On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems
Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи ха...
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| author | Skrypnik, W. I. Скрипник, В. І. |
| author_facet | Skrypnik, W. I. Скрипник, В. І. |
| author_sort | Skrypnik, W. I. |
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| datestamp_date | 2020-03-18T19:43:50Z |
| description | Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ. |
| first_indexed | 2026-03-24T02:35:05Z |
| format | Article |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat Acad. Sci. Ukraine, Kyiv)
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE
GKS-TYPE OSCILLATOR EQUILIBRIUM SYSTEMS
ПРО НЕОБМЕЖЕНИЙ ПАРАМЕТР ПОРЯДКУ У ГРАТКОВИХ
РIВНОВАЖНИХ СИСТЕМАХ ОСЦИЛЯТОРIВ ТИПУ ГКШ
An unbounded order parameter (magnetization) is established to exist for a wide class of lattice Gibbs
(equilibrium) systems of linear oscillators interacting via a strong pair near neighbor polynomial potential and
other many-body potentials. The considered systems are characterized by a general polynomial short-range
interaction potential energy generating Gibbs averages that satisfy two generalized GKS inequalities.
Встановлено iснування необмеженого параметра порядку (намагнiченостi) для широкого класу ґрат-
кових гiббсiвських (рiвноважних) систем лiнiйних осциляторiв, що взаємодiють завдяки сильному пар-
ному полiномiальному потенцiалу близьких сусiдiв та iншим багаточастинковим потенцiалам. Розгля-
нутi системи характеризуються загальною полiномiальною близькодiйовою потенцiальною енергiєю,
що породжує середнi, якi пiдкоряються двом нерiвностям ГКШ.
1. Introduction and main result. In this paper we consider Gibbs classical systems of
one-dimensional oscillators (unbounded spins) on the d-dimensional hyper-cubic lattice
Zd, with a polynomial ferromagnetic GKS (Griffiths – Kely – Sherman)-type translation-
invariant potential energy U(qΛ) = U(−qΛ) on a hypercube Λ with the finite cardinality
|Λ| centered at the origin, where qΛ is an array of (qx, x ∈ Λ), qx is the oscillator
coordinate taking values in R.
For a wide class of oscillator systems with a polynomial ferromagnetic n-n (near
neighbor) pair potential whose strength is g there exists the unit spin long-range order
(lro), that is the following inequality holds [1 – 4]
〈sxsy〉Λ ≥ 1− o(λ), sx = signσx, σx(qΛ) = qx, (1.1)
where 〈., .〉Λ denotes the Gibbs average, λ is either g−1 or the temperature and o(λ) is
a continuous function tending to zero in the limit of zero λ. Such the lro generates the
bounded ferromagnetic order parameter (bounded magnetization) mΛ = |Λ|−1
∑
x∈Λ
sx,
which due to (1.1) has a non-zero average when it is squared, that is 〈m2
Λ〉Λ ≥ 1− o(λ).
If there is short-range order, that is the average in (1.1) decreases when the (Euclidean)
distance |x− y| grows, then the bounded magnetization is zero and there is no order in
the system.
An existence of the unit spin lro for oscillator systems with a n-n pair (non-
polynomial) potential for oscillator systems has been proven earlier in the paper [5],
in which the reflection positive Pieirls argument was employed (see also [6, 7]). The
non-trivial problem to derive the similar bound for 〈σxσy〉Λ was not considered in
the mentioned papers. We solve this problem in this paper. Our result implies that
the system is ordered and has the non-zero unbounded ferromagnetic order parameter
MΛ = |Λ|−1
∑
x∈Λ
σx at a large g. Our technique can be characterized as a GKS-type
Pieirls argument strengthened by the Ruelle superstability bound. It is based on the
facts that the basic constant (independent of oscillator variables) present in the Ruelle
super-stability bound [8, 9] grows polynomially in g at infinity and that the average〈
e−(g/2)[σl
xσk
y+σk
xσl
y]
〉
Λ
decreases exponentially in g at infinity, where the expression
c© W. I. SKRYPNIK, 2009
538 ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE GKS-TYPE OSCILLATOR ... 539
under the sign of the exponent is proportional to the pair n-n potential and x, y are
nearest neighbors. Non-triviality of MΛ at low temperatures was proved earlier in [10]
for lattice systems of oscillators with the interaction generated by the pair bilinear
nearest neighbor potential. The technique of [10] is not connected with a generalized
Pieirls argument. A review of results concerning lro in lattice oscillator systems a reader
may find in [4]. New results concerning existence of the magnetization in Ising models
can be found in [11, 12].
Our potential energy is given by (see also [3, 4])
U(qΛ) =
∑
x∈Λ
u0(qx)− g
∑
〈x,y〉∈Λ
(qk
xql
y + ql
xqk
y ) + U ′(qΛ), (1.2)
where 〈x, y〉 means nearest neighbors, u0 is a bounded below polynomial of the 2n-th
degree such that u0 −
1
2
q2n is also bounded from below, k + l = 2n0 < 2n,
U ′(qΛ) = −
∑
A⊆Λ
φA(qA), φA(qA) =
∑
〈n(|A|)〉<2n
JA;n(|A|)Sq
n|A|
[A] , JA;n(|A|) ≥ 0,
the first sum is performed over subsets of Λ with the number of sites |A| ≤ n and
the second one over the sequence of positive integers such that the number 〈n(|A|)〉 =
=
∑|A|
j=1
nj < 2n is even, n(|A|) = (n1, . . . , n|A|), JA∪x;n(|A|),n0 = JA−x∪0;n(|A|),n0
(translation invariance). If A = (x1, . . . , xk) = x(k), |A| = k then JA;n(|A|) =
= Jx(k), n(k) = Jn(k)
(
|x1 − xk|, . . . , |xk−1 − xk|
)
, where |x| is the Euclidean norm
of the site x and q
n|A|
[A] =
∏k
j=1
qnj
xj
. Here S means symmetrization. There is another
representation for U, given by (3.1), in which its interacting part is zero for coinciding
arguments (this part differs from U by a “boundary” term generated by an external field).
We assume that the interaction is short-range, that is
J−l =
∑
0∈A
1
|A|
∑
〈n(|A|)〉=2l
JA;n(|A|) < ∞,
where the summation is performed over all subsets of Zd which contain the origin. Our
main result is formulated as follows.
Theorem 1.1. Let d ≥ 2 and l > 2n0 − l then there exists a positive number
g0 > 1 such that for g > g0 the following uniform in Λ bound is valid
〈σxσy〉Λ ≥ σ̄g−θ − σ0g−α, (1.3)
where θ = 2 +
d + 1
n− n0
l
2n0 − l
, σ̄, σ0 > 0 depend on β and α is an arbitrary positive
number.
Corollary 1.1. For an arbitrary temperature there exists a positive number g∗
such that for g ≥ g∗ the left-hand side of (1.3) is positive uniformly in Λ, implying
the existence of lro and that the unbounded order parameter MΛ is non-zero in the
thermodynamic limit.
The proof of (1.3) demands the bounds which were not employed in [1 – 4] for the
proof of (1.1), namely, the Ruelle superstability bound [8]
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
540 W. I. SKRYPNIK
ρΛ(qX) ≤ exp
{
−β
∑
x∈X
(uε(qx)− β−1c)
}
, uε = u− 3εv, (1.4)
where c is the basic superstability constant, ε is an arbitrary arbitrary small number(
ε <
1
3
)
and
u(q) = u0(q)− 2dgq2n0 −
n−1∑
l=1
J−l q2l, v(q) =
n−1∑
j=1
q2j . (1.5)
The Ruelle superstability bound (1.4) is proved if one establishes the following
superstability and regularity conditions for the potential energy
U(qX) ≥
∑
x∈X
u(qx), (1.6)
∣∣W (qX1 ; qX2)
∣∣ ≤ 1
2
∑
x∈X1, y∈X2
Ψ(|x− y|)
[
v(qx) + v(qy)
]
, X1 ∩X2 = ∅, (1.7)
where
W (qX ; qY ) = U(qX∪Y )− U(qY )− U(qX), ‖Ψ‖1 < ∞,
‖Ψ‖1 =
∑
x
Ψ(|x|) (the summation is performed over Zd).
We prove Theorem 1.1 with the help of Theorem 1.2 and Proposition 1.1.
Theorem 1.2. Let d ≥ 2, l > 2n0 − l, c be the basic superstability constant and
c ≤ c̄gκ + o(g−1), where c̄ is a positive constant. Then inequality (1.3) holds in which
either θ = 2 +
κ
n
l
2n0 − l
for κ ≥ n
n− n0
or θ = 2 +
1
n− n0
l
2n0 − l
for κ <
n
n− n0
.
If κ =
n
n− n0
(1.3) holds for sufficiently large β.
From the analytical structure of the basic superstability constant presented in [4, 9]
one easily derives the following proposition proved in the last section.
Proposition 1.1. Let c be the basic superstability constant then κ =
n(d + 1)
n− n0
.
We shall rely, also, on the following proposition whose proof can be found in [4].
Proposition 1.2. Let U0 be a bounded from below even polynomial of the 2n-th
degree, U(q) = U0(q)− 2dgq2n0 , n0 < n. Then there exists positive constants g0 > 1,
κ0, µ̄, ē such that for g ≥ g0 the potential U has the the unique deepest minimum e0
and the following inequalities hold
e0 ≤ ēg1/2(n−n0),
∣∣U(e0)
∣∣ ≤ µ̄gn/(n−n0),
∫
e−βU(q)dq ≤ κ0e
−βU(e0),
where the integration is performed over R.
The first two bounds in this proposition are equalities for the simplest potential
U(q) = u(q) = ηq2n − 2dgq2n0 and its unique positive minimum
e0 =
(
2d
n0
ηn
g
)1/(2(n−n0))
, u(e0) = −η
n− n0
n0
(
2d
gn0
ηn
)n/(n−n0)
.
Our paper is organized as follows. In the next section we give a proof of Theorem 1.2.
In the third section (1.5) – (1.7) and Proposition 1.1 are proved.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE GKS-TYPE OSCILLATOR ... 541
2. Proof of Theorem 1.2. The Gibbs averages for a measurable function FX on
R|X| are 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,
where
∫
dqX denotes the integral over R|X|, β is the inverse temperature. We assume
that
u0(q) = ηq2n + u1(q), u1(q) =
n−1∑
j=1
ηjq
2j , η ≥ 1.
The proof of Theorem 1.2 begins from a derivation of the inequality
〈σxσy〉Λ ≥ r2 − 2
(〈
χ+
x χ−y
〉1/2
Λ
+ 〈χ−x χ+
y 〉
1/2
Λ
)
〈σ4
x〉
1/4
Λ 〈σ4
y〉
1/4
Λ −
− r2
[
2
(
〈χx,[−r,r]〉Λ + 〈χy,[−r,r]〉Λ
)
+ 〈χ+
x χ−y 〉Λ + 〈χ+
y χ−x 〉Λ
]
, (2.1)
where χx,[r,r′](qΛ) = χ[r,r′](qx), χ+
x = χx,[0,∞], χ−x = χx,[−∞,0] and χ[r,r′] is the
characterisctic function of the interval [r, r′].
Inequality (2.1) is an analog of the inequality for the two point spin Gibbs average for
the bounded spin systems from [5]. It is known from [3, 4] that 〈χ+
x χ−y 〉Λ exponentially
tends to zero at infinity in g. In order to derive (1.3) from (2.1) for r polynomially
decreasing in g at infinity one has to establish that the Gibbs average 〈σ4
x〉Λ tends only
polynomially to infinity in growing g and that 〈χx,[−r,r]〉Λ tends exponentially to zero
at the same time. We will establish that 〈χx,[−r,r]〉Λ tends to zero at infinity in g with
the help of the equality
〈χx,[−r,r]〉Λ = 〈χx,[−r,r]χx∗,[−r,r]〉Λ + 〈χx,[−r,r]χx∗,[−r,r]c〉Λ =
= 〈χx,[−r,r]χx∗,[−r,r]〉Λ + 〈χx,[−r,r]χx∗;r,r′〉Λ + 〈χx,[−r,r]χx∗,[−r′,r′]c〉Λ, (2.2)
where x∗ ∈ Λ is one of the nearest neighbors of x, χx;r,r′(qΛ) = χr,r′(qx) =
= χ[r,r′](qx) + χ[−r′,−r](qx), bounds (2.3′), (2.3′′), (2.4) and r, r′ chosen in a special
way. The last term in the right-hand side of (2.2) and 〈σ4
x〉 will be estimated with the
help of the superstability bound.
The following bound has been already employed by us in [3, 4] for a proof of (1.1)
χ+(qx)χ−(qy) ≤ e−
g
2 [ql
xqk
y+qk
xql
y], k + l = 2n0. (2.3)
For l, k = 1 it was proposed in [13]. An exposition of the two generalized GKS
inequalities can be found in [14, 15].
In this paper we introduce the following new bounds for estimates of the summands
in the right-hand side of (2.1)
χ[−r,r](qx)χ[−r,r](qy) ≤ egr2n0
e−(g/2)[ql
xqk
y+qk
xql
y], (2.3′)
χ[−r,r](qx)χr,r′(qy) ≤ e(g/2)(rlr′k+rkr′l)e−(g/2)[ql
xqk
y+qk
xql
y], r′ > r, (2.3′′)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
542 W. I. SKRYPNIK
〈
χx,[−r,r]χx∗,[−r′,r′]c
〉
Λ
≤ 〈χx∗,[−r′,r′]c〉Λ ≤ ec
∫
|q|≥r′
e−βuε(q)dq ≤
≤ ec−βr′2n/4
∫
|q|≥r′
e−βũ(q)dq, (2.4)
〈σ4
x〉Λ ≤ r′4 + ec
∫
|q|≥r′
e−βuε(q)q4dq ≤ r′4 + κ4(4β−1)4ec−βr′2n/4
∫
|q|≥r′
e−βũ(q)dq,
(2.5)
where κ4 = max
q≥0
q4e−q, ũ(q) = uε(q) −
1
2
q2n. We applied the superstability bound
for ρΛ(qy), y = x, x∗ in (2.4) and (2.5) and the estimate
∫
|qx|≤r′
q4
xρΛ(qx)dqx ≤
≤ r′4
∫
ρΛ(q)dq = r′4 in (2.5). Thus all the averages containing characteristic functions
in (2.1) will be estimated with the help of the average
〈
e−
g
2 [σl
xσk
y+σk
xσl
y]
〉
Λ
for the
nearest neighbors x, y and the superstability bound. Let’s apply the third inequality in
Proposition 1.2 for U(q) = ũ(q), e0 = ẽ, κ0 = κ̃. Then the last integral in (2.4) and
(2.5) is less than
ec−βr′2n/4
∫
e−βũ(q)dq ≤ exp
{
c− β
(
1
4
r′2n −
∣∣ũ(ẽ)
∣∣)} κ̃ ≤
≤ exp
{
c− β
(
1
4
r′2n − µ̄gn/(n−n0)
)}
κ̃. (2.6)
Let κ ≤ n
n− n0
in Theorem 1.2 and put r′ = (8µ̄)1/2ng1/2(n−n0). Then the expression
in the round brackets in the right-hand side of the last inequality is equal to βµ̄gn/(n−n0)
and the right-hand side of (2.4) tends to zero in the limit of infinite g
(
if µ̄β > c̄ for
κ =
n
n− n0
)
. Let κ >
n
n− n0
and put r′ = (8c̄gκβ−1)1/2n. Then 4−1βr
′2n − c ≥
≥ c̄gκ − o(g−1) and the right-hand side of (2.4) together with the second term in the
right-hand side of (2.5) tends to zero in the limit of infinite g once more. Let’s put
r = g−1r′−l(2n0−l)−1
.
Then the exponents in (2.3′), (2.3′′), containing r, r′, are bounded in g since k =
= 2n0 − l < l, l ≥ 1. Hence Theorem 1.2 is proved if the inequality (2.1) is valid since
the average
〈
e−(g/2)[σl
xσk
y+σk
xσl
y]
〉
Λ
for the nearest neighbors x, y exponentially tends to
zero in the limit of infinite g [3, 4] and r′ grows as g to some finite power. Now, to
prove Theorem 1.2 we have to prove (2.1).
Proof of (2.1):
1 = χ[−∞,−r](q) + χ[r,∞](q) + χ[−r,r](qΛ) = χ[−r,r]c(q) + χ[−r,r](qΛ).
Let’s insert this decomposition in qx, qy into the two point Gibbs average. We obtain the
following bound:
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE GKS-TYPE OSCILLATOR ... 543
〈σxσy〉Λ ≥
≥ r2
[
−
〈
χx,[−r,r]χy,[−r,r]
〉
Λ
+
〈
χx,[r,∞]χy,[r,∞]
〉
Λ
+
〈
χx,[−∞,−r]χy,[−∞,−r]
〉
Λ
]
−
−2
(
〈χ+
x χ−y 〉
1/2
Λ + 〈χ−x χ+
y 〉
1/2
Λ
)
〈σ2
xσ2
y〉
1/2
Λ . (2.7)
Here we applied the inequalities
χx,[r,∞]χy,[−∞,−r] ≤ χ+
x χ−y ,
σxσy
(
χx,[−∞,−r] + χx,[r,∞]
)
χy,[−r,r] ≥
≥ −|σxσy|
(
χx,[−∞,−r]χy,[0,r] + χx,[r,∞]χy,[−r,0]
)
,
χx,[−∞,−r]χy,[0,r] ≤ χ−x χ+
y , χx,[r,∞]χy,[−r,0] ≤ χ+
x χ−y ,
and the Schwartz inequality〈
|σxσy|χx,[r,∞]χy,[−∞,−r]
〉
Λ
≤ 〈σ2
xσ2
y〉
1/2
Λ
〈
χx,[r,∞]χy,[−∞,−r]
〉1/2
Λ
,〈
|σxσy|χx,[r,∞]χy,[−r,0]
〉
Λ
≤ 〈σ2
xσ2
y〉
1/2
Λ
〈
χx,[r,∞]χy,[−r,0]
〉1/2
Λ
.
Further
〈χx,[r,∞]χy,[r,∞]〉Λ =
〈
χx,[r,∞](1− χy,[−r,r] − χy,[−∞,−r])
〉
Λ
≥
≥ 〈χx,[r,∞]〉Λ − 〈χy,[−r,r]〉Λ − 〈χx,[r,∞]χy,[−∞,−r])〉Λ ≥
≥ 〈χx,[r,∞]〉Λ − 〈χy,[−r,r]〉Λ − 〈χ+
x χ−y 〉Λ. (2.8)
Since our systems are invariant under the transformation of changing of all oscillator
variables signs we have
〈χx,[r,∞]〉Λ = 〈χx,[−∞,−r]〉Λ, 〈χx,[−∞,−r]χy,[−∞,−r]〉Λ = 〈χx,[∞,r]χy,[∞,r]〉Λ.
As a result the first equality and the equality
〈χx,[r,∞]〉Λ + 〈χx,[−∞,−r]〉Λ + 〈χx,[−r,r]〉Λ = 1
give two equalities
〈χx,[r,∞]〉Λ =
1
2
− 1
2
〈χx,[−r,r]〉Λ, 〈χx,[−r,−∞]〉Λ =
1
2
− 1
2
〈χx,[−r,r]〉Λ.
Substituting the first equality into (2.8) one obtains
〈χx,[r,∞]χy,[r,∞]〉Λ ≥ 1
2
− 1
2
〈χx,[−r,r]〉Λ − 〈χy,[−r,r]〉Λ − 〈χ+
x χ−y 〉Λ.
The same inequality holds for the second term in the first square bracket in (2.7) with
the permuted x, y . Hence (2.7), the last inequality, the inequality
χx,[−r,r]χy,[−r,r] ≤
1
2
(
χx,[−r,r] + χy,[−r,r]
)
and the Schwartz inequality complete the proof.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
544 W. I. SKRYPNIK
3. Estimates for potential energy. Let us derive the following new representation
for the potential energy from which (1.6) is easily derived
U(qΛ) =
∑
x∈Λ
u(qx) + g
∑
〈x,y〉∈Λ
(qx − qy)2Q(qx, qy) + U−(qΛ) + U∂Λ(qΛ), (3.1)
where U−(q, . . . , q) = 0, U− ≥ 0, Q ≥ 0, U∂Λ(qΛ) ≥ 0, U∂Λ(qΛ) is a boundary term,
generated by a boundary external field, and u is determined by (1.5). It will be derived
with the help of the following proposition.
Proposition 3.1. Let |A| be an arbitrary positive integer and 〈n(|A|)〉 =
=
∑|A|
j=1
nj = 2l. Then there exists a positive polynomial Qx,y such that the following
equality holds
Sq
n|A|
[A] =
1
|A|
∑
x∈A
q2l
x −
∑
x6=y∈A
(qx − qy)2Qx,y(qA). (3.2)
Proof. We will use induction. Let A = (1, . . . , k) and Pn(k)(q(k)) = Sq
n(k)
[k] =
= S
∏n
j=1
q
nj
j , nk < nj , nk−1 = n′ − nk. Let’s introduce the function
Pn(k−2);r = Pn(k−2),n′−r,r,
where r ∈ R+. Then the following equalities are true
Pn(k−2);nk
= Pn(k) , n′ +
k−2∑
j=1
nj = 2l, (3.3)
Pn(k−2);0(q(k)) = Pn(k−2);n′(q(k)) =
1
k
k∑
j=1
Pn(k−2);n′(q(k\j)), (3.4)
where q(k\j) is the sequence (1, . . . , k) without the positive integer j ≤ k. Let
1
k − 1
k−1∑
j=1
q2l
j − Pn(k−1)(q(k−1)) ≥ 0.
The following equality is easily derived
1
k
k∑
j=1
q2l
j − 1
k
k∑
j=1
Pn(k−2);n′(q(n\j)) =
=
1
k
k∑
j=1
1
k − 1
k∑
l=1,l 6=j
q2l
l − Pn(k−2);n′(q(k\j))
≥ 0.
The same inequality holds with 0 substituted instead of n′. This and (3.4) mean that the
function
1
k
∑k
j=1
q2l
j −Pn(k−2);r(qk)) is positive at the end points of the interval [0, n′].
Its second derivative is negative. This and the inequality nk ≤ n′ imply that
1
k
k∑
j=1
q2l
j − Pn(k)(q(k)) ≥ 0.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE GKS-TYPE OSCILLATOR ... 545
The first derivative in qj of the left-hand side of this inequality is equal to zero for
coinciding variables. This proves the proposition.
Proof of (3.1). The expression for U ′ is rewritten as
U ′(qΛ) = −
∑
l≤n−1
∑
A⊆Λ
φA;l(qA), φA;l(qA) =
∑
〈n(|A|)〉=2l
JA;n(|A|)Sq
n|A|
[A] ,
Let
φ−n(|A|)
(qA) =
∑
x6=y∈A
(qx − qy)2Qx,y(qA).
Substituting (3.2) into (1.2) we obtain (3.1) with
U−(qΛ) =
∑
A⊆Λ
φ−A(qA), φ−A(qA) =
∑
〈n(|A|)〉<2n
JA;n(|A|)φ
−
n(|A|)
(qA),
U∂Λ(qΛ) =
∑
x∈Λ
ul;∂Λ(qx) + g
∑
x∈∂Λ
q2n0
x , ul;∂Λ(qx) =
n−1∑
l=1
Jl;∂Λq2l
x ,
Jl;∂Λ =
∑
x∈A⊂Λc
1
|A|
∑
〈n(|A|)〉=2l
JA;n(|A|) ,
where Λc = Xd\Λ. Here we took into account that every boundary point has 2d − 1
nearest neighbors.
Proof of (1.7). Let 〈n(|A|)〉 = 2l then following bound is valid
∣∣∣Sq
n|A|
[A]
∣∣∣ ≤ 1
|A|!
(∑
x∈A
|qx|
)2l
≤ |A|2l−1
|A|!
∑
x∈A
|qx|2l. (3.5)
From the definition of W we obtain
W (qX ; qY ) =
∑
A1∈X,A2∈Y,Aj 6=∅
φA1∪A2(qA1 , qA2) =
∑
l≤n−1
Wl(qX ; qY ),
where
Wl(qX ; qY ) =
∑
A1∈X,A2∈Y,Aj 6=∅
φA1∪A2;l(qA1 , qA2). (3.6)
Inequality (3.5) yields
φA;l(qA) ≤ |A|2l−1
|A|!
∑
x∈A
q2l
x
∑
〈n(|A|)〉=2l
JA;n(|A|) ,
∣∣Wl(qX ; qY )
∣∣ ≤ 1
2
∑
x∈X,y∈Y
Ψ′(|x− y|
)(
q2l
x + q2l
y
)
,
Ψ′(|x− y|
)
=
∑
l≤n−1
Ψl(|x− y|),
(3.7)
Ψl
(
|x− y|
)
= 4
∑
x,y∈A
|A|2l−1
|A|!
∑
〈n(|A|)〉=2l
JA;n(|A|) < ∞,
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
546 W. I. SKRYPNIK
where the first summation is performed over all subsets of Zd which contain x, y. From
(3.5) one derives
qk
xql
y + ql
xqk
y ≤ 22(n0−1)(q2n0
x + q2n0
y ).
Hence, (3.6), (3.7) prove (1.7) with Ψ(|x−y|) = 22n0gδ1,|x−y|+Ψ′(|x−y|), where δ1,k
is the Kronecker symbol and |x| is the Euclidean norm of the lattice site x. ‖Ψ‖1 < ∞
due to the condition J−l < ∞ and the fact that summations in A are always performed
over sets whose numbers of sites are less than n.
Proof of Proposition 1.1. The basic constant c is a function of an arbitrary positive
number r and a number ε <
1
3
. That is, c = c(ε, I−1
r , I(ε)) (see [8, 9]) and the integrals
I(ε), Ir are determined as follows:
Ir = e−
1
2 β‖Ψ‖1v̄rI0, I0 =
∫
|q|≤r
e−βū(q)dq, I(ε) =
∫
exp
{
− βuε(q)
}
dq,
where ū(q) = u(q) + ‖Ψ‖1v(q), v̄r = sup
|q|≤r
v(q). Moreover,
c(ε, z′, z) = c0 + ln
(
1 + ξz′ + f(ε, zz′)
)
,
f(ε, z) =
∑
j≥0
e−εlj(1+2lj)
d
(2z)(1+2lj)
d
, z ≥ 1,
(3.8)
where positive constants c0, ξ may depend on ε, lj = (1 + 2α)j , α is proportional to ε
to some positive power.
From the bounds (1 + 2lj)d ≥ 1 + (2lj)d, (1 + 2lj)d ≤ 2d(1 + (2lj)d) we obtain
f
(
ε,
z
2
)
≤ z2d ∑
j≥0
e−ε(2lj)
d+1
z2d(2lj)
d
≤
≤ z2d
sup
x≥0
e−(1/2)εxd+1+2d(ln z)xd ∑
j≥0
e−(1/2)ε(2lj)
d+1
≤
≤ z2d
exp
{
ε
2
(
d + 1
dε
2d+1 ln z
)d+1
}∑
j≥0
e−(1/2)ε(2lj)
d+1
. (3.9)
Here we found the maximum of the function −1
2
εxd+1 + 2d(ln z)xd equating its deri-
vative in x to zero. Further, the following simple bound is true:
Ir ≥ exp
{
−β
[
3
2
‖Ψ‖1v̄r − g2−2n0r2n0
]} ∫
2−1r≤|q|≤r
e−β(u0(q)+u1(q))dq,
where −u1 coincides with the third summand in the expression for u in (1.5). Since
‖Ψ‖ > g, v̄r > r2n the coefficient in front of the last integral decreases exponentially
in g and the integral does not depend on g. That is, taking into account the second and
third bounds from Proposition 1.2 with U(q) = uε(q) for the estimate of I(ε) one sees
that there exists a positive numbers Ī , µ̄ independent of g such that
I−1
r I(ε) ≤ Ī exp
{
gn/(n−n0)µ̄
}
.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
ON THE UNBOUNDED ORDER PARAMETER IN LATTICE GKS-TYPE OSCILLATOR ... 547
This bound, (3.8) and (3.9) yield that there exists a positive number c̄ independent of g
such that
c ≤ c̄gn(d+1)/(n−n0) + o(g−1).
The proposition is proved.
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Received 09.09.08
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 4
|
| id | umjimathkievua-article-3038 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:05Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/82/f4c51f09eb2bf3265c513381b2dab782.pdf |
| spelling | umjimathkievua-article-30382020-03-18T19:43:50Z On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems Про необмежений параметр порядку у граткових рівноважних системах осциляторів типу ГКШ Skrypnik, W. I. Скрипник, В. І. Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ. Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ. Institute of Mathematics, NAS of Ukraine 2009-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3038 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 4 (2009); 538-547 Український математичний журнал; Том 61 № 4 (2009); 538-547 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3038/2825 https://umj.imath.kiev.ua/index.php/umj/article/view/3038/2826 Copyright (c) 2009 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title | On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title_alt | Про необмежений параметр порядку у граткових рівноважних системах осциляторів типу ГКШ |
| title_full | On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title_fullStr | On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title_full_unstemmed | On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title_short | On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems |
| title_sort | on an unbounded order parameter in lattice equilibrium gks-type oscillator systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3038 |
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