On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction
We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of osc...
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irk-123456789-1653262020-02-14T01:26:10Z On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction Skrypnik, W.I. Статті We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions. Запропоновано нове коротке доведення з6іжності високотемпературних полімєрних розкладів термодинамiчної границі канонічних кореляційних функцій ґраткових класичних та квантових гіббсівських систем осциляторів, що взаємодіють завдяки парному та тернарному потенціалам, а також нерівноважних стохастичних систем осциляторів, які взаємодіють завдяки парному потенціалу з гіббсівськими початковими кореляційними функціями. 2013 Article On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction / W.I. Skrypnik // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 689–697. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165326 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Skrypnik, W.I. On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction Український математичний журнал |
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We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions. |
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Skrypnik, W.I. |
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Skrypnik, W.I. |
| author_sort |
Skrypnik, W.I. |
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On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction |
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On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction |
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On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction |
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On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction |
| title_full_unstemmed |
On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction |
| title_sort |
on polymer expansions for generalized gibbs lattice systems of oscillators with ternary interaction |
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Інститут математики НАН України |
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2013 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165326 |
| citation_txt |
On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction / W.I. Skrypnik // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 689–697. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT skrypnikwi onpolymerexpansionsforgeneralizedgibbslatticesystemsofoscillatorswithternaryinteraction |
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2025-07-14T18:19:51Z |
| last_indexed |
2025-07-14T18:19:51Z |
| _version_ |
1837647467276926976 |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON POLYMER EXPANSIONS FOR GENERALIZED GIBBS LATTICE SYSTEMS
OF OSCILLATORS WITH TERNARY INTERACTION
ПРО ПОЛIМЕРНI РОЗКЛАДИ ДЛЯ УЗАГАЛЬНЕНИХ ГIББСIВСЬКИХ
ҐРАТКОВИХ СИСТЕМ ОСЦИЛЯТОРIВ З ТЕРНАРНОЮ ВЗАЄМОДIЄЮ
We propose a new short proof of the convergence of high-temperature polymer expansions for the thermodynamic limit
of canonical correlation functions of classical and quantum Gibbs lattice systems of oscillators interacting via pair and
ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial
correlation functions.
Запропоновано нове коротке доведення збiжностi високотемпературних полiмерних розкладiв термодинамiчної
границi канонiчних кореляцiйних функцiй ґраткових класичних та квантових гiббсiвських систем осциляторiв, що
взаємодiють завдяки парному та тернарному потенцiалам, а також нерiвноважних стохастичних систем осциляторiв,
якi взаємодiють завдяки парному потенцiалу з гiббсiвськими початковими кореляцiйними функцiями.
We consider in the canonical ensemble generalized Gibbs systems on the lattice Zd, whose sites
index variables from the measure space (Ω, P 0), with the potential energy Uc, which is a measurable
function, expressed through the one-particle (external) potential u(ω) and the two-particle complex-
valued potential ux−y(ωx, ωy)
Uc(ωΛ) =
∑
x∈Λ
u(ωx) +
∑
x,y∈Λ
ux−y(ωx, ωy),
where Λ is a finite set with the cardinality |Λ|. P 0 is a positive σ-finite measure (it is finite on
compact sets if Ω is a complete metric space) and P 0(Ω) =∞.
The Gibbs canonical correlation functions are given by
ρΛ(ωX) = Z−1
Λ
∫
e−βUc(ωΛ)P 0(dωΛ\X), ZΛ =
∫
e−βUc(ωΛ)P 0(dωΛ) > 0, β ∈ R+.
Here β is an inverse temperature, the integration is performed over Ω|Λ\X|, Ω|Λ|, respectively, and
P 0(dωX) =
∏
x∈X
P 0(dωx).
In [1, 2] we showed that three different choices of the measure space Ω and the potentials
correspond to Gibbs classical, Gibbs quantum systems of lattice oscillators, interacting via the pair
u0
x−y(qx, qy) = J0(|x− y|)u0(qx, qy)
and the factorized ternary
ux,y,z(qx, qy, qz) = J1(|x− y|)J1(|y − z|)u1(qx, qy)u1(qy, qz)
potentials such that Js ∈ L1(Zd) and nonequilibrium gradient stochastic systems of lattice oscilla-
tors interacting via the pair potential 2u0
x−y (their initial states are Gibbsian with a pair potential).
The correlations functions of the latter systems are represented as correlation functions of a lattice
diffusion Gibbs path oscillator system with pair and ternary interaction.The potential energy of the
c© W. I. SKRYPNIK, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5 689
690 W. I. SKRYPNIK
stochastic oscillators, which is present in the Smoluchowski equation (the forward Kolmogorov equa-
tion), is a sum of two terms generated by an external and pair potentials. The considered potentials
are polynomials in oscillator variables qx ∈ R. The external potentials are bounded from below
even polynomials u(q) and u0(q), q ∈ R, of the 2n and 2n0-the degrees for classical, quantum and
stochastic systems, respectively. Note that we used in [2] the following notations: u0(q) = 2u0(q)
and u0(q, q′) = 2u0(q, q′).
The spaces Ω for these systems are represented as Ω0×Ω∗ (ω ∈ Ω: ω = (ω0, ω∗), ω
0 ∈ Ω0, ω∗ ∈
∈ Ω∗) such that Ω0 = Ω∗ = R for classical Gibbs systems and Ω0 = R × C(R+), Ω∗ = C(R+)
for the remaining systems, where C(R+) is the space of continuous functions on R+.The measure
P 0 is factorized on Ω0 × Ω∗, i.e., P 0 = P̃ ⊗ P0 and P0 is a probability measure. Besides the
following equalities are true: P̃ (dq) = dq, P0(ds) = (
√
2π)−1e−s
2/2ds for Gibbs classical systems,
P̃ (dw) = dqP βq,q(dw), P̃ (dw) = dqPq(dw) for Gibbs quantum and stochastic systems, respectively,
where Pq(dw) is the Wiener measure and P βq,q(dw) is the conditional Wiener measure concentrated
on continuous paths starting and arriving into a point q at a “time” β (see the Remark 1 in the end
of the paper).
The goal of this paper is to find the thermodynamic limit Λ→ Zd of the correlation functions and
simplify the technique proposed for that in [1, 2] based on proving of a convergence of the polymer
high-temperature expansion for the correlation functions (see the Remark 2). The expansion is given
by
ρΛ,X(ωX) = z̄|X|e
β
∑
x∈X
u(ωx)
ρΛ(ωX) =
∑
Y ∈Λ
ρ̄Λ(X ∪ Y )
∫
P (dωY )FωX (ωY ),
where FωX (ωY ) are the truncated Boltzmann functions satisfying the KS recursion relation with the
interaction potential energy, ρ̄Λ(X) = z̄|X|ZΛ\XZ
−1
Λ and
P (dωY ) = z̄−|Y |
∏
y∈Y
e−βu(ωy)P 0(dωy), z̄ =
∫
e−βu(ω)P 0(dω).
To perform the thermodynamic limit one has to guarantee the absolute convergence of this expansion
uniformly in Λ. The important role in this is played by the superstability condition for he real part of
the pair potential
|Reux−y(ωx, ωy)| ≤
1
2
J ′(|x− y|)(v(ωx) + v(ωy)),
∫
eγβv
1+ζ(ω)−βu(ω)P 0(dω) <∞, (1)
where |x| is the Euclidean norm of x, J ′, v, ζ ≥ 0, γ > 0, ||J ′||1 =
∑
x
J ′x <∞ (J ′ ∈ Zd) and the
summation is performed over Zd. To formulate our results we need the following notations:
B = ess sup
ω
∑
x
bx(ω), D =
∫
eβṽ(ω)P (dω),
bx(ω) = e−β(ṽ(ω)−||J ′||v(ω))||
√
J ||−1
1
√
J(|x|)
∫
eβṽ(ωx)|e−βux(ω,ωx) − 1|P (dωx).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON POLYMER EXPANSIONS FOR GENERALIZED GIBBS LATTICE SYSTEMS . . . 691
Theorem 1. Let (1) hold, J(|x|) ≥ 0, ||
√
J ||1 <∞ and
lim
β→0
B = 0, lim
β→.0
BD = 0.
Then for sufficiently small β there exists the thermodynamic limit ρ̄ of ρ̄Λ, the polymer cluster expan-
sion for the correlation functions ρΛ(ωX) converges absolutely uniformly in Λ, their thermodynamic
limit is also represented by the polymer cluster expansion in which the summation is performed
over Zd and the thermodynamic limit of ρ̄ is substituted instead of ρ̄Λ. Moreover there exist positive
constants c, C such that c−|X| exp
{
−β
∑
x∈X
ṽ(ωx)
}
|ρΛ(ωX)| ≤ C.
The proof of this theorem is standard [3, 4]: at first one proves the bound∑
Y :|Y |=m
∫
P (dωY )|FωX (ωY )| ≤ e|X|(eB)me
β
∑
x∈X
ṽ(ωx)
and then shows that the polymer correlation functions ρ̄Λ, whose sequence satisfies the polymer KS
equation, are appropriately bounded. Note that due to the law of conservation of probability in the
stochastic systems ρ̄Λ coincides with the same expression for the Gibbs initial distribution generated
by a pair potential and an external field, that is, the existence of its thermodynamic limit follows
from the results of [3].
We have to choose such ṽ, J that the condition of the Theorem 1 is satisfied. The following
lemma gives the first step in this direction.
Lemma 1. Let (1) hold, ζ > 0,
Imux−y(ωx, ωy) = J̃(|x− y|)φ(ωx, ωy), ||J̃ ||1 <∞, (2)
and
ṽ(ω) = (v1+ζ(ω) + βr−1b(ω))γ, b2(ω) =
∫
|φ(ω, ω′)|2P (dω′), 0 < r < 1. (3)
Let alsoD(γ) coincide withD and J in the expression for B in the Theorem 1 be given by J = J ′+J̃ .
Then there exist positive constants a1, a2 such that
B ≤ a1β
1−r√D(2γ) + a2β
ζ
1+ζD(2γ + 2−1|J ′|0).
The proof of the lemma will be given in the end of the paper.
Corollary. The conditions for B,D in the Theorem 1 are satisfied if D(γ) is bounded in β > 0
in a neighborhood of zero.
In the three considered cases the real part of the pair potential and the external field u depend
only on ω0. This fact and the fact that the measure P 0 is factorized on Ω0 × Ω∗ yield the result
D =
∫
eγβv
1+ζ(ω0)D∗(ω
0)P ′(dω0), P ′(dω0) =
(∫
e−βu(ω0)P̃ (dω0)
)−1
e−βu(ω0)P̃ (dω0),
where
D∗(ω
0) =
∫
eβ
rγb(ωa,ω∗)P0(dω∗).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
692 W. I. SKRYPNIK
Now we are going to prove that D is finite for the considered systems. In order to do this we have
to write down the explicit expressions of the potentials and characterize their properties.
For the classical Gibbs systems the real-valued part of the complex potential is given by (ω =
= (q, s) ∈ R2)
Reux−y(ωx, ωy) = ux−y(qx, qy) = J0(|x− y|)u0(qx, qy)− J2
1 (|x− y|)u2
1(qx, qy).
And in its turn for Gibbs quantum and stochastic systems (ω = (q, w) ∈ R × C(R+)) it is given,
respectively, by
Reux−y(ωx, ωy) = β−1
β∫
0
ux−y(wx(τ)− wy(τ))dτ,
Reux−y(ωx, ωy) = u0
x−y(qx, qy) + u1,x−y(wx(tβ−1), wy(tβ
−1)) +
tβ−1∫
0
u2,x−y(wx(τ)− wy(τ))dτ,
where t is the time. For our purposes here the explicit expressions for the the pair potential u2,x is
not needed (it is expressed in terms of the interaction pair potential u0,x) and we advise readers to
find them in the Section 3 in [2].
For the classical Gibbs systems J̃ = J1 and the following equality is true:
φ(qx, sx, qy, sy) =
1√
2
(sx + sy)u1(qx, qy).
The potential φ(ωx, ωy) is expressed in terms of the stochastic integrals in w∗x, w
∗
y ∈ C(R+)
φ(wx, w
∗
x, wy, w
∗
y) = 4−1(β−12)κ
t′∫
0
dw∗x(τ)u′(wx(τ), wy(τ)) +
t′∫
0
dw∗y(τ)u′(wy(τ), wx(τ))
,
(3′)
where u′(q, q′) = u1(q, q′), κ = 1, t′ = β, J̃ = J1 and u′(q, q′) = 2∂u0(q, q′), κ = 0, t′ = tβ−1,
J̃ = J0, for quantum and stochastic systems, respectively (u0, u1 are symmetric functions).
We impose as in [1, 2] the conditions
|u0(q, q′)| ≤ 1
2
(v0(q) + v0(q′)), |u′(q, q′)| ≤ 1
2
(v′(q) + v′(q′)). (4)
They are more general than their Kunz version considered in [1] (there is a product of the square
roots of the two functions instead of the half of their sums in [1]). Here v0(q), v′(q) are positive
polynomials in |q| of the 2m0-th, m1-th degrees, respectively, such that for classical and quantum
systems m0 < n, 2m1 < n . Note that the similar inequalities hold for ∂u0(q, q′), ∂2u0(q, q′) in
which the functions from the right-hand side which coincide with polynomials in |q| with the degrees
2m0− 1, 2m0− 2, respectively. This follows from the inequality albk ≤ 2k+l(ak+l + bk+l), a, b > 0
since the potentials u0(q, q′), u1(q, q′) are linear combinations of the elementary polynomials qlq′k,
l + k ∈ Z+. As a result m1 = 2m0 − 1 for the stochastic systems for which m0 < n0. These
conditions allow one to prove the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON POLYMER EXPANSIONS FOR GENERALIZED GIBBS LATTICE SYSTEMS . . . 693
Theorem 2. Let (4) hold. Let also the inequality 1 + ζ <
n
m
, m = max (m0, 2m1) hold for
the classical and quantum systems and the inequalities 1+ ζ < min
(
2n0 − 1
n0 +m0 − 1
,
n0
m0
)
, t ≤ t0β2
hold for the stochastic systems, where t is the time and t0 is a positive constant. Then D(γ) is
bounded in β in a neighborhood of zero for the classical systems if r =
m1
n
. The same is true for
the quantum and stochastic oscillator systems if r =
1
2
+
m1
n
.
Hence we proved the main result of our paper.
Theorem 3. Let (4), the conditions for ζ, r, t in the Theorem 2 and for J in the Theorem 1
and Lemma 1 hold. Then the conclusion of the Theorem 1 is true for ρ̄Λ and the correlation functions
ρΛ(ωX) of the classical, quantum and stochastic oscillator systems.
Proof of Theorem 2. For the classical and quantum systems (1) is valid with J ′ = J0 + J2 and
v(q) = v0(q) + v2
1(q), v(w) = β−1
∫ β
0
v(w(τ))dτ, respectively. The second condition in (1) will be
satisfied if 1 + ζ <
n
m
. For the classical systems we have
b2(q, s) = (2
√
2π)−1
(∫
e−βu(q)dq′
)−1 ∫
(s+ s′)2u2
1(q, q′)e−
s′2
2 e−βu(q′)dq′ds′,
where the integration is performed over R2 and
b2(q, s) ≤ (c2
4s
2 + c2
3)v′2(q) + β−
2m1
n (c2
1s
2 + c2
0),
where c3, c4 does not depend on β and c1, c0 are bounded functions in nonnegative finite β. Here we
used the inequality (s+ s′)2 ≤ 2(s2 + s′2) and rescaled the variables in the corresponding integrals
in the numerators and denominators by β−
1
2n . As a result
b(q, s) ≤ (c4|s|+ c3)v′(q) + β−
m1
n (c1|s|+ c0).
Hence we can put r =
m1
n
and D∗(q) is easily estimated as a Gaussian integral. That is
D∗(q) ≤ 2 exp
{
1
2
[
c3γβ
m1
n v′(q) + γ2(β
m1
n c4v
′(q) + c1)2
]
+ c0γ
}
.
As a result the condition 1 + ζ <
n
m
guarantees that
D =
∫
eγβv
1+ζ(q)D∗(q)P
′(dq)
is finite for nonnegative finite β after a rescaling of the variables in the corresponding integrals in
the numerator and denominator by β−
1
2n . This concludes the proof for the classical systems.
Now let’s consider the quantum and stochastic systems characterized, respectively, by the external
potentials u(q), 2u0(q). In an estimate of D∗(ω0) , based on the estimates from the Section 2 from
[3], we will use the inequalities
η−q
2n0 − ū ≤ u0(q) ≤ η+q
2n0
+ ū, η−q
2n − ū ≤ u(q) ≤ η+q
2n + ū, η+ ≤ 3
2
η−.
The proof is based on application of the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
694 W. I. SKRYPNIK
Lemma 2. Let b(w,w∗) correspond to φ in (3′). Then∫
eγβ
rb(w,w∗)P0(dw∗) ≤ κ∗
[√
IP + eσβ
2(1−κ)
∫ t′
0 w2n(τ)dτ
]
,
IP =
∫
P ′(dw)e2σβ2(1−κ)
∫ t′
0 w2n(τ)dτ ,
where 2σ < η−, σ < 2n2η2
− for the quantum (κ = 1) and stochastic systems (κ = 0), respectively,
n = 2n0 − 1 for the stochastic systems, κ∗ does not depend on w and is an entire function of
(β2r−2
m1
n
(1−κ)t′−2κ+1−m1
n )
1
2 .
The proof repeats all the steps of the proof of Lemma 2.1 in [5] which coincides with the Lemma
2 for quantum systems. Note that m1 corresponds to n1 in [5]. For quantum and stochastic systems
the function κ∗ is bounded in nonnegative finite β if 2r − 1 − m1
n
≥ 0 and therefore one can put
r =
1
2
+
m1
n
. Note that for the stochastic systems 2m1 < 2(n− 1) following from m1 = 2m0 − 1,
m0 < n0.
Now in order to show that D is bounded for nonnegative finite β one has to establish the same
fact for IP . It was done for quantum systems in [5] ( see the proof of the Lemma 2.2) with the
help of the Golden – Thompson and Jensen inequalities, applied for the numerator and denominator,
and a rescaling of the simple integrals in a variable from R. For them u(w) = β−1
∫ β
0
u(w(τ))dτ,
where u(q), q ∈ R, is the external potential from the expression of the potential energy of quantum
systems. The analogous simplified technique will be applied for the stochastic systems. Instead of
the Golden – Thompson inequality the law of conservation of probability will be utilized by us and
a rescaling of continuous (Wiener) paths as in [2] will not be considered. For the stochastic systems
we have ω = (q, w) and
u(q, w) = u0(q) + u1(w(t′)) +
t′∫
0
u2(w(τ))dτ,
where
u1(q) = (2η0 − 1)u0(q) + u1(q), u2(q) = −∂2u0(q) + β(∂u0(q))2, η0 >
1
2
,
and u1 is an even polynomial of the degree less than 2n0. The function v is given by (see the proof
of the Proposition 4.1 in [2])
v(q, w) = v0(q) + v1(w(t′)) +
t′∫
0
v2(w(τ))dτ, v2 = v′2 + βv′′2 ,
where v0, v1, v
′
2, v
′′
2(q) are positive polynomials with the degrees 2m0 < 2n0, 2m0, 2(m0 − 1),
2(n0 + m0 − 1), respectively (n0, m0 are denoted by n, m, respectively, in the Section 2 in [2]).
From elementary inequality (a1 + . . .+ ak)
n
m ≤ kn(an1 + . . .+ ank)
1
m ≤ kn(a
n
m
1 + . . .+ a
n
m
k ), where
n,m ∈ Z+, n > m, aj ≥ 0 and the Helder inequality it follows that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON POLYMER EXPANSIONS FOR GENERALIZED GIBBS LATTICE SYSTEMS . . . 695
v1+ζ(q, w) ≤ 3〈1+ζ〉
v1+ζ
0 (q) + v1+ζ
1 (w(t′)) + t′ζ
t′∫
0
v1+ζ
2 (w(τ))dτ
,
where 〈1 + ζ〉 is the numerator integer in the fractional representation of 1 + ζ and 1 + ζ <
< min
(
2n0 − 1
n0 +m0 − 1
,
n0
m0
)
. The last condition implies the second inequality in (1).
The following law of conservation of probability holds∫
dq
∫
e−βu(q,w|a)Pq(dw) =
∫
e−β(u1(q)+2η0au0(q))dq, (5)
where u(q, w|a) is equal to u(q, w) if one substitutes au0 instead of u0 into its expression. It follows
from the gradient character of the Smoluchowski equation. We remind that the function under the
sign of the integral in q in the left-hand side of this equality coincides with the solution of the
Smoluchowski equation with the initial function which coincides with the function under the sign of
the integral in its right-hand side. Let a = 2−1 and in addition 2σ + n2η2
+ < 4n2η2
− then
eγβv
1+ζ(q,w)e−β(u(q,w)−u(q,w| 1
2
))e2β2σ
∫ t′
0 w2n(τ)dτ ≤ eC(β,t′), (6)
where
C(β, t′) = β(t′β−1C0 + t′ζ+1(C1 + β−2C2) + C3 + C4t
′), β < 1,
and Cj are constants. Here one has to use the fact that the coefficient before
∫ t′
0
w2n(τ) dτ in the
exponent in the left-hand side of (6) is negative, and apply the inequality qk ≤ εql + ε−
k
l−k (see the
proof below) for q > 0, k ≤ l and small ε < 1. That is
t′∫
0
wk(τ) dτ ≤ βε
t′∫
0
wl(τ)dτ + (βε)−
k
l−k t′.
C3 is the contribution of terms depending on q, w(t′), C0 is the contribution of the terms in∫ t′
0
u2(w(τ))dτ containing the second derivative of u0, since β
− m0−1
2n0−m0 ≤ β−1. The integrals of v′′2 ,
v′2 contributed the constants C1, C2 > 0, respectively, since β
− (m0−1)(1+ζ)
n−(m0−1)(1+ζ) ≤ β
− n0
2n0−1−n0 ≤ β−2.
The terms containing the first derivative of u0 in the integral
∫ t′
0
u2(w(τ))dτ contribute C4. (6) is
valid since we choose ε such that sum of all the terms in the exponent in the left-hand side of (5) with
powers in q, w less than the senior powers in u(q, w) − u
(
q, w
∣∣∣1
2
)
yield the negative coefficients
before the terms with the senior powers . Using these bounds, (6) and the law of conservation of
probability we see that (
√
IP ≤ IP )
D ≤ 2κ∗e
C(β,t′)
∫
e−β(u1(q)+η0u0(q))dq
(∫
e−β(u1(q)+2η0u0(q))dq
)−1
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
696 W. I. SKRYPNIK
Hence D is bounded in nonnegative finite β (recall that t′ = tβ−1) if t ≤ t0β
2 since
2 + ζ
1 + ζ
≤ 2,
where t0 is independent of β.
The inequality qk ≤ εqn + ε−
k
n−k is proved if one proves that for q ≥ ε−
1
n−k the inequality
qk ≤ εqn holds. This inequality is true if (a + ε−
1
n−k )k ≤ ε(a + ε−
1
n−k )n for a ≥ 0. The last
inequality is checked comparing the coefficients near al in its both sides. They equal C lkε
− k−l
n−k ,
C lnεε
− n−l
n−k = C lnε
− k−l
n−k , respectively. Hence all the terms in its left-hand side are smaller than the
corresponding terms in its right-hand side. This concludes the proof of the theorem.
Proof of Lemma 1. From
|ea − 1| = |eiIm a(eRe a − 1) + (eiIm a − 1)| ≤ |Re a|e|Re a| + 2|Im a|,
(1) and the superstability condition it follows that (note that ω0
0 = ω0)
|e−βux(ω,ωx) − 1| ≤ β|Reux(ω, ωx)|eβ|Reux(ω,ωx)| + 2β|Imux(ω, ωx)| ≤
≤ β
[
1
2
J ′(|x|)(v(ω0
x) + v(ω0))e
β
2
J ′(|x|)(v(ω0
x)+v(ω0)) + 2J̃(|x|)φ(ωx, ω)
]
.
Let b0x and b1x are the contributions to bx of the first and second terms in the square brackets. Let also
B0 and B1 be the corresponding parts from B. Then the Schwartz inequality gives
b1x(ω) ≤ 2β
√
D(2γ)J̃(|x|)e−γβr||
√
J ||−1
1
√
J(|x|)b(ω)b(ω)e−β(γv1+ζ(ω)−||J ′||v(ω))||
√
J ||−1
1
√
J(|x|).
That is
B1 ≤ 2
√
D(2γ)γ−1β1−rκ1κ||
√
J ||21, κn = sup
a≥0
ane−a,
κ = sup
a≥0,x
e−β(γa1+ζ−||J ′||a)||
√
J ||−1
1
√
J(x) = e
β||
√
J ||−1
1 |
√
J |0
(
||J′||γ
− 1
1+ζ
1+ζ
)1+ 1
z
, |J |0 = sup
x
J(|x|).
Multiplying b0x by eγβv
1+ζ(ω0
x)e−γβv
1+ζ(ω0
x) we obtain, using the equalities 1− 1
2(1 + ζ)
=
1 + 2ζ
2(1 + ζ)
,
1
2
− 1
2(1 + ζ)
=
ζ
2(1 + ζ)
, the following inequality:
B0 ≤ 2−1β
ζ
1+ζ γ
− 1
1+ζ [κ1κ(0)||J ′||1 + κ0κ(1)||J
1+2ζ
2(1+ζ) ||1]D(2γ + 2−1|J ′|0),
where
κ(n) = sup
a≥0
an exp
{
−a1+ζ + (γ−1βζ ||
√
J ||1)
1
1+ζ (2−1|J
1+2ζ
2(1+ζ) |0 + ||J ′||1|J
ζ
2(1+ζ) |0)a
}
.
We have B = B0 +B1 and the last bounds for B0, B1 prove the lemma.
Remark 1. We assume that the generator of the semigroup, producing the Wiener measure,
coincides with ∂2, where ∂2 is the operator of the second derivative. This obliges us to assume that
the mass of the quantum particles is equal to
1
2
. The transition to the quantum systems with the mass
equal to 1, considered in [1], is produced by a simple rescaling of variables.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON POLYMER EXPANSIONS FOR GENERALIZED GIBBS LATTICE SYSTEMS . . . 697
Remark 2. Our main result (Theorem 3) permits us to obtain the conclusions of the Theorems
3 – 4.1 in [1], Theorem 2.1 in [2] easily. The conditions and the proof of the Theorem 3 are simpler
than those of the theorems in [1, 2] based on a rescaling of wiener paths. The proposed in this paper
technique is based on another choice of ṽ, the bound for B in the Lemma 1, on the bound of the
Lemma 2, the Golden – Thompson inequality (for the quantum systems) and the law of conservation
of probability (for the stochastic systems). No rescaling of wiener paths is needed in this paper. The
grand canonical analog of the result of [2] can be found in [6].
1. Skrypnik W. On polymer expansions for Gibbs lattice systems of oscillators with ternary interaction // Ukr. Math. J. –
2001. – 53, № 11. – P. 1532 – 1544.
2. Skrypnik W. On polymer expansion for Gibbsian states of non-equilibrium systems of interacting Brownian oscillators
// Ukr. Math. J. – 2003. – 55, № 12.
3. Kunz H. Analiticity and clustering properties of unbounded spin systems // Communs Math. Phys. – 1978. – 59. –
P. 53 – 69.
4. Gruber G., Kunz H. General properties of polymer systems // Communs Math. Phys. – 1971. – 22. – P. 133 – 161.
5. Skrypnik W. Kirkwood – Salsburg equation for lattice quantum systems of oscillators with manybody interaction
potentials // Ukr. Math. J. – 2009. – 61, № 5.
6. Skrypnik W. On evolution of Gibbs states of lattice gradient stochastic dynamics of interacting oscillators, Random
operators and stochastic dynamics // Theory Stochast. Processes. – 2009. – 15(31), № 1. – P. 61 – 82.
Received 20.09.11,
after revision — 15.02.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
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