On the Exponential Decay of the n-point Correlation Functions and the
  Analyticity of the Pressure
On the Exponential Decay of the n-Point
Correlation Functions and the Analyticity of the
Pressure.
Assane Lo
April 1st, 2007
Abstract
The goal of this paper is to provide estimates leading to a direct proof
of the exponential decay of the n-point correlation functions for certain
unbounded models of Kac type. The methods are based on estimating
higher order derivatives of the solution of the Witten Laplacian equation
on one forms associated with the hamiltonian of the system. We also
provide a formula for the Taylor coefficients of the pressure that is suitable
for a direct proof the analyticity.
1 Introduction
In recent publications [66] we have given a generalization to the higher dimen-
sional case of the exponential decay of the two-point correlation functions for
models of Kac type. In this paper, we shall establish a weak exponential decay
of the n-point correlation functions, and provide an exact formula suitable for
a direct proof the analyticity of the pressure.
Let Λ be a finite subset of Zd, and consider a Hamiltonian Φ of the phase
space RΛ. We shall focus on the case where Φ = ΦΛ is given by
ΦΛ(x) =
+ Ψ(x), (1)
under suitable assumptions on Ψ.
Recall that if 〈f〉 denote the mean value of f with respect to the Gibbs
measure
−Φ(x)
the covariance of two functions g and h is defined by
cov(g, h) = 〈(g − 〈g〉)(h− 〈h〉)〉 . (2)
If one wants to have an expression of the covariance in the form
cov(g, h) = 〈∇h ·w〉L2(Rn,Rn;e−Φdx) , (3)
http://arxiv.org/abs/0704.1332v3
for a suitable vector field w we get, after observing that ∇h = ∇(h− 〈h〉), and
integrating by parts,
cov(g, h) =
(h− 〈h〉)(∇Φ−∇) ·we−Φ(x)dx. (4)
(Here we have assumed that g and h are functions of polynomial growth ).
This leads to the question of solving the equation
g − 〈g〉 = (∇Φ−∇) ·w. (5)
Now, trying to solve this above equation with w = ∇f, we obtain the equation
g − 〈g〉 = (−∆+∇Φ ·∇) f
〈f〉 = 0.
The existence and smoothness of the solution of this equation were established
in [8] (see also [66]) under certain assumptions on Φ. Now taking gradient on
both sides of (6), we get
∇g = [(−∆+∇Φ ·∇)⊗ Id+HessΦ]∇f. (7)
We then obtain the emergence of two differential operators:
Φ := −∆+∇Φ ·∇ (8)
Φ := A
Φ ⊗ Id+HessΦ. (9)
Here the tensor notation means that A
Φ acts diagonally on the vector field
solution to produce a system of equations.
cov(g, h) =
(1)−1
Φ ∇g ·∇h
−Φ(x)
dx. (10)
The operators A
Φ and A
Φ are called the Helffer-Sjöstrand’s operators. These
are unbounded operators acting on the weighted spaces
L2(RΛ, e−Φdx) and L2(RΛ,RΛ, e−Φdx)
respectively.
The formula (10) was introduced by Helffer and Sjöstrand and in some sense
is a generalization of Brascamp-Lieb inequality as already pointed out in [1].
The unitary transformation
UΦ : L
2(RΛ) → L2(RΛ, e−Φdx)
u 7−→ e
will allow us to work with the unweighted spaces L2(RΛ) and L2(RΛ,RΛ) by
converting the operators A
Φ and A
Φ into equivalent operators
Φ = −∆+
⊗ I+HessΦ. (12)
respectively.
The equivalence can be seen by observing that
Φ = e
−Φ/2 ◦A
Φ ◦ e
. (13)
The operators W
Φ and W
Φ are unbounded operators acting on
2(RΛ) and L2(RΛ,RΛ)
respectively. These are in fact, the euclidean versions of the Laplacians on zero
and one forms respectively, already introduced by E. Witten [18] in the context
Morse theory.
The equivalence between the operators A
Φ and Witten’s Laplacians was
first observed by J. Sjöstrand [13] in 1996.
2 Higher Order Exponential Estimates
We shall consider a Hamiltonian of the form
Φ(x) = ΦΛ(x) =
+ Ψ(x), x ∈ RΛ.
where
|∂α∇Ψ| ≤ Cα, ∀α ∈ N
. (14)
g will denote a smooth function on RΓ with lattice support Sg = Γ (& Λ) . We
shall identify g with g̃ defined on RΛ and shall assume that
|∂α∇g| ≤ Cα ∀α ∈ N
|Γ|. (15)
As in [66] , we shall momentarily assume that Ψ is compactly supported in RΛ
and g is compactly supported in RΓ but these assumptions will be relaxed later
Let M be the diagonal matrix
M = (δijρ(i))i,j∈Λ
where ρ is a weight function on Λ satisfying
e−λ ≤
ρ (i)
≤ eλ, if i ∼ j for some λ > 0. (16)
Assume also that for every M as above, there exists δo ∈ (0, 1) such that
−1HessΦ(x)Ma, a
≥ δoa
, ∀x ∈ RΛ, ∀a ∈ RΛ. (17)
For instance, the d−dimensional nearest neighbor Kac model
ΦΛ(x) =
ln cosh
(xi + xj)
satisfies this assumption for ν small enough. See [66] for details.
The following theorem has been proved in [66]:
Theorem 1 (A. Lo [66]) Let g be a smooth function with compact support on
RΓ satisfying
|∂α∇g| ≤ Cα ∀α ∈ N
|Γ| (18)
and Φ is as above. If f is the unique C∞−solution of the equation
−∆f +∇Φ ·∇f = g − 〈g〉
〈f〉L2(µ) = 0,
then ∑
f2xi(x)e
2κd(i,Sg) ≤ C ∀x ∈ RΛ.
κ and C are positive constants. C could possibly depend on the size of the
support of g but does not depend on Λ and f.
We now propose to generalize this theorem to higher order derivatives.
Proposition 2 If in addition to the assumptions of theorem 1, Φ satisfies the
following growth condition: for κ > 0 as above,
j,i1,...,ik∈Λ
Φ2xjxi1 ...xik
(x)e2κd({i1,...,ik},Sg) ≤ Ck ∀x ∈ R
Λ, for k ≥ 2 (19)
for some Ck > 0 not dependent on Λ and f , then for any k ≥ 1, f satisfies
i1,...,ik∈Λ
f2xi1 ...xik
(x)e2κd({i1,...,ik},Sg) ≤ Ck,g ∀x ∈ R
Λ (20)
where Ck,g > 0 is a constant that depends on the size of the support of g but not
on Λ and f.
Proof.
The case k = 1 being theorem 1, we assume for induction that the result is
true when k is replaced by k̂ < k with k̂ ≥ 2.
For k ≥ 2(see [8] for details), we have
g, t1 ⊗ ...⊗ tk
= (∇Φ ·∇−∆)
f, t1 ⊗ ...⊗ tk
f, t1 ⊗ ...⊗HessΦtj ⊗ ...⊗ tk
A∪B={1,...,k},A∩B=∅
#B≤k−2
〈tA(∂x)∇Φ, tB(∂x)∇f〉 .
In the right hand side of this last above equality, we have used the notation
tJ(∂x)u :=
u, t1 ⊗ ...⊗ t#J
Now fix i2, ..., ik ∈ Λ. Because ∇
f(x) → 0 as |x| → ∞ (see [66]), we consider
xo ∈ R
Λ that maximizes
x 7−→
f2xi1 ...xik
ρ2(i1, ..., ik)
where
ρ(i1, ..., ik) = e
κd({i1,...,ik},Sg).
Observe here that xo could possibly depend on i2, ..., ik ∈ Λ.
Choose
ρ(i1, ..., ik)fxi1 ...xik (xo)
tj = eij if j = 2, ..., k
Let M1 be the diagonal matrix
M1 = (δsi1ρ(i1, ..., ik))si1
Mj = I if j 6= 1 (21)
in particular, we have
g,M1t1 ⊗ ...⊗Mktk
= (∇Φ ·∇−∆)
f,M1t1 ⊗ ...⊗Mktk
f,M1t1 ⊗ ...⊗HessΦMjtj ⊗ ...⊗Mktk
A∪B={1,...,k},A∩B=∅
#B≤k−2
〈tMA(∂x)∇Φ, tMB(∂x)∇f〉
tMA(∂x)u :=
f,M1tij1 ⊗ ...⊗M#Atij#A
, ji ∈ A.
As in [66], the function
x 7−→
f(x),M1t1 ⊗ ...⊗Mktk
achieves its maximum at xo. Using the notation ΦxiA = Φxiℓ1 ...xiℓr
if A =
{ℓ1, ...ℓr} ⊂ {1, ...k} , we therefore have
gxi1 ...xik
(xo)ρ(i1, ..., ik)
2fxi1 ...xik
fxi1 ...xik
(xo)fxsxi2 ...xik (xo)ρ(i1, ..., ik)
2Φxsxi1 (xo)
fxi1 ... xs
...xik
(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
2Φxsxij (xo)
A∪B={1,...,k},A∩B=∅
#B≤k−2
∇ΦxiA fxi1 ...xik (xo)ρ(i1, ..., ik)
,∇fxiB (xo)
A∪B={1,...,k},A∩B=∅
#B≤k−2
∇ΦxiA ,
∇fxiB (xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
Equivalently
gxi1 ...xik
(xo)ρ(i1, ..., ik)
2fxi1 ...xik
fxi1 ...xik
(xo)fxs...xik (xo)ρ(i1, ..., ik)
2Φxsxi1 (xo)
fxi1 ... xs
...xik
(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
2Φxsxij (xo)
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik)
fxiBxs(xo)
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
Now taking summation over i2, ..., ik, we get
i2,...,ik∈Λ
gxi1 ...xik
(xo)ρ(i1, ..., ik)
2fxi1 ...xik
i2,...,ik∈Λ
fxi1 ...xik
(xo)fxs...xik (xo)ρ(i1, ..., ik)
2Φxsxi1 (xo)
i2,...,ik∈Λ
fxi1 ... xs
...xik
(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
2Φxsxij (xo)
i2,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik)
2fxiBxs(xo)
i2,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
Next, we propose to estimate each term of the right hand side of this above
inequality.
i2,...,ik∈Λ
fxi1 ...xik
(xo)fxs...xik (xo)ρ(i1, ..., ik)
2Φxsxi1 (xo)
i2,...,ik∈Λ
∇fxi2 ...xik
(xo),HessΦM1t1
i2,...,ik∈Λ
M1∇fxi2 ...xik
(xo),M
1 HessΦM1t1
i2,...,ik∈Λ
1 HessΦM1t1
i2,...,ik∈Λ
i1,...ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
Similarly, it is easy to see that
i2,...,ik∈Λ
fxi1 ... xs
...xik
(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
2Φxsxij (xo)
≥ (k − 1)δ0
i1,...ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
To estimate the last two terms, we have
i2,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
∣ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik)
2fxiBxs(xo)
i1,...,ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
i1,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
∣ΦxiAxsρ(i1, ..., ik)fxiBxs(xo)
To estimate the second factor of the right hand side of this last above inequality,
we have
i1,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
∣ΦxiAxs(xo)ρ(i1, ..., ik)fxiBxs(xo)
i1,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxs(xo)ρ(i1, ..., ik)fxiBxs(xo)
i1,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
Φ2xiAxs
(xo)ρ
2(i1, ..., ik)
ρ2(i1, ..., ik)f
xiBxs
i1,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
Φ2xiAxs
(xo)e
2κd({ij :j∈A},Sg)
e2κd({ij :j∈B}∪{s},Sg)f2xiBxs
≤ Ck.
This last inequality above follows from the induction assumption and that of
Thus,
i2,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik)
2fxiBxs(xo)
≥ −Ck
i1,...,ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
Similarly, we have
i2,...,ik∈Λ
A∪B={1,...,k},A∩B=∅
#B≤k−2
ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik)
≥ −Ck
i1,...,ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
We then finally get
i1,...,ik∈Λ
gxi1 ...xik
(xo)ρ(i1, ..., ik)
2fxi1 ...xik
≥ kδo
i1,...ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
i1,...,ik∈Λ
xi1 ...xik
(xo)ρ(i1, ..., ik)
i1,...,ik∈Λ
xi1 ...xik
(xo)ρ(i1, ..., ik)
2 = 0
then there is nothing to prove, otherwise we have, after using Cauchy-Schwartz
and dividing by
i1,...,ik∈Λ
f2xi1 ...xik
(xo)ρ(i1, ..., ik)
i1,...ik∈Λ
xi1 ...xik
(xo)ρ(i1, ..., ik)
i1,...,ik∈Λ
g2xi1 ...xik
+ Ck,g
≤ Ck,g. �
3 Relaxing the Assumptions of Compact sup-
As in [8], we consider the family cutoff functions
χ = χε (22)
(ε ∈ [0, 1]) in C∞o (R) with value in [0, 1] such that
χ = 1 for |t| ≤ ε−1
∣χ(k)(t)
∣ ≤ Ck
for k ∈ N .
We then introduce
Ψε(x) = χε(|x|)Ψ(x) x ∈ R
Λ (23)
gε(x) = χε(|x|)g(x) x ∈ R
. (24)
A straightforward computation (see [66]) shows that Ψε(x) and gε(x).satisfy
|∂α∇Ψε| ≤ Cα +Oα,Λ(ε), ∀α ∈ N
|Λ|. (25)
|∂α∇gε| ≤ Cα +Oα,Λ(ε), ∀α ∈ N
, (26)
and that
M−1HessΦε(x)M ≥ δ
, 0 < δ′ < 1. in the sense of (17) (27)
It then only remains to check that
j,i1,...,ik∈Λ
Ψ2εxjxi1 ...xik
(x)e2κd({i1,...,ik},Sg) ≤ Ck + Ok,Λ(ε) ∀x ∈ R
Λ, ∀k ≥ 2
where Ck is a positive constant that does not depend on f and Λ.
Ψε(x) = χε(r)Ψ(x)
Let α be such that |α| ≥ 3. Using Leibniz’s formula, we have
|∂αΨε| ≤
∣∂βχε(r)∂
∣ (29)
≤ |∂αχε(r)Ψ|+ |∂
β 6=0
∣∂βχε(r)∂
∣ . (30)
Assuming that Ψ(0) = 0 and write
Ψ(x) =
x ·∇Ψ(sx)ds
|∂αχε(r)Ψ(x)| ≤
∣xj1∂
αχε(r)Ψxj1 (sx)
≤ C |r∂αχε(r)| .
Now using the fact that
χε(r) = Oα(ε),
we have
|∂αχε(r)Ψ(x)| = Oα,Λ(ε).
Finally, using the fact that
χε(r) = Oβ(ε) for every |β| ≥ 1, (31)
it is then easy to see that
β 6=0
χε(r)∂
= Oα,Λ(ε). (32)
j,i1,...,ik∈Λ
Ψ2εxjxi1 ...xik
(x)e2κd({i1,...,ik},Sg) ≤ Ck,g+ Ok,Λ(ε) ∀x ∈ R
Λ, ∀k ≥ 2.
Now using the arguments developed in [8] (see also [66]) about the conver-
gence of the corresponding solutions as ε → 0, we obtain:
Proposition 3 If g(0) = Ψ(0) = 0, then Proposition 2 holds without the as-
sumptions of compact support on Ψ and g.
4 The n-Point Correlation Functions
The higher order correlation is defined as
〈g1, ..., gk〉 := 〈(g1 − 〈g1〉) ... (gk − 〈gk〉)〉 . (34)
For simplicity we shall take k = 3 and Φ is as in proposition 2.
Let g1, g2, and g3 be smooth functions satisfying (15) and fi i = 1, 2, 3 shall
denote the unique solution of the system
−∆fi +∇Φ ·∇fi = gi − 〈gi〉
L2(µ)
〈fi〉L2(µ) = 0.
Recall that
∇fi = A
(1)−1
Φ ∇gi.
For an arbitrary smooth function c, it is easy to see that
〈c(x) (gi − 〈gi〉)〉 = 〈∇fi ·∇c〉 .
A direct computation shows that
〈g1, g2,g3〉 = 〈∇f3 · (Hessf1)∇g2〉+ 〈∇f3 · (Hessg2)∇f1〉
+ 〈∇f2 · (Hessf1)∇g3〉+ 〈∇f2 · (Hessg3)∇f1〉 .
Let us now estimate each term of the right and side of this equality.
Using Cauchy-Schwartz, and proposition 2, it is easy to see that
|〈∇f3 · (Hessf1)∇g2〉| ≤ Ce
−κ1d(Sg2 ,Sg1)
|〈∇f3 · (Hessg2)∇f1〉| ≤ Ce
−κ1d(Sg2 ,Sg1),
|〈∇f2 · (Hessf1)∇g3〉| ≤ Ce
−κ1d(Sg3 ,Sg1)
|〈∇f2 · (Hessg3)∇f1〉| ≤ Ce
−κ1d(Sg3 ,Sg1)
Here the constants C only depends on the size of the support of the gi’s. and
κ1 > 0.
|〈g1, g2,g3〉| ≤ C
−κ1d(Sg2 ,Sg1) + e−κ1d(Sg3 ,Sg1)
If g1 = xi, g2 = xj , and g3 = xk, we obtain
|〈(xi − 〈xi〉) (xj − 〈xj〉) (xk − 〈xk〉)〉| ≤ C
e−κ1d(i,j) + e−κ1d(i,k)
Thus if d > 1, we obtain this weak exponential decay of the truncated correla-
tions in the sense that the exponential decay occurs as you simultaneously pull
the spins away from a fixed one. Note that in the one dimensional case, we
obtain a stronger exponential decay due to the fact that
i ≤ j ≤ k =⇒ d(i, k) = d(i, j) + d(j, k).
This was already pointed out in [8].
5 The Analyticity of the Pressure
In this section, we attempt to study a direct method for the analyticity of the
pressure for certain classical convex unbounded spin systems. It is central in
Statistical Mechanics to study the differentiability or even the analyticity of the
pressure with respect to some distinguished thermodynamic parameters such as
temperature, chemical potential or external field. In fact the analytic behavior of
the pressure is the classical thermodynamic indicator for the absence or existence
of phase transition. The most famous result on the analyticity of the pressure
is the circle theorem of Lee and Yang [28]. This theorem asserts the following:
consider a {−1, 1}−valued spin system with ferromagnetic pair interaction and
external field h and regard the quantity z = eh as a complex parameter, then all
zeroes of all partition functions (with free boundary condition), considered as
functions of z lie in the complex unit circle. This theorem readily implies that
the pressure is an analytic function of h in the region h > 0 and h < 0. Heilmann
[29] showed that the assumption of pair interaction is necessary. A transparent
approach to the circle theorem was found by Asano [30] and developed further by
Ruelle [31],[32], Slawny [33], and Gruber et al [34]. Griffiths [35] and Griffiths-
Simon [36] found a method of extending the Lee-Yang theorem to real-valued
spin systems with a particular type of a priory measure. Newman [37] proved
the Lee-Yang theorem for every a priory measure which satisfies this theorem in
the particular case of no interaction. Dunlop [38],[39] studied the zeroes of the
partition functions for the plane rotor model. A general Lee-Yang theorem for
multicomponent systems was finally proved by Lieb and Sokal [40]. For further
references see Glimm and Jaffe [41].
The Lee-Yang theorem and its variants depend on the ferromagnetic char-
acter of the interaction. There are various other way of proving the infinite
differentiability or the analyticity of the pressure for (ferromagnetic and non
ferromagnetic) systems at high temperatures, or at low temperatures, or at
large external fields. Most of these take advantage of a sufficiently rapid decay
of correlations and /or cluster expansion methods. Here is a small sample of rele-
vant references. Bricmont, Lebowitz and Pfister [42], Dobroshin [43], Dobroshin
and Sholsman [44],[45], Duneau et al [46],[47],[48], Glimm and Jaffe [41],[49], Is-
rael [50], Kotecky and Preiss [51], Kunz [52], Lebowitz [53],[54], Malyshev [55],
Malychev and Milnos [56] and Prakash [57]. M. Kac and J.M. Luttinger [58]
obtained a formula for the pressure in terms of irreducible distribution functions.
We propose a new way of analyzing the analyticity of the pressure for certain
unbounded models through a representation by means of the Witten Laplacians
of the coefficients in the Taylor series expansion. The methods known up to
now rely on complicated indirect arguments.
6 Towards the analyticity of the Pressure
Let Λ be a finite domain in Zd (d ≥ 1) and consider the Hamiltonian of the
phase space given by,
Φ(x) = ΦΛ(x) =
+ Ψ(x), x ∈ RΛ. (36)
where
|∂α∇Ψ| ≤ Cα, ∀α ∈ N
|Λ|, (37)
HessΦ(x) ≥ δo, 0 < δo < 1. (38)
Let g is a smooth function on RΓ with lattice support Sg = Γ. We identified
with g̃ defined on RΛ by
g̃(x) = g(xΓ) where x = (xi)i∈Λ and xΓ = (xi)i∈Γ (39)
and satisfying
|∂α∇g| ≤ Cα ∀α ∈ N
|Γ| (40)
Under the additional assumptions that Ψ is compactly supported in RΛ and g is
compactly supported in RΓ, it was proved in [66] (see also [8]) that the equation
−∆f +∇Φ ·∇f = g − 〈g〉
〈f〉L2(µ) = 0
has a unique smooth solution satisfying ∇kf(x) → 0 as |x| → ∞ for every
k ≥ 1.
Recall also that ∇f is a solution of the system
(−∆+∇Φ ·∇)∇f +HessΦ∇f = ∇g in RΛ. (41)
As in [66] and [8], these assumptions will be relaxed later on.
ΦtΛ(x) = Φ(x) − tg(x), (42)
where x = (xi)i∈Λ, and assume additionally that g satisfies
Hessg ≤ C. (43)
We consider the following perturbation
θΛ(t) = log
−ΦtΛ(x)
. (44)
Denote by
dxe−Φ
Λ(x) (45)
< · >t,Λ=
· dxe−Φ
. (46)
7 Parameter Dependency of the Solution
From the assumptions made on Φ and g, it is easy to see that there exists T > 0
such that or every t ∈ [0, T ), ΦtΛ(x) satisfies all the assumptions required for the
solvability, regularity and asymptotic behavior of the solution f(t) associated
with the potential ΦtΛ(x). Thus, each t ∈ [0, T ) is associated with a unique
C∞−solution, f(t) of the equation
f(t) = g − 〈g〉
L2(µ)
〈f(t)〉L2(µ) = 0.
Hence,
v(t) = ∇g (47)
where v(t) = ∇f(t). Notice that the map
t 7−→ v(t)
is well defined and
{v(t) : t ∈ [0, T )}
is a family of smooth solutions on RΛ satisfying
∂αv(t) → 0 as |x| → ∞ ∀α ∈ N|Λ| and for each t ∈ [0, T )
and corresponding to the family of potential
ΦtΛ : t ∈ [0, T )
. (48)
Let us now verify that v is a smooth function of t ∈ (0, T ).We need to prove
that for each t ∈ (0, T ), the limit
v(t+ ε)− v(t)
exists. Let
vε(t) =
v(t + ε)− v(t)
We use a technique based on regularity estimates to get a uniform control of
vε(t) with respect to ε.
With ε small enough, we have
0 = −∆
v(t+ ε)− v(t)
∇Φt+ε ·∇v(t+ ε)−∇Φt ·∇v(t)
HessΦt+εv(t+ ε)−HessΦtv(t)
Equivalently,
v(t + ε)− v(t)
∇Φt+ε ·∇ [v(t + ε)− v(t)]
+HessΦt+ε
v(t + ε)− v(t)
HessΦt+ε −HessΦt
v(t)−
∇Φt+ε −∇Φt
·∇v(t)
= Hessgv(t) +∇g ·∇v(t)
−∆vε(t) +∇Φt+ε ·∇vε(t) +HessΦt+εvε(t)
= Hessgv(t) +∇g ·∇v(t)
Let w(t) be the unique C∞−solution of the system
−∆w(t) +∇Φt ·∇w(t) +HessΦtw(t) = Hessgv(t) +∇g ·∇v(t). (49)
Recall that the unitary transformation UΦt+ε, allows us to reduce
−∆vε(t) +∇Φt+ε ·∇vε(t) +HessΦt+εvε(t)
= Hessgv(t) +∇g ·∇v(t)
into (
|∇Φt+ε|
∆Φt+ε
Vε +HessΦt+εVε =
[Hessgv(t) +∇g ·∇v(t)] e−Φ
t+ε/2
where Vε = vε(t)e−Φ
t+ε/2.
Remark 4 This unitary transformation already mentioned in the introduction
was introduced in the proof of the existence of solution (see [66] ) to avoid work-
ing with the weighted spaces L2(RΛ,RΛ, e−Φdx). The proof was based on Hilbert
space method. The method consists of determining an appropriate function space
and an operator which is a natural realization of the problem. In this particular
problem, the function spaces to be considered are the Sobolev spaces BkΦ(R
defined by
BkΦ(R
u ∈ L2(RΛ) : ZℓΦ∂
αu ∈ L2(RΛ) ∀ ℓ+ |α| ≤ k
where
These are subspaces of the well known Sobolev spaces W k,2(RΛ), k ∈ N.
Taking scalar product with Vε on both sides of (51), we get
∇Φt+ε
HessΦt+εVε ·Vεdx =
[Hessgv(t) +∇g ·∇v(t)] e−Φ
t+ε/2 ·Vεdx.
Now using the uniform strict convexity on the left hand side and Cauchy-
Schwartz on the right hand side, we obtain
‖Vε‖B0 ≤ Ct for small enough ε. (54)
We then deduce that
|∇Φt+ε|
Vε = q̃ε (55)
where
q̃ε = [Hessgv(t) +∇g ·∇v(t)] e
−Φt+ε/2+
∆Φt+ε
Vε −HessΦt+εVε (56)
is bounded in B0 uniformly with respect to ε for ε small enough.
Taking again scalar product with Vε on both sides of (55) and integrating
by parts, we obtain
‖∇Vε‖
|∇Φt+ε|
≤ ‖q̃ε‖L2 ‖V
ε‖L2 (57)
It follows that Vε is uniformly bounded with respect to ε in B1
for ε small
enough.
Next, observe that
|∇Φt|
Vε = q̂ε (58)
where
q̂ε = q̃ε −
|∇Φt+ε −∇Φt|
(∇Φt+ε −∇Φt) ·∇Φt
Vε (59)
= q̃ε −
ε2 |∇g|
ε∇g ·∇Φt
Vε (60)
is uniformly bounded in B0 with respect to ε for small enough ε. Using regu-
larity, it follows that for small enough ε, Vε is uniformly bounded in B2Φt with
respect to ε.This implies that q̂ε is uniformly bounded in B
Φt for ε small enough.
Again, we can continue by a bootstrap argument to consequently get that for ε
small enough, Vε is uniformly bounded in BkΦt with respect to ε for any k.
V = w(t)e−Φ
We have (
|∇Φt|
V +HessΦtV
= [Hessgv(t) +∇g ·∇v(t)] e−Φ
. (61)
Now combining this equation with (51), we obtain
|∇Φt|
(Vε −V) +HessΦt (Vε −V)
= − [Hessgv(t) +∇g ·∇v(t)] e−Φ
t/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ
t+ε/2
|∇Φt|
|∇Φt+ε|
∆Φt+ε
+(HessΦt −HessΦt+ε)Vε.
Now let us check that for small enough ε, the right hand side of (62) is O(ε)
in B0.
For the first term, we have
− [Hessgv(t) +∇g ·∇v(t)] e−Φ
t/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ
t+ε/2
= [Hessgv(t) +∇g ·∇v(t)] e−Φ
eεg/2 − 1
[Hessgv(t) +∇g ·∇v(t)] ge−Φ
Thus for ε small enough
∥− [Hessgv(t) +∇g ·∇v(t)] e
−Φt/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ
t+ε/2
≤ Cε.
For the second term, we have
|∇Φt|
|∇Φt+ε|
∣∇Φt+ε
∣∇Φt+ε
∣∇Φt+ε
∣+ ε |∇g|
Using now the fact that Vε is uniformly bounded in BkΦt with respect to ε
for any k, we see that the second term of the right hand side of (62) is O(ε)
in B0Φt . The last two terms of the right hand side of (62) are obviously O(ε) in
B0Φt .
From the same regularity argument as above, we get that Vε −V is O(ε) in
B2Φt . Again iterating the regularity argument, we obtain that for small enough
ε, Vε −V is O(ε) in BkΦt for every k. We have proved:
Proposition 5 Under the above assumptions on Φ and g, there exists T > 0
so that for each t ∈ (0, T ), vε(t) converges to w(t) in C∞.
Remark 6 The proposition establishes that v(t) is differentiable in t and
is given by the unique C∞−solution w(t) of the system
−∆w(t) +∇Φt ·∇w(t) +HessΦtw(t) = Hessgv(t)−∇g ·∇v(t). (63)
Iterating this argument, we easily get that, v(t) is smooth in t ∈ (0, T ).
Now we are ready for the following:
8 A Formula for the Taylor Coefficients
First observe that for an arbitrary suitable function f(t) = f(t, w)
< f(t) >t,Λ=< f
′(t) >t,Λ +cov(f, g). (64)
Hence,
< f(t) >t,Λ=< f
′(t) >t,Λ + < A
(1)−1
(∇f) ·∇g >t,Λ . (65)
Agf := A
(1)−1
(∇f) ·∇g. (66)
Thus,
< f(t) >t,Λ=<
f >t,Λ . (67)
The linear operator
+Ag will be denoted by Hg.
To obtain a formula for the coefficients in the Taylor expansion of
θΛ(t) = log
−ΦtΛ(x)
, (68)
we first the derivatives of θΛ(t) in terms of Hg
Λ(t) =< g >t,Λ=<
g >t,Λ=< H
gg >t,Λ;
Λ(t) =
< g >t,Λ=< A
(1)−1
(∇g) ·∇g >t,Λ=<
g >t,Λ;
Λ (t) =
(1)−1
(∇g) ·∇g >t,Λ=<
(1)−1
(∇g) ·∇g
(1)−1
(1)−1
(∇g) ·∇g
·∇g >t,Λ
g >t,Λ .
By induction it is easy to see that
Λ (t) =<
g >t,Λ=< H
(n−1)
g g >t,Λ (∀n ≥ 1)
Next, we propose to find a simpler formula for θ
Λ (t) that only involves Ag.
Hgg = A
(1)−1
Φt (∇g) ·∇g
= Agg
∇f ·∇g +
(1)−1
(1)−1
Φt (∇g) ·∇g
·∇g (69)
where f satisfies the equation
∇f = A
(1)−1
(∇g) . (70)
With v(t) = ∇f, as before, we get
∇f ·∇g = A
(1)−1
Φt (Hessgv(t) +∇g ·∇v(t)) ·∇g
and H2g becomes
H2gg = A
(1)−1
(Hessgv(t) +∇g ·∇v(t)) +∇
(1)−1
(∇g) ·∇g
(1)−1
2∇ (Agg) ·∇g
= 2A2gg.
Proposition 7 If
θΛ(t) = log
−Φt(x)
where
Φt(x) = ΦΛ(x)− tg(x)
is as above then θ
Λ (t), the nth− derivative of θΛ(t) is given by the formula
Λ(t) =< g >t,Λ,
and for n ≥ 1
Λ (t) = (n− 1)! < A
g g >t,Λ .
Proof. We have already established that
Λ (t) =< H
g g >t,Λ for n ≥ 1.
It then only remains to prove that
Hn−1g g = (n− 1)!A
g g for n ≥ 1.
The result is already established above for n = 1, 2, 3, . By induction, assume
Hn−1g g = (n− 1)!A
g g .
if n is replaced by ñ ≤ n.
g g =
(n− 1)!An−1g g
= (n− 1)!
An−1g g +A
An−1g g =
(1)−1
An−2g g
= ∇ϕn ·∇g
where
∇ϕn =
(1)−1
We obtain,
∇ϕn = A
(1)−1
∇An−2g g +Hessg∇ϕn +∇g ·∇ (∇ϕn)
We then have
An−1g g =
∇ϕn ·∇g
(1)−1
∇An−2g g +Hessg∇ϕn +∇g ·∇ (∇ϕn)
(1)−1
∇An−2g g +∇ (∇ϕn ·∇g)
An−2g g +Ag
An−2g g
= AgHg
= AgHg
(n− 2)!
H(n−2)g g
(from the induction hypothesis)
(n− 2)!
(n−1)
(n− 2)!
(n− 1)!An−1g g
(still by the induction hypothesis)
= (n− 1)Ang g.
Thus,
Hng g = (n− 1)! (n− 1 + 1)A
= n!Ang g
Proposition 8 If g(0) = 0, then the formula
Λ (t) = (n− 1)! < A
g g >t,Λ, n ≥ 2
still holds if we no longer require Ψ and g to be compactly supported in RΛ.
Proof. As in [8], consider the family cutoff functions
χ = χε (71)
(ε ∈ [0, 1]) in C∞o (R) with value in [0, 1] such that
χ = 1 for |t| ≤ ε−1
∣χ(k)(t)
∣ ≤ Ck
for k ∈ N
We could take for instance
χε(t) = f(ε ln |t|)
for a suitable f .
We then introduce
Ψε(x) = χε(|x|)Ψ, x ∈ R
Λ (72)
gε(x) = χε(|x|)g x ∈ R
Γ (73)
One can check that both Ψε(x) and gε(x) satisfies the assumptions made above
on Ψ and g. Now consider the equation
−∆fε +∇Φ
ε ·∇fε = gε− < gε >t,Λ. (74)
which implies
−∆+∇Φtε ·∇
⊗ vε +HessΦ
εvε = ∇gε (75)
where
vε= ∇fε
It was proved in [8] that vε = A
(1)−1
∇gε converges in C
∞ to A
(1)−1
∇g as
ε → 0.
Proposition 9 Let
PΛ(t) =
θΛ(t)
be the finite volume Pressure.
Denote by an (n ≥ 2) the nth Taylor coefficient. We have
< An−1g g >Λ
n |Λ|
Remark 10 This formula for an gives a direction towards proving the analytic-
ity of the pressure in the thermodynamic limit. In fact one only needs to provide
a suitable Cn estimate for < An−1g g >Λ .
9 Some Consequences of the Formula for nth−Derivative
of the Pressure.
In the following, we shall additionally assume that
∇g(0) = 0, and
∇ΦtΛ(0) = 0 for all t ∈ [0, T ).
When n = 1, we recall that A0gg = g,
Λ(t) =< g >t,Λ
and if
v(t) = ∇f = A
(1)−1
then we have
−∆+∇ΦtΛ ·∇
⊗ v(t) +HessΦtΛv(t) = ∇g
Again the tensor notation means that (−∆+∇ΦtΛ ·∇) acts diagonally on the
components of v(t).
As in [8] v(t) is a solution of the equation
g =< g >t,Λ +v(t) ·∇Φ
Λ − divv(t). (76)
Using the assumptions above, we have
Λ(t) = < g >t,Λ
= divv(t)(0).
Similarly, the formula
Λ (t) = (n− 1)! < A
g g >t,Λ,
implies that
Λ (t) = (n− 1)!divvn(t)(0),
where
vn(t) = A
(1)−1
An−1g g
Acknowledgements. I would like to thank Professor. Haru Pinson and Pro-
fessor. Tom Kennedy for accepting to discuss with me the ideas developed in
this paper. I also would like to acknowledge Professor Bruno Nachtergaele for
his constructive suggestions and all members of the mathematical physics group
at the University of Arizona for their support.
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	Introduction
	Higher Order Exponential Estimates
	Relaxing the Assumptions of Compact support
	The n-Point Correlation Functions
	The Analyticity of the Pressure
	Towards the analyticity of the Pressure
	Parameter Dependency of the Solution
	A Formula for the Taylor Coefficients
	Some Consequences of the Formula for nth-Derivative of the Pressure.
