies the Log-Sobolev inequality
Λ⊂⊂Zd,ω∈Ω
|f|2
ν|f|2log ≤ C ν|∇f|2
ν|f|2
for some positive constant C.
The main assumption about the local specification has been that the one site
measure Ei,ω satisfies the Log-Sobolev Inequality with a constant uniformly to the
boundary conditions, while the main assumption about the interactions is that the
phase φ(x) dominates over the interactions, in the sense that
k
|∇ V(x ,ω )|2 ≤ k +k(ds(x )+ds(ω )) ≤ k + (∂ φ(x )+∂ φ(ω ))
j i j i j k d(xi) i d(ωj) j
0
for s ≤ p−1. Then the Log-Sobolev inequality is extended to the infinite dimen-
sional Gibbs measure. In other words, what we roughly require is for a phase of
order dp the interaction to be the most of order dp+1.
2
As an example of a measure Ei,ω that satisfies (H0), that is the Log-Sobolev
Inequality with a constant c uniformly with respect to the boundary conditions
ω with non quadratic interaction, one can think the following measure on the
Heisenberg group
HΛ,ω(x ) = α dp(x )+ε (d(x )+ρd(ω ))s
Λ i i j
i∈Λ {i,j}∩Λ6=∅,j:j∼i
X X
6 IOANNISPAPAGEORGIOU
for α > 0, ε,ρ ∈ R, and p > s > 2, where as above x = ω for i 6∈ Λ. The proof of
i i
this follows with the use of uniform U-Bounds (see [I-P]).
We briefly mention some consequences of this result. The first follows directly
from Remark 1.1.
Corollary 2.2. Let ν be as in Theorem 2.1. Then ν satisfies the Spectral Gap
inequality
ν|f −νf|2 ≤ Cν|∇f|2
where C is as in Theorem 2.1.
The proofs of the next two can be found in [B-Z].
Corollary 2.3. Let ν be as in Theorem 2.1 and suppose f : Ω → R is such that
k|∇f|2k < 1. Then
∞
ν eλf ≤ exp λν(f)+Cλ2
for all λ > 0 where C is as in Theorem 2.1. Moreover, by applying Chebyshev’s
(cid:0) (cid:1) (cid:8) (cid:9)
inequality, andoptimisingoverλ, we arriveatthe following’decayof tails’ estimate
1
ν f − fdν ≥ h ≤ 2exp − h2
C
(cid:26)(cid:12) Z (cid:12) (cid:27) (cid:26) (cid:27)
(cid:12) (cid:12)
for all h > 0. (cid:12) (cid:12)
(cid:12) (cid:12)
Corollary 2.4. Suppose that our configuration space is actually finite dimensional,
so that we replace Zd by some finite graph G, and Ω = (M)G. Then Theorem 2.1
still holds, and implies that if L is a Dirichlet operator satisfying
ν(fLf) = −ν |∇f|2 ,
then the associated semigroup P = etL is ul(cid:0)tracont(cid:1)ractive.
t
Remark 2.5. In the above we are only considering interactions of range 1, but we
can easily extend our results to deal with the case where the interaction is of finite
range R.
Proof of Theorem 2.1. We want to extend the Log-Sobolev Inequality from
the single-site measure E{i},ω to the Gibbs measure for the local specification
{EΛ,ω} on the entire lattice. To do so, we will follow the iterative method
Λ⊂⊂Zd,ω∈Ω
developed by Zegarlinski in [Z1] and [Z2] (see also [Pa2] and [I-P] for similar appli-
cation). Without loose of generality in proof of the theorem we will assume that
d = 2, that is, that the configuration space is Ω = MZ2. Define the following sets
Γ = (0,0)∪{j ∈ Z2 : dist(j,(0,0)) = 2m for some m ∈ N},
0
Γ = Z2 rΓ .
1 0
where dist(i,j·) refers to the distance of the shortest path (number of vertices)
between two nodes i and j. Note that dist(i,j) > 1 for all i,j ∈ Γ ,k = 0,1 and
k
LS INEQUALITIES WITH NON QUADRATIC INTERACTIONS. 7
Γ ∩Γ = ∅. Moreover Z2 = Γ ∪Γ . As above, for the sake of notation, we will
0 1 0 1
write E = Eω for k = 0,1. Denote
Γk Γk
(2.1) P = EΓ1EΓ0
In order to prove the Log-Sobolev Inequality for the measure ν, we will express the
entropy with respect to the measure ν as the sum of the entropies of the measures
EΓ0 and EΓ1 which are easier to handle. We can write
f2 f2 EΓ0f2
ν(f2log ) =νEΓ0(f2log )+νEΓ1(EΓ0f2log )+
νf2 EΓ0f2 EΓ1EΓ0f2
(2.2) ν(EΓ1EΓ0f2logEΓ1EΓ0f2)−ν(f2logνf2)
According tohypothesis (H0), theLog-SobolevInequality issatisfiedforthesingle-
state measures E{j} and the sets Γ are unions of one dimensional sets of distance
k
greater than the length of the interaction one. Thus, as we mentioned in Re-
mark 1.1 in the introduction, the (LS) holds for the product measures EΓk with
the same constant c. If we use the LS for EΓi,i = 0,1 we get
q
(2.2) ≤ cν(EΓ0 |∇0f|2)+cνEΓ1 ∇Γ1(EΓ0f2)21 +
(2.3) ν(P1f2logP1f2)−(cid:12) ν(f2logνf2)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
For the third term of (2.3) we can write
P1f2 P2f2
ν(P1f2logP1f2) =νEΓ2(P1f2log )+νEΓ3(P2f2log )+
EΓ2P1f2 EΓ3P2f2
ν(EΓ3P2f2logEΓ3P2f2)
If we use again the Log-Sobolev Inequality for the measures EΓi,i = 2,3 we get
q 2
(2.4) ν(P1f2logP1f2) ≤ cν ∇Γ2(P1f2)21 +cν ∇Γ3(P2f2)12 +ν(P3f2logP3f2)
(cid:12) (cid:12) (cid:12) (cid:12)
If we work similarly for the (cid:12)last term ν(P(cid:12)3f2log(cid:12)P3f2) of (2.4(cid:12)) and inductively for
any term ν(Pkf2logPkf2), t(cid:12)hen after n s(cid:12)teps (2(cid:12).3) and (2.4)(cid:12)will give
f2
ν(f2log ) ≤ν(Pnf2logPnf2)−ν(f2logνf2)+
νf2
n
2
(2.5) cν|∇0f|2 +c ν ∇Γk(Pk−1f2)21
Xk=1 (cid:12) (cid:12)
(cid:12) (cid:12)
In order to calculate the third and fourth term on(cid:12) the right-hand(cid:12) side of (2.5) we
will use the following proposition
Proposition 2.6. Suppose that hypothesis (H0) and (H1) are satisfied. Then the
following bound holds
2
(2.6) ν ∇Γi(EΓjf2)12 ≤ C1ν|∇Γif|2 +C2ν ∇Γjf 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 IOANNISPAPAGEORGIOU
for {i,j} = {0,1} and constants C ∈ (0,∞) and 0 < C < 1.
1 2
The proof of Proposition 2.6 will be the subject of Section 4. If we apply
inductively relationship (2.6) k times to the third and the fourth term of (2.5) we
obtain
2
(2.7) ν ∇Γ0(Pkf2)12 ≤ C22k−1C1ν|∇Γ1f|2 +C22kν|∇Γ0f|2
(cid:12) (cid:12)
and
(cid:12) (cid:12)
(cid:12) (cid:12)
2
(2.8) ν ∇Γ1(EΓ0Pkf2)12 ≤ C22kC1ν|∇Γ1f|2 +C22k+1ν|∇Γ0f|2
(cid:12) (cid:12)
If we plug (2.7(cid:12)) and (2.8) in (2(cid:12).5) we get
(cid:12) (cid:12)
f2
ν(f2log ) ≤ν(Pnf2logPnf2)−ν(f2logνf2)+
νf2
n−1 n−1
c( C2k−1)C ν|∇ f|2 +c( C2k)ν|∇ f|2+
2 1 Γ1 2 Γ0
k=0 k=0
X X
n−1 n−1
(2.9) c( C2k)C ν|∇ f|2 +c( C2k+1)ν|∇ f|2
2 1 Γ1 2 Γ0
k=0 k=0
X X
If we take the limit of n to infinity in (2.9) the first two terms on the right hand
side cancel with each other, as explained on the proposition bellow.
Proposition 2.7. Under hypothesis (H0) and (H1), Pnf converges ν-almost ev-
erywhere to νf.
The proof of this proposition will be presented in Section 3. So, taking the limit
of n to infinity in (2.9) leads to
|f|2 C
ν(|f|2log ) ≤ cA 1 +C +C ν|∇ f|2 +cAν|∇ f|2
ν|f|2 C 2 1 Γ1 Γ0
(cid:18) 2 (cid:19)
where A = lim n−1C2k < ∞ for C < 1, and the theorem follows for a
n→∞ k=0 2 2
constant C = max{cA C1 +C +C ,cA} (cid:3)
P C2 2 1
(cid:16) (cid:17)
3. Proof of Proposition 2.7.
Before proving Proposition 2.7 we will present the key proposition of this paper,
Proposition 3.2. This proposition will also be used in the next section 4 where
Proposition 2.6 is proved.
In the case of quadratic interactions V(x,y) = (x−y)2 one can calculate
Ei,ω f2(∇ V(x −x )−Ei,ω∇ V(x −x ))2
j i j j i j
(see [B-H] and [H]) w(cid:0)ith the use of the relative entropy ine(cid:1)quality (see [D-S])
and the Herbst argument (see [L] and [H]). Herbst’s arguement states that if a
LS INEQUALITIES WITH NON QUADRATIC INTERACTIONS. 9
probability measure µ satisfies the LS inequality and a function F is Lipschitz
continues with kFk ≤ 1 and such that µ(F) = 0, then for some small ǫ we have
Lips
µeǫF2 < ∞
For µ = Ei,ω and F = ∇jV(xi−xj)−Ei,ω∇jV(xi−xj) we then obtain
2
Ei,ωe4ǫ(∇jV(xi−xj)−Ei,ω∇jV(xi−xj))2 < ∞
uniformly on the boundary conditions ω, because of hypothesis (H0). In the more
general case however of non quadratic interactions that we examine in this work,
the Herbst argument cannot be applied. In this and next sections we show how
one can handle exponential quantities like the last one with the use of U-bound
inequalities and hypothesis (H1) for the interactions.
U-bound inequalities introduced in [H-Z] are used to prove Logarithmic Sobolev
and Logarithmic Sobolev type inequalities. In the aforementioned paper, the U-
bound inequalities
(3.1) µ(fqdr) ≤ Cµ|∇f|q +Dµ|f|q
where used to prove Log-Sobolev q inequalities for q ∈ (1,2], the spectral Gap
inequality, as well as F-Sobolev inequalities. In particular, the U-bound inequality
(3.1) for q ∈ (1,2] and r bigger than the conjugate of q was used to prove that
the measure e−dr(x)dx/ e−dr(x)dx satisfies the Log-Sobolev q inequality. In the
context of the typical Log-Sobolev inequality, this implies that the measure
R
e−dr(x)dx
e−dr(x)dx
forr > 2satisfiestheLog-Sobolevinequality. In[I-P],theU-Boundinequality(3.1)
R
was shown fortheonesite measure Ei,ω uniformlyontheboundaryconditions fora
specific example of a measure onthe Heisenberg groupwith quadratic interactions.
However, it appears that this is very difficult to obtain in general when boundary
conditions are involved. In this paper however, where the Log-Sobolev inequality
is assumed for the single node measure Ei,ω uniformly on the boundary conditions,
as stated in hypothesis (H0), the strong U-Bound inequality (3.1) for r > 2 and
q = 2 is not necessary. Instead we will prove the weaker version of it
(3.2) Ei,ωds(x )f2 ≤ CEi,ω|∇ f|2 +CEi,ωf2
i i
for s < p−1. This will be then used in order to control the interactions and prove
sweeping out relations as in Proposition 2.6 and Lemma 3.3. Weaker U-Bound
inequalities than (3.1) have been used in the past for measures without interaction
in [H-Z] and [Pa1] in order to prove weaker inequalities than the Log-Sobolev, like
F-Sobolev and Modified Log-Sobolev inequalities. In effect, we focus on bounding
Ei,ωds(x )f2 instead of Ei,ωdp(x )f2 that would had been the appropriate analogue
i i
U-bound for the Log-Sobolev inequality, than the inferior (3.2), since it contains
constants independent of the boundary conditions. Furthermore for p larger than
10 IOANNISPAPAGEORGIOU
the interactions power we can control the boundary conditions. Following closely
on the proof of U-bound inequalities for the free boundary measure in [H-Z], the
main U-bound inequality is proven in the following proposition.
Denote
Di,ω = ∂ φ(x )+ J ∂ V(x ,ω )
d(xi) i ij d(xi) i j
j∼i
X
and
Bi,ω = ∂2 φ(x )+ J ∂2 V(x ,ω )
d(xi) i ij d(xi) i j
j∼i
X
Lemma 3.1. For any ω ,j ∼ i, denote B a set such that for all x ∈ B to have
j N i N
J ∂ V(x ,ω ) < N
i,j d(xi) i j
j∼i
X
There exist large enough constants M > 0 and N > 0 such that for every x ∈
i
O := {∂ φ(x ) > M}∪ {∂ φ(x ) < M}∩Bc the following to hold
d(xi) i d(xi) i N
ζ
|Di,ω|(∆(cid:8) d)+Bi,ω|∇ d|2 ≤ D(cid:9)i,ω 2|∇ d|2
i i i
2
for some ζ < 1. (cid:12) (cid:12)
(cid:12) (cid:12)
Proof. To prove the lemma we will distinguish two main cases. At first we assume
x : ∂ φ(x ) > M. In this case we will consider two sub cases, (a) x ∈ Bc and
i d(xi) i i N
(b) x ∈ B . Then we will examine the third case (c) x ∈ {∂ φ(x ) < M}∩Bc .
i N i d(xi) i N
a) x ∈ {∂ φ(x ) > M}∩Bc .
i d(xi) i N
Because of (H1.2) and (H1.4) we get
Bi,ω ≤ k +k ∂ φ(x )+k +k J ∂ V(x ,ω )
1 1 d(xi) i 2 2 i,j d(xi) i j
j∼i
X
We can then compute
Di,ω(∆ d)+Bi,ω|∇ d|2 <(k +k )τ2 +(θ+k τ2)∂ φ
i i 1 2 1 d(xi)
(3.3) +(θ+k τ2) J ∂ V(x ,ω )
2 i,j d(xi) i j
j∼i
X
We also have
2
1 ξ2
(3.4) Di,ω 2|∇ d|2 ≥ ∂ φ+ J ∂ V(x ,ω )
2 i 2 d(xi) i,j d(xi) i j
!
j∼i
(cid:12) (cid:12) X
where above we(cid:12) used(cid:12) the fact that
0 < ξ < |∇d| ≤ τ
for some τ,ξ ∈ (0,∞), and
|∆d| < θ