Long-range order in Gibbs lattice classical linear oscillator systems
The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generali...
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irk-123456789-1649672020-02-12T01:29:00Z Long-range order in Gibbs lattice classical linear oscillator systems Skrypnik, W.I. Статті The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools. Доведено існування феромагнітного далекого порядку для гіббсівської класичної ґраткової системи лінійних осциляторів, що взаємодіють завдяки сильному парному поліноміальному феромагнітному потенціалу близьких сусідів та іншим (непарним) потенціалам, які слабкі, якщо не феромагнітні. При цьому використано узагальнений аргумент Пайєрлса та дві контурні нерівності. 2006 Article Long-range order in Gibbs lattice classical linear oscillator systems / W.I. Skrypnik // Український математичний журнал. — 2006. — Т. 58, № 3. — С. 388–405. — Бібліогр.: 21 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164967 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Skrypnik, W.I. Long-range order in Gibbs lattice classical linear oscillator systems Український математичний журнал |
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The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools. |
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Skrypnik, W.I. |
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Skrypnik, W.I. |
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Skrypnik, W.I. |
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Long-range order in Gibbs lattice classical linear oscillator systems |
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Long-range order in Gibbs lattice classical linear oscillator systems |
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Long-range order in Gibbs lattice classical linear oscillator systems |
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Long-range order in Gibbs lattice classical linear oscillator systems |
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Long-range order in Gibbs lattice classical linear oscillator systems |
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long-range order in gibbs lattice classical linear oscillator systems |
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Інститут математики НАН України |
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2006 |
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Статті |
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Long-range order in Gibbs lattice classical linear oscillator systems / W.I. Skrypnik // Український математичний журнал. — 2006. — Т. 58, № 3. — С. 388–405. — Бібліогр.: 21 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT skrypnikwi longrangeorderingibbslatticeclassicallinearoscillatorsystems |
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2025-07-14T17:42:59Z |
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2025-07-14T17:42:59Z |
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1837645147954741248 |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL
LINEAR OSCILLATOR SYSTEMS
DALEKYJ PORQDOK U HIBBSIVS\KYX ÌRATKOVYX
KLASYÇNYX SYSTEMAX LINIJNYX OSCYLQTORIV
Existence of the ferromagnetic long range order (lro) is proven for Gibbs classical lattice systems of linear
oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other
(nonpair) potentials which are weak if they are not ferromagnetic. A generalized Peierls argument and two
different contour bounds are our main tools.
Dovedeno isnuvannq feromahnitnoho dalekoho porqdku dlq hibbsivs\ko] klasyçno] ©ratkovo] systemy
linijnyx oscylqtoriv, wo vza[modigt\ zavdqky syl\nomu parnomu polinomial\nomu feromahnitnomu
potencialu blyz\kyx susidiv ta inßym (neparnym) potencialam, qki slabki, qkwo ne feromahnitni. Pry
c\omu vykorystano uzahal\nenyj arhument Paj[rlsa ta dvi konturni nerivnosti.
1. Introduction and main result. Let’s consider Gibbs classical and quantum systems of
one-dimensional oscillators on the d-dimensional hypercubic lattice Z
d, with the potential
energy U(qΛ) = U(−qΛ) on a set Λ with the finite cardinality |Λ|, where qΛ is an array
of (qx, x ∈ Λ), qx is the oscillator coordinate taking value in R. The potential energy is
assumed to be a growing function at infinity. It is invariant under the simplest discrete
symmetry, namely Z2, which is realized as a transformation of changing of all the signs
of the oscillator variables. We tacitly assume, also, that the correlation functions exist in
the thermodynamic limit.
Let 〈 〉Λ denote the Gibbs average for the system confined to Λ (see (2.1)) and
χ+
x (qΛ) = χ(0,∞)(qx), χ−
x (qΛ) = χ(−∞,0)(qx), σx(qΛ) = qx, sx = signσx,
where χ(a,b) is the characteristic function of the open interval (a, b). Taking into account
that χ+(−)
x =
1
2
[1 ± sx] one obtains
4〈χ+
x χ
−
y 〉Λ = 1 + 〈sx〉Λ − 〈sy〉Λ − 〈sxsy〉Λ.
Since the systems are invariant under the transformation of changing signs of the
oscillator variables the third and the second terms in the right-hand side of last equality
are equal to zero and
〈sxsy〉Λ = 1 − 4〈χ+
x χ
−
y 〉Λ.
Hence if
〈χ+
x χ
−
y 〉Λ <
1
4
(1.1)
then the ferromagnetic lro for the unit spins (unit spin lro) sx occurs, i.e.,
〈sxsy〉Λ ≥ a > 0, (1.2)
where a is independent of Λ. (1.1) may be derived from the generalized Peierls principle.
Theorem 1.1. Let the contour bound holds〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤ e−|Γ|E , (1.3)
c© W. I. SKRYPNIK, 2006
388 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 389
where Γ is the set of pairs of nearest neighbors and e−2E <
1
6
. Then there exists a
constant a independent Λ such that
〈χ+
x χ
−
y 〉Λ ≤ ae−2dE .
Corollary 1.1. If
e−2E < min
(
6−1, (4a
1
d )−1
)
then the unit spin ferromagnetic lro occurs, that is (1.2) holds.
The first application of the Peierls argument for proving of the existence of the unit
spin lro was given in [1] for Euclidean two-dimensional quantum boson field system
which can be approximated by a lattice linear oscillator system (the analog of sx was
a smeared over unit cubes, centered at the sites of the square lattice, boson field). A
wide range of applicability of the argument was demonstrated later in [2] for anisotropic
Heisenberg systems and simplest classical systems of many component oscillators inter-
acting via a pair n-n potential (see Remark 1). In [3 – 6] we extended the range to lattice
linear oscillator systems with a strong ferromagnetic n-n belinear interaction and nonequi-
librium simplest lattice systems of interacting Brownian linear oscillators, respectively.
In this paper, we prove the contour bound (1.3), infinite growth of E in the parameter
g, and existence of the lro for two lattice linear oscillator systems with the following
potential energy (Λ is a hypercube with cardinality |Λ| ) which is the most general in the
class of energies with polynomial external fields and pair n-n potentials
U(qΛ) =
∑
x∈Λ
(u0(qx) − gη0q
2n0
x ) +
∑
〈x,y〉∈Λ
φ(qx, qy) + U ′(qΛ), η0 = 0, 1, (1.4)
where U ′ equals zero for coinciding arguments, 〈x, y〉 means nearest neighbors, u is a
bounded below polynomial of the 2n-th degree n > n0,
u0(q) = ηq2n + u1(q), u1(q) =
n−1∑
j=1
ηjq
2j ,
φ is given either by
η0 = 0, φ(qx, qy) = −g(qkxqly + qlxq
k
y ), k + l = 2n0,
or
η0 = 1, φ(qx, qy) = g0(qx − qy)2n1Q′(qx, qy), g0 = g
ξ
2(n−n0) = z−ξ,
k, l are odd positive numbers, Q′ is a positive symmetric even homogeneous polynomial
of the 2n2-th degree, i.e., is the sum of the terms qsqm, s + m = 2n2 with positive
coefficients, n1 + n2 < n, such that Q′(1, 1) = 1 and n′2 = max(s,m); for η0 = 0 the
function U ′ is such that the Gibbs averages satisfy the GKS inequality [7, 8]. Such the
systems we’ll call GKS-type systems (see Remark 2). Prime will not mean differentiation
in our notations.
The parameter g determines depthes of minima of the potential u(q) = u0(q) −
− gη0q
2n0 (see Remark 3).
It will be proven in Appendix C the following important equality:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
390 W. I. SKRYPNIK
qkxq
l
y + qlxq
k
y = q2n0
x + q2n0
y − (qx − qy)2Q(qx, qy), (1.5)
where Q is a positive homogeneous polynomial with the degree 2(n0 − 1) (it is a sum of
Q′ and a positive homogeneous noneven polynomial). This equality transforms the case
η0 = 0 into the case η0 = 2d. By rescaling of variables the case η0 > 0 may be mapped
into the case η0 = 1.
Our first result, generalizing the result of [6], is formulated as the following theorem.
Theorem 1.2. For classical oscillator GKS-type systems with the potential energies
(1.4) with η0 = 0 the contour bound (1.3) holds with the positive function E in g, which
is growing at infinity as g
n
n−n0 , and the unit spin lro occurs at sufficiently large g.
This theorem follows from the explicit expression for E in (1.3) given by (2.2) – (2.4)
and Theorem 2.1.
Since the corner stone of our technique for η0 = 1 is the generalized Ruelle super-
stability bound we have to introduce the superstability and regularity conditions for the
rescaled potential energy. We rescale all the variables by zn−1 and put ug(q) = u(zn−1q),
φg(q, q′) = φ(zn−1q, zn−1q′),
u∗(q) = ug(q + e0) − ug(e0), φ∗(q, q′) = φg(q + e0, q
′ + e0),
Ug(qΛ) = U(zn−1qΛ), U∗(qX) = Ug(qX + e0) − |X|ug(e0),
where e0 is the deepest minimum of ug which grows in g at infinity (see Section 3), e0
is easily found for the potential u0 = ηq2n leading to the effective potential u which
has only two symmetric wells. One may see that U∗ has a finite limit when g tends to
infinity. This fact plays an important role in our method since we’ll obtain bounds for the
correlation functions generated by U∗.
Let’s put
W∗(qX ; qY ) = U∗(qΛ) − U∗(qY ) − U∗(qX), Λ = X ∪ Y.
The same relation will hold, also, for U ′
∗ and W ′
∗.
We require validity of the following superstability and regularity conditions:
U∗(qΛ) ≥
∑
x∈Λ
u−∗ (qx), u−∗ (q) = u∗(q) − ζv0(znq) − ζ0, (1.6)
|W ′
∗(qX ; qY )| ≤
∑
x∈X,y∈Y
Ψ′(|x− y|)(v0(qx) + v0(qy)),
v0(q) =
n−1∑
j=1
q2j ,
(1.7)
where the positive L1-function Ψ′, and numbers ζ, ζ0 ≥ 0 do not depend on g. The
second term in the expression for u−∗ is a contribution of the negative (nonferromagnetic)
term in U ′ which shows that the latter is always small, that is, it depends on positive
powers of g−1. Condition (1.6) can be substituted by the inequality for U coinciding with
(1.6) in which v0(zq) is inserted instead of v0(znq). Conditions (1.6), (1.7) allow positive
(ferromagnetic) interaction terms in U to be large. Interaction is stronger for translation
invariant potentials. For translation invariant interaction two conditions for the functions
U ′,W ′ as in [3, 4] may be postulated in such the way that they will imply (1.6), (1.7),
that is (1.6), (1.7) generalize the superstability and regularity conditions from [3, 4].
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 391
Theorem 1.3. Let the potential energy of the classical oscillator systems in (1.4)
with η0 = 1 satisfy (1.6), (1.7) with Ψ′ having a finite range and
n− 2(n1 + n2) < ξ ≤ 2(n− 1)n1 − 2n2. (1.8)
Then the contour bound (1.3) holds with the positive function E in g, which is growing at
infinity as g
n
2(n−n0) , and the unit spin lro occur at sufficiently large g.
This theorem is based on the derivation of the E from (1.3) in the form
E = βe0 − e∗(g) + 2 ln I∗(g,Q0), (1.9)
where the integral I∗(g,Q0) is given by (2.13) and e∗(g) is extracted from our general-
ization of the Ruelle superstability bound proposed in Theorem 2.2. The corner stone of
its proof is check of the condition that the second and third terms have to grow at infinity
in g not so fast as e0.
These theorems establish existence of phase transitions for the classical systems with
pair short-range Kunz potentials since in the high-temperature phase there is a decrease
of correlations [9]. They imply that the behavior of the strength of the n-n pair interaction
has to be correlated with the depth of the deepest minimum of the effective potential and
a character of the potential itself (this is done through the formula (2.8)).
The right-hand side of (1.8) guarantee that ug is a bounded function in g and that
W∗ satisfies (1.7) with Ψ(|x − y|) = Ψ′(|x − y|) + C̃δ|x−y|,1 instead of Ψ′
|x−y|, where
δx,y is the Kronecker symbol (see Appendix A). The left-hand side of (1.8) arises from
application of the exponential bound (2.8) for the product of the characteristic functions
in (1.3) and the necessity of suppression of the first unbounded term in the expression Qg
in it (Qg in (2.10) does not allow the strength of the nn ferromagnetic interaction g0 be
arbitrary small). As a result of this bound asymptotics of β−1E in g coincides with the
asymptotics of the deepest minimum of the external potential u in the last theorem.
Theorem 1.3 generalizes our previous results in [3, 4] for finite-range interactions
with the help of the new version of the superstability bound that has appeared already
in [5]. It was applied in [5] for a proof of lro in nonequilibrium systems of Brownian
oscillators with Gibbsian initial states and it contains an exponent with a potential energy,
generated by the pair ferromagnetic n-n potential, before the right-hand side of the usual
superstability bound.
The main ideas of our method are exposed in the next section where it is shown how
Theorems 2.1, 2.2 allow to prove Theorems 1.2, 1.3. In Section 3 we formulate and prove
Lemma 3.1 which gives bounds necessary for proofs of Theorems 2.1, 2.2. In Section 4
our basic superstability bound is presented which implies validity of the superstability
bounds in Theorem 2.2, Lemma 4.1 and Corollary 4.2 establish asymptotics of e∗ from
Theorem 2.2. Proof of Theorem 1.1 and Theorem 4.1 may be found in [5].
Development of our approach demanded a clarification of the question in what way
the Ruelle superstability bound [10] depends on the behavior of the function Ψ from at
infinity in the regularity condition (4.2). It turned out that the usual superstability bound
does not feel such the behavior. We introduce the condition∑
x
|x|kΨ(|x|) <∞
which coincide with the Ruelle condition for k = 1 and establish the asymptotics of
the main constant c of the new superstability bound in an arbitrary small constant ε in
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
392 W. I. SKRYPNIK
(4.3) in terms of k (see Corollary 4.1). For finite-range interactions c0 grows very slow
at zero in ε and this helps to find asymptotics of E in (1.9) after putting ε = z2(n−1)n.
A generalization of Theorem 1.3 to infinite-range interaction demands a more profound
change of the Ruelle arguments which concern asymptotics of c0 in ε.
We expect that that our results may be generalized to the potentials u exp{ũ}, where
u, ũ are polynomials, for which e0 is easily found.
2. Contour bounds. For classical linear oscillator systems with the potential energy
U and the inverse temperature β the Gibbs average is given by
〈FX〉Λ = Z−1
Λ
∫
FX(qX)e−βU(qΛ)dqΛ =
∫
FX(qX)ρΛ(qX)dqX ,
ρΛ(qX) = Z−1
Λ
∫
e−βU(qΛ)dqΛ\X , ZΛ =
∫
e−βU(qΛ)dqΛ.
(2.1)
Here the integrations are performed over R|Λ|, R|X|, R|Λ\X| and ρΛ are the correlation
functions, where |X| is the cardinality of X .
The contour bound for η0 = 1 is derived from the new generalized Ruelle superstabil-
ity bound for correlation functions and reduced density matrices. The contour bound for
η0 = 0 is obtained with a help of a generalized Bricmont – Fontaine argument [11] and
results from the inequality for k = 2s+ 1, l = 2(n0 − s) − 1
χ+
x χ
−
y ≤ e−
gβ
2 (σk
xσ
l
y+σl
xσ
k
y ). (2.1a)
Its proof is easy
χ+
x χ
−
y = e−
gβ
2 (σk
xσ
l
y+σl
xσ
k
y )e
gβ
2 (σk
xσ
l
y+σl
xσ
k
y )χ+
x χ
−
y ≤
≤ e−
gβ
2 (σk
xσ
l
y+σl
xσ
k
y )χ+
x χ
−
y ≤ e−
gβ
2 (σk
xσ
l
y+σl
xσ
k
y ).
Here one takes into account that σx ≥ 0, σy ≤ 0. The last inequality is employed for the
proof of the contour bound as follows:〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤
〈
e−
gβ
2
∑
〈x,y〉∈Γ(σk
xσ
l
y+σl
xσ
k
y )
〉
Λ
=
=
〈
e
gβ
2
∑
〈x,y〉∈Γ(σk
xσ
l
y+σl
xσ
k
y )
〉
Λ[Γ]
≤
≤ e−
gβ
2
∑
〈x,y〉∈Γ〈(σ
k
xσ
l
y+σk
yσ
l
y)〉Λ[Γ] = eEΓ ,
where 〈., .〉Λ[Γ] is the average corresponding to the potential energy
UΓ(qΛ) = U(qΛ) +
g
2
∑
〈x,y〉∈Γ
(
qkxq
l
y + qlxq
k
y
)
.
In last line, we applied the Jensen inequality. Taking into account that the Gibbs
average is a monotone increasing function in J(A,nA) (a result of the GKS inequality, see
Remark 2) we obtain
EΓ ≥ E|Γ|, E = βg〈σkσ′l + σlσ′
k〉, (2.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 393
where
〈σkσ′l + σlσ′
k〉 = Z−1(2)
∫
(qk1q
l
2 + ql1q
k
2 )e−βu(q1,q2)dq1dq2, (2.3)
Z(2) =
∫
e−βu(q1,q2)dq1dq2,
u(q1, q2) = u0(q1) + u0(q2) − g(qk1q
l
2 + ql1q
k
2 ),
(2.4)
k, l are positive integers, k = 2s − 1, l = 2(n0 − s) − 1, i.e., k + l = 2n0 and β is
the inverse temperature. Formula (1.3) together with (2.4) is the generalized BF contour
bound.
The asymptotics of E in (2.2) is determined with the help of the following theorem.
Theorem 2.1. There exists a positive function in β C+ independent of g and a
positive constant c− such that for sufficiently large g two following inequalities hold:
〈σ2n0 + σ′
2n0〉 ≥ C+g
−1g
n
n−n0 , (2.5)〈
| − σ2n0 − σ′
2n0 + σkσ′
l + σ′
k
σl|
〉
≤ (gβ)−1C−, k + l = 2n0. (2.6)
The most important implication of this theorem is the following statement.
Corollary 2.1. Let E be given by (2.2). Then E grows in g at infinity as g
n
n−n0 .
This corollary is a result of (2.5), (2.6) and the obvious inequality
〈σkσ′l + σ′
k
σl〉 ≥ 〈σ2n0 + σ′
2n0〉 −
∣∣∣〈−σ2n0 − σ′
2n0 + σkσ′
l + σ′
k
σl〉
∣∣∣.
Theorem 1.2 is proved.
Main idea of proof of Theorem 1.3. Let’s put
ρΛ
∗ (qX) = ρΛ
g (qX + e0).
From the translation invariant character of the Lebesque measure it follows that ρΛ
∗ is
expressed in terms of U∗. The dependence of u∗, superstability and regularity conditions
on g enable us to prove Theorem 1.3 with the help of the following theorem.
Theorem 2.2. Let the conditions of Theorem 1.3 be satisfied. Then for the correla-
tion functions ρΛ
∗ the following supertability bound is valid:
ρΛ
∗ (qX) ≤ exp
{
|X|e∗(g) − β
[
U+
∗ (qX) +
∑
x∈X
u+
∗ (qx)
]}
, (2.7)
where
u+
∗ = u−∗ − 3εv0, ε = z2n(n−1),
e∗(g) depends neither on oscillator variables nor Λ, U+
∗ is the positive part of U∗ gener-
ated by a pair potential and for arbitrary δ > 0 and sufficiently large g
δe0 ≥ e∗(g). (2.7a)
Inequality (2.7) differs from the usual superstability bound from [10] by the presence
of U+
∗ in its right-hand side. This term plays a significant role in proving (1.1) with
positive E(g) increasing at infinity in g: it forbids the strength g0 of the positive n-n pair
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
394 W. I. SKRYPNIK
interaction uncontrollably tend to zero after applying (2.8). Calculation of asymptotics of
e∗(g) is quite a nontrivial task that can be solved only with the help of the generalization
of the Ruelle superstability bound proposed in Theorem 4.1 ((4.4) determines the structure
of the main constant c0 in the new superstability bound).
The exponential term appears in the bounds of the correlation functions of classi-
cal oscillator systems in the high-temperature phase if one applies the technique of the
Kirkwood – Saltsburg relations considered in [9, 12]. The similar term is present in the
superstability bound in [13] for classical oscillator ferromagnetic systems with a special
pair potential.
We derive the contour bound (1.3) for the potential energy (1.4) for η0 = 1 with the
help of (2.7) and the following bound which is a generalization of the similar bound from
[3 – 5]: ∏
〈x,x′〉∈Γ
χ+(qx)χ−(qx′) ≤ exp
{
−β
[
e0|Γ| −Qg,Γ(qΛ)
]}
, (2.8)
where
Qg,Γ(qΛ) =
∑
〈x,x′〉∈Γ
Qg(qx, qy),
Qg(qx, qy) = e
−2(n1+n2)+1
0 ×
×
[
22n2g−1
0 φ(qx, qy) +
(
4
3
)n1+n2(
|q2x − e20|n1+n2 + |q2y − e20|n1+n2
)]
.
The idea to apply the analogue of (2.8) goes back to [1]. (2.8) is a consequence of the
following inequality:
χ+(qx)χ−(qy) ≤ ec[−R
2(n1+n2)+Q(qx,qy|R)], R, c > 0, (2.9)
where
Q(qx, qy|R) = 22n2g−1
0 φ(qx, qy) +
(
4
3
)n1+n2(
|q2x −R2|n1+n2 + |q2y −R2|n1+n2
)
.
It is derived easily from two inequalities
χ+(qx)χ−(qy) ≤ ec[−R
2(n1+n2)+g−1
0 22n2φ(qx,qy))], |qx|, |qy| ≥ 2−1R,
χ+(qx)χ−(qy) ≤ ec[−R
2(n1+n2)+( 4
3 )n1+n2 (|q2x−R2|n1+n2+|q2y−R2|n1+n2 )],
|qx|, |qy| ≤ 2−1R.
To derive the first inequality one has to account the inequalities for qx ≥ R
2
, qy ≤ −R
2
(qx − qy)2n1 ≥ R2n1 ,
Q′(qx, qy) = Q′(|qx|, |qy|) ≥ Q′
(
R
2
,
R
2
)
=
(
R
2
)2n2
Q′(1, 1) =
(
R
2
)2n2
.
For |qx| ≤ R
2
, |qy| ≥ R
2
the second term in the expression for Q is not less than
R2(n1+n2). Inequality (2.8) results from (2.9) putting
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LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 395
Qg(q, q′) = Q(q, q′|R), c = βe
−2(n1+n2)+1
0 , R = e0.
Let’s substitute (2.8) into (1.3), translate all the variables by e0 in the Gibbs average
and apply (2.7)〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤ e−βe0|Γ|
∫
ρΛ
∗ (qΓ) exp{βQg,Γ(qΓ)}dqΓ ≤
≤ e−(βe0−e∗(g))|Γ|
∫
exp
{
− β
[ ∑
x∈Γ
u+
∗ (qx)+
+
∑
〈x,y〉∈Γ
(φg(qx + e0, qy + e0) −Qg(qx + e0, qy + e0)
]}
dqΓ. (2.10)
We’ll show in the next section that e0 = µz−n, where the µ is a bounded function of
g . For u0(q) = ηq2n this function is a constant. Now, if
z−ξz2(n−1)(n1+n2) > e
−2(n1+n2)+1
0 , (2.11)
or (we use the equality φg = z2(n1+n2)(n−1)φ)
−ξ + 2(n− 1)(n1 + n2) < (2(n1 + n2) − 1)n,
then the terms with φg disappear in the right-hand side of (2.10) and we obtain for
I∗(g,Q0) > 1 〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤ e−(βe0−e∗(g))|Γ|(I∗(g,Q0))2|Γ|, (2.12)
where
I∗(g,Q0) =
∫
e−β[u+
∗ (q)−e−2(n1+n2)+1
0 Q0(q;e0)]dq,
Q0(q; e0) =
(
4
3
)n1+n2
|q(q + 2e0)|n1+n2 .
(2.13)
If I∗(g,Q0) ≤ 1 then (2.12) holds without the last term in its right-hand side. The
condition I∗(g,Q0) > 1 is necessary in (2.12) to account the fact that in pairs from Γ
there may be coinciding sites. As a result (1.9) holds.
From Proposition 3.1 and (2.7a) it follows that E in (1.9) grows at infinity in g and
the unit spin lro occurs in the system. This will conclude the proofs of Theorems 1.2, 1.3.
3. Asymptotics of integrals. Our main aim in this sections is to determine the asymp-
totics of E in g at infinity in formulas (1.9) and (2.2). For this purpose let’s introduce the
new potential h and describe properties of the minima of ug and u in terms of its minima
u(z−1q) = z−2nh(q), ug(q) = u(zn−1q) = z−2nh(znq),
where
h(q) = ηq2n − q2n0 + h1(q), h1(q) =
n−1∑
j=1
ηjz
2(n−j)q2j = z2nu1(z−1q).
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396 W. I. SKRYPNIK
Let e′(j), µ(j), e(j) be the minima of u, h, ug , respectively. The deepest among the
sequences will be denoted by e+, µ, e0, respectively. Then
e′(j) = z−1µ(j), e(j) = z−nµ(j). (3.1)
It is so since for the equations for the minima we have
∂u(z−1q) = ∂h(q) = 0, ∂ug(q) = ∂h(znq) = 0, ∂ =
∂
∂q
.
µ is the only root of ∂h which converges to the unique positive root µ0 of ∂h0
h0(q) = ηq2n − q2n0 , ∂h0(q) = 2q2n0−1(ηnq2(n−n0) − n0)
when z tends to zero. It is simple and an analytical function of z in the neighborhood of
zero (see Section 1 in [14]), that is
µ = µ0 +
∑
k≥1
zkµk, µ0 =
(
n0
nη
) 1
2(n−n0)
. (3.2)
It is the deepest minimum of h since other roots µj of ∂h converge to zero for vanishing
z. For these roots there is the following convergent expansion in the neighborhood of
zero [14]:
µ(j) =
∑
k≥1
z
k
lj µk,j ,
∑
s
ls = 2n0 − 1, 1 ≤ lj ≤ 2n0 − 1. (3.3)
Validity of these expansions can be confirmed for a special choice of u1 (see Appen-
dix B).
It is not difficult to check that
∂sug(q) = z−2n+sn∂sh(znq), ∂2ug(q) = ∂2h(znq). (3.4)
Relations (3.3), (3.4) yield
∂sug(e(j)) = z−2n+sn∂sh(µ(j)), ∂2ug(e(j)) = ∂2h(µ(j)). (3.5)
This implies z the derivatives of ∂sug at the minima of ug tend to zero if g tends to infinity
and s > 2. That is
ug(q + e0) − ug(e0) = aq2 + pg(q), a = ∂2h(µ), lim
q→∞
pg(q) = 0. (3.5a)
We’ll restrict z to the neighborhood of zero, i.e., the interval [0, z0], for which z ≤ 1,
zn ≤ 1
4
µ0, µ >
1
2
µ0, µj <
1
2
µ. Let θ =
1
4
min
z∈[0,z0]
∂h2(µ). θ > 0 since ∂h2(µ0) > 0
and µ is continuous in z.
From (3.5) and the Taylor expansion it follows that (∂2h(µ) > 0, z ≤ 1)
ug(q + e0) − ug(e0) ≤ C̃, |q| ≤ 1, C̃ = max
0≤z≤z0
2n∑
s=2
z2n−sn|∂sh(µ)|(s!)−1.
(3.5b)
Taylor expansion gives possibility to choose sufficiently small ε such that
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LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 397
h(q + µ) − h(µ) ≥ θq2, −εµ ≤ q ≤ εµ. (3.5c)
µj are shallow minima with respect to the deepest minimum µ, that is, µj and h(µj)
tends to zero for vanishing z (h is continuous). As a result for ε <
1
2
(
we assumed that
µj ≤
1
4
µ
)
h(q) − h(µ) ≥ θ∗, q ∈ R
+\[(1 − ε)µ, (1 + ε)µ],
(3.5e)
θ∗ = min
z∈[0,z0]
{
[h((1 + ε)µ) − h(µ), h((1 − ε)µ) − h(µ)
}
> 0.
Lemma 3.1. Let e+ = z−1µ, µ be the deepest minima of u, h, respectively. Then
there exitsts a positive constant κ− such that
κ−z−(2l−n+1) ≤
∫
q2l exp
{
−β[u(q) − u(e+)]
}
dq, (3.6)
and there exist positive constants κ+ and g0 such that for g ≥ g0 the following inequality
holds: ∫
q2l exp
{
−β[u(q) − u(e+) − z−nQ̃(zq)]
}
dq ≤
≤ κ+z−(2l−n+1)eβãz
−n(εa1+z
ka2), ε > 0, (3.6a)
where
Q̃(q) = a1Q
0(q − µ;µ) + a2z
kv0(q − µ),
ã = Q0(µ;µ) + v0(3µ), k > 0, as ≥ 0,
the integration is performed over R and µ is given by (3.4).
Proof. From the definition of h we derive the following formula:∫
q2le−β[u(q)−z−nQ̃(zq)]dq = z−1−2l
∫
q2le−βz
−2n[h(q)−znQ̃(q)]dq. (3.7)
Here we rescaled the variable in the first integral by z.
Let 0 ≤ ε < 1 and R be such that
h(q) − h(µ) − Q̃(q) ≥ q2, q ≥ R ≥ µ.
Let’s decompose the positive half-line into three sets (the first set is in the round
brackets)
R
+ = ([0, (1 − ε)µ] ∪ [(1 + ε)µ,R]) ∪ [(1 − ε)µ, (1 + ε)µ] ∪ [R,∞]
change sign of q in the integral in (3.7) and make estimates of the integral over these sets.
For the integral over the second set we have
(1+ε)µ∫
(1−ε)µ
q2le−βz
−2n[h(q)−znQ̃(q)]dq ≤
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398 W. I. SKRYPNIK
≤ (µ(1 + ε))2l
εµ∫
−εµ
e−βz
−2n[h(q+µ)−znQ̃(q+µ)]dq ≤
≤ (µ(1 + ε))2leβãz
−n(εa1+z
ka2)
εµ∫
−εµ
e−βz
−2nh(q+µ)dq. (3.8)
Here we applied the inequalities Q0(±εµ;µ) ≤ εn1+n2Q0(µ;µ), εn1+n2 ≤ ε,
v0(±εµ) ≤ v0(3µ). (3.5c) implies that the above integral in the right-hand side of (3.8)
is less than
εµ∫
−εµ
e−βz
−2nh(q+µ)dq ≤ e−βz
−2nh(µ)
εµ∫
−εµ
e−βθz
−2nq2dq ≤
≤ e−βz
−2nh(µ)
∫
e−βθz
−2nq2dq = (π−1βθ)−
1
2 zne−βz
−2nh(µ). (3.9)
Inequality (3.5e) implies that the integral in (3.7) over the first set is less than
R2l+1e−βz
−2nh(µ)e−βz
−2n[θ∗−znQ̃(R)].
For the integral over the third set we have
∞∫
R
q2le−βz
−2n(h(q)−znQ̃)dq ≤ e−βz
−2nh(µ)
∫
q2le−βz
−2nq2dq =
= (β− 1
2 zn)1+2l
∫
q2le−q
2
dq. (3.10)
Combining (3.7) – (3.10) we obtain the right-hand side of (3.6a) with sufficiently small ε
(and arbitrary ε)
κ+ = (µ̄(1 + ε))2l(π−1βθ)−
1
2 +R2l+1κ′ + β− 1+2l
2
∫
q2le−q
2
dq, µ̄ = max
z∈[0,z0]
µ,
where κ′ = supz∈[0,z0],θ∗−znQ̃(R)≥ε0 z
−n+1e−βz
−2nε0 < ∞ and g0 = {g : θ∗ −
− znQ(R) = ε0 > 0}.
Now we have to prove (3.6). Let’s rescale the integral by zn−1 (the new variable is
z1−nq) ∫
q2le−βu(q)dq = z(n−1)(2l+1)
∫
q2le−βug(q)dq. (3.10a)
Let e0 be the deepest minimum of ug then
∫
q2le−βug(q)dq ≥
e0+1∫
e0−1
q2le−βug(q)dq =
1∫
−1
(q + e0)2le−βug(q+e0)dq ≥
≥ (e0 − 1)2le−βz
−2nh(µ)
1∫
−1
e−β(ug(q+e0)−ug(e0))dq.
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LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 399
Substituting (3.5b) into the last inequality we obtain∫
q2le−βug(q)dq ≥ (e0 − 1)2le−βz
−2nh(µ)eC̃ . (3.11)
From (3.11), (3.1), (3.3) we deduce that
∫
q2le−βug(q)dq ≥ z−2nl(µ−zn)2le−βz
−2nh(µ)e−βC̃ = z−2nlκ−e−βz
−2nh(µ), (3.12)
where κ− =
µ0
4
e−βC̃ . Formulae (3.10a) and (3.12) prove (3.6) for the integral over the
positive real line. The same inequality holds for integral over the negative real. This
follows easily by changing the sign of q in the integral. The bounds for Q̃(−q) are the
same for Q̃ (since Q0(q − µ;µ) is even iq). This concludes the proof of the lemma.
Now we can derive the bound for the integral in (2.12).
Proposition 3.1. For g ≥ g0 there exist positive constants κ, κ′ such that the fol-
lowing inequality is valid:
I∗(g,Q0) ≤ κeβe0ε∗ ,
∫
q2le−βu
+
∗ (q)dq ≤ κ′z−2ln, ε > 0.
Proof. From the definition of h we see that ug(z−nq) = z−2nh(q). So after the
rescaling of the variable by z−n in the integral I∗(g,Q), taking into account that ε =
= z2n(n−1),
εv0(z−nq) ≤ v0(q), e
−2(n1+n2)+1
0 Q0(z−nq; e0) = z−nµ−2(n1+n2)+1Q0(q;µ)
(3.13)
we obtain
I∗(g,Q) ≤ z−neβ(ζ0+1)
∫
exp
{
−βz−2n
[
h(q + µ) − h(µ) − znQ̃µ(q)
]}
dq, (3.14)
where
Q̃µ(q) = µ−2(n1+n2)+1Q0(q;µ) + (ζ + 1)znv0(q).
Formulae (3.14), (3.7), (3.6a) prove the inequality for I∗(g,Q0), in which
κ = eβ(ζ0+1)κ+, ε∗ = 2ãεµ0, a1 = µ−2(n1+n2), a2 = 1 + ζ, k = n, since in this
case Q̃µ(q − µ) = Q̃(q).
By the same argument one derives∫
q2le−βu
+
∗ (q)dq ≤
≤ z−n(2l+1)eβ(ζ0+1)
∫
q2l exp
{
−βz−2n
[
h(q + µ) − h(µ) − znQ̃µ(q)
]}
dq.
Translating q by −µ and applying (3.7) and (3.6a) we obtain the needed bound with
κ′ = κ(1 + µ)2l.
Proof of Theorem 2.1. In obtaining bounds for integrals in (2.3), (2.4) we rely on
(1.5) from which it follows that:
u(q1, q2) = u(q1) + u(q2) + g(qx − qy)2Q(qx, qy). (3.15)
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400 W. I. SKRYPNIK
Let’s rescale the variables in the integrals by zn−1. As a result
〈σ2n0 + σ′
2n0〉 = z2n0(n−1)Z−1
g (2)
∫
(q2n0
1 + q2n0
2 )e−βug(q1,q2)dq1dq2,
Zg(2) =
∫
e−βug(q1,q2)dq1dq2 = z−2(n−1)Z(2), (3.16)
ug(q1, q2) = ug(q1) + ug(q2) − 2ug(e0) + gz2n0(n−1)(qx − qy)2Q(qx, qy).
The following inequality is true:
(q1 − q2)2Q(q1, q2) ≤ z−2(n0−1)nQ̄, qj ∈ [e0 − 1, e0 + 1],
where
Q̄ = 4 max
0≤z≤1
Q(1 + znµ, 1 + znµ).
The last inequality yields (g = z−2(n−n0))
ug(q1, q2) ≤ ug(q1) + ug(q2) − 2ug(e0) + Q̄, qj ∈ [e0 − 1, e0 + 1]. (3.17)
As a result (see the derivation of (3.11))
Zg(2) ≥
e0+1∫
e0−1
e0+1∫
e0−1
e−βug(q1,q2)dq1dq2 ≥
≥ e−βQ̄
e0+1∫
e0−1
e−β(ug(q)−ug(e0))dq
2
≥ e−β(Q̄−C̄). (3.18)
The same argument leads to∫ (
q2n0
1 + q2n0
2
)
e−βug(q1,q2)dq1dq2 ≥
≥ (e0 + 1)2n0
e0+1∫
e0−1
e0+1∫
e0−1
e−βug(q1,q2)dq1dq2 ≥
≥ 2(e0 + 1)2n0e−βQ̄
e0+1∫
e0−1
e−β(ug(q)−ug(e0))dq
2
≥
≥ 2(e0 + 1)2n0e−β(Q̄−C̄). (3.19)
From (3.19) and right-hand side of (3.6a) for l = 0, we see that for sufficiently small
z there exists the constant C+ such that〈
σ2n0 + σ′
2n0
〉
≥
≥ (κ+)−1z−2(n−1)z2n0(n−1)z2(n−1)z−2n0n(µ+ zn)2n0e−β(Q̄−C̄) ≥ z−2n0C+.
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LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 401
Hence, (2.5) is proven since −2n0 = 2(n− n0) − 2n and z2(n−n0) = g−1.
To prove (2.6) it is sufficient now to use the bound〈
−σ2n0 − σ′
2n0 + σkσ′
l + σ′
k
σl
〉
=
〈
(σ − σ′)2Q(σ, σ′)
〉
≤
≤ z2n0(n−1)Z−1
g (2)
∫
(q1 − q1)2Q(q1, q2)e−βug(q1,q2)dq1dq2 ≤
≤ (gβ)−1κ1
Z0
g (2)
Zg(2)
, (3.20)
where
κ1 = max
q≥0
qe−q, Z0
g (2) =
(∫
e−β(ug(q)−ug(e0))dq
)2
.
From (3.6a) for l = 0, Q̃ = 0, (3.1) (one has to take into account the equality u(e+) =
= ug(e0) = z−2nh(µ), rescale the variable in the integral by zn−1) and (3.18) it follows
thatZ0
g (2) ≤ (κ+)2. This together with (3.18) yields (2.6) withC− = κ1(κ+)2e−β(Q̄−C̄).
This concludes the proof of Theorem 1.2.
4. Basic superstability bound. A measurable function U(qX) is required to satisfy
the superstability and regularity conditions
U(qX) − U+(qX) ≥
∑
x∈X
u−(qx), (4.1)
|W (qX1 ; qX2)| ≤
1
2
∑
x∈X1,y∈X2
Ψ(|x− y|)[v(qx) + v(qy)],
X1 ∩X2 = ∅, v ≥ 0,
(4.2)
where
U+(qX) =
∑
x,y∈X
ϕ+
x,y(qx, qy), ϕ+ ≥ 0,
and all the functions are measurable. The following integrals are necessary attributes of
the superstability bound since its main parameter c depends on them
ū(q) = u(q) + ‖Ψ‖1v(q), Ir = e−
1
2β‖Ψ‖1v̄rI0, I0 =
∫
|q|≤r
e−βū(q)dq,
I(ε) =
∫
exp
{
−β[u−(q) − 3εv(q)]
}
dq, v̄r = sup
|q|≤r
v(q).
Theorem 4.1. Let ρΛ(qX) be the correlation functions corresponding to the poten-
tial energy U(qΛ) which satisfies (4.1), (4.2). Let’s put ψ(x) = |x|k, lj = (1 + 2α)j and
require that
‖ψΨ‖1 ≤ ∞, ‖Ψ‖1
[
(1 + 3α)2(d+k) − 1
]
≤ ε
2
,
where 0 < 3ε < 1, ‖Ψ‖p =
∑
x
|Ψ(|x|)|p and the summation is performed over Z
d.
Then the following superstability bound is valid:
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402 W. I. SKRYPNIK
ρΛ(qX) ≤ exp
{
−β
[
U+(qX) +
∑
x∈X
(u−(qx) − 3εv(qx))
]
+ |X|c(ε, I−1
r , I(ε))
}
(4.3)
and there exist the positive numbers c0, ξ such that for Vj = (1 + 2lj)d the following
representation is true:
c(ε, z′, z) = c0 + ln(1 + ξz′ + f(ε, zz′)), f(ε, z) =
∑
j≥0
e−εψ(lj)VjzVj . (4.4)
Moreover, if Ψ has a finite support then c0 does not depend on ε and ξ is bounded
in ε.
The bound (4.3) differs from the Ruelle bound by the presence of e−βU
+
, more gen-
eral ψ and the condition for α (k = 1 in the Ruelle condition) and, also, less general lj .
Our ψ enables to control an asymptotics of c in ε needed for providing the right
asymptotics of e∗(g) in g in Theorem 2.2 after putting ε = z2(n−1)n = (µ−1e0)−2(n−1).
The character of the asymptotics of f and c is depicted in the following corollary proven
in the appendix.
Corollary 4.1. Let the conditions of Theorem 4.1 be satisfied and k ≥ d.
(a) Then the following inequality is true:
lim
ε→0
ε
2d
k+d ln f(ε, z) ≤ 2d
2k+d
k+d (ln z)2.
(b) Moreover, if Ψ has a finite support, then the following inequality holds:
lim
ε→0
ε
2d
k+d c(ε, z, z′) ≤ 2d
2k+d
k+d (ln zz′)2.
Assertion (b) follows from (a) and and the last statement of Theorem 4.1.
The next lemma reduces proof of Theorem 2.2 to a determination of the asymptotics
of c in ε = z2n(n−1) for fixed z, z′.
Lemma 4.2. Let u−∗ , v
0 be the functions in the superstability and regularity con-
ditions (1.6), (1.7). Let, also, the integrals I(z2n(n−1)), Ir(z2n(n−1)), associated to
them (instead of u−, v) be denoted by I∗(g), I∗r(g), respectively. Then the functions
(I∗r)−1(g), I∗(g), are bounded in g.
Proof. The integral I∗(g) is bounded due to Proposition 3.1. All the functions in the
expressions for I∗r(g) tend to a finite limit in g at infinity. Indeed, u∗ tends to a finite
limit due to (3.5a). For the inverse power of this integral the Jensen inequality is used
and the Lebesque dominated convergence theorem is easily applied since the integration
is performed over a compact space. This concludes the proof of the lemma.
For short-range interaction k in Corollary 4.1 may be arbitrary. Thus, Lemma 4.1 and
Corollary 4.1 show that e∗ satisfies the property indicated in Theorem 2.2.
Corollary 4.2. Let the conditions of Theorem 2.2 be satisfied and c be given in
Theorem 4.1. Then the function e∗ from Theorem 2.2 is given by
e∗(g) = c
(
z2n(n−1), I∗r(g), I∗(g)
)
(4.5)
and behave as eθ0 asymptotically in g at infinity where θ is an arbitrary small number.
This corollary completes the proof of Theorem 2.2.
Proof of Corollary 4.1 can be found in [5].
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LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 403
Appendix A. To prove that φ∗ satisfies (1.7) one has to use the inequalities (k, l are
positive integers, q, q′ ≥ 0)
(q ± q′)l ≤ 2l(ql + q′
l), qkq′
l + qlq′
k ≤ 2k+l(qk+l + q′
k+l).
The second inequality and the equality Q′(1, 1) = 1 lead to
Q′(q, q′) ≤ 22n2(q2n2 + q′
2n2).
As a result of the first previous and last inequalities one obtains for r = cz−1, c > 0,
z ≤ 1
24n2(q − q′)2n1(q2n2 + q′
2n2 + 2r2n2) ≤
≤ 24n2+2n1(q2n1 + q′
2n1)(q2n2 + q′
2n2 + 2r2n2) ≤
≤ 24n2+2n1
[
q2n2+2n1 + q′
2n2+2n1 + q2n1q′
2n2+
+ q2n2q′
2n1 + 2(q2n1 + q′
2n1)r2n2
]
≤
≤ 24n2+2n1
[
(1 + 22n1+2n2)(q2n2+2n1 + q′
2n2+2n1) + 2(q2n1 + q′
2n1)r2n2
]
≤
≤ 24n2+2n1(1 + 22n1+2n2)(1 + c2n2)z−2n2+2(n−1)n1×
×
[
v0(z−(n−1)q) + v0(z−(n−1)q′)
]
.
From the equality e0 = µz−n (see the third section) it follows that (µ = c)
|φ∗(q, q′)| = |φg(q + e0, q
′ + e0)| ≤
≤ 24n2+2n1(1 + 22n1+2n2)(1 + µ2n2)z−2n2+2(n−1)n1 [v0(q) + v0(q′)].
The derivation of the regularity condition (1.7), now, is straightforward since µ is bonded
in z ≤ 1.
Appendix B. Let’s put
u1(q) = η2lq
2l, h1(q) = η2lz
2(n−l)q2l,
2n− 1 = 2s+ 2l − 1, 2n0 − 1 = s+ 2l − 1, s ∈ Z
+.
In this case root equation is given by
2−1∂h(q) = nηq2n−1 − n0q
2n0−1 + lη2lz
2(n−l)q2l−1 = 0,
or
q2l−1(nηq2s − n0q
s + lη2lz
2(n−l)) = 0,
q = 0, nηx2 − n0x+ lη2lz
2(n−l) = 0, x = qs.
The nontrivial roots of this quadratic equation are given by
2−
1
s
(
µ0 ± µ′
) 1
s , µ′ =
(
µ0 − 4l
nη
z2(n−l)η2l
)1
2
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
404 W. I. SKRYPNIK
If η2l < 0 then there is only one non-zero root
µ = 2−
1
s (µ0 + µ′)
1
s , µ′ =
(
µ0 +
4l
nη
z2(n−l)|η2l|
) 1
2
.
If η2l > 0 then there are two non-zero roots µ, µ− for sufficiently small z
µ = 2−
1
s (µ0 + µ′)
1
s , µ′ =
(
µ0 − 4l
nη
z2(n−l)|η2l|
) 1
2
,
µ− = 2−
1
s (µ0 − µ′)
1
s = z
2(n−l)
s (µ̃(z))
1
s , µ̃(0) > 0.
For µ, µ− the expansions (3.4), (3.5) are valid, respectively. The trivial root has multi-
plicity 2l − 1, the root µ− is simple and other roots are complex.
Appendix C. Let’s consider the polynomial
x2n + y2n − xky2n−k − ykx2n−k.
If it is positive for positive x, y then it is positive for all x, y. It is not difficult to check that
for positive arguments the second derivative in k of the polynomial is negative, that is, the
polynomial is a convex function in k on the interval [0, 2n]. Besides, It is has a root of the
second order for the value x = y, i.e., the first and the second derivative are equal to zero
for the value. This proves (1.5), i.e., existence of Q. These remarkably simple arguments
were proposed by the referee. The initial proof was very lengthy but it was constructive
and gave the explicit expression for Q. Since we do not use it in our subsequent bounds
this proof was omitted by us.
Remark 1. Another method of proving of existence of lro for the classical ferromag-
netic oscillator systems was proposed in [15]. In [16] the static quantum Holstein model
was reduced to a classical linear oscillator system with a non-n-n nonpair interaction and
for such the system occurrence of the antiferromagnetic lro was proven with a help of a
special generalized Peierls argument. In [17], the classical n-n linear oscillator systems
were treated in terms of the Pirogov – Sinai theory (see [18, 19]) and most profound re-
sults concerning phase diagrams were obtained. These ideas were applied in the lattice
Higgs gauge field models in [20].
Remark 2. The potential energy for ferromagnetic GKS systems is given as follows:
U ′(qΛ) =
∑
(A,nA),A⊆Λ
JA,nA
qnA
[A] , qnA
[A] =
∏
x∈A
qnx
x ,
nA = (nx, x ∈ A), nx ∈ Z
+,
JA,nA
≥ 0,
∑
x∈A
nx = 2n(A), nx, n(A) ∈ Z
+.
In the last line the intermediate condition implies that the potential energy is invariant
under the change of the signs of all the oscillator variables.
In order to guarantee finiteness of the partition function and correlation function it has
to be assumed that
sup
A
n(A) < n, max
x
∑
A,x∈A
JA,nA
<∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
LONG-RANGE ORDER IN GIBBS LATTICE CLASSICAL LINEAR OSCILLATOR SYSTEMS 405
Remark 3. In general case a depth of the potential in (1.5) is controlled by n − 1
parameters, i.e., coefficients of the polynomial u. The different subsets of this space can
be described with the help of one parameter g and the representation
u(q) = ηq2n −
n−1∑
s=1
ηsg
nsq2s.
To deal with this case one has to find an appropriate rescaling reducing the polynomial
to a sum of a polynomial with easily calculated minima and an additional polynomial
depending on g−1.
Remark 4. Asymptotics of the integral in (3.6) can be found with the help of the usual
one-dimensional Laplace method [21]. It has to be generalized for finding asymptotics of
the integral in (3.6a). Our arguments in the proof of (3.6a) are close to the arguments of
the usual one-dimensional Laplace method but not identical to them. To find asymptotics
of integrals in (2.5), (2.6) one has do employ the two-dimensional Laplace method. It
is less known than its one-dimensional version. Our proofs are optimal and there is no
need for a reader to check conditions in a general statement of the Laplace method for
establishing asymptotics of our integrals.
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Received 19.10.2004,
after revision — 26.04.2005
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