Gibbs states of lattice spin systems with unbounded disorder
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irk-123456789-321282012-04-10T12:11:32Z Gibbs states of lattice spin systems with unbounded disorder Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. 2010 Article Gibbs states of lattice spin systems with unbounded disorder / Yu. Kondratiev, Yu. Kozitsky, T. Pasurek // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43601:1-12. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 61.43.Fs, 64.60.De, 64.70.kj http://dspace.nbuv.gov.ua/handle/123456789/32128 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. Gibbs states of lattice spin systems with unbounded disorder Condensed Matter Physics |
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Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. |
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Kondratiev, Yu. |
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Gibbs states of lattice spin systems with unbounded disorder |
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Gibbs states of lattice spin systems with unbounded disorder |
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Gibbs states of lattice spin systems with unbounded disorder |
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Gibbs states of lattice spin systems with unbounded disorder |
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Gibbs states of lattice spin systems with unbounded disorder |
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gibbs states of lattice spin systems with unbounded disorder |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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Gibbs states of lattice spin systems with unbounded disorder / Yu. Kondratiev, Yu. Kozitsky, T. Pasurek // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43601:1-12. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kondratievyu gibbsstatesoflatticespinsystemswithunboundeddisorder AT kozitskyyu gibbsstatesoflatticespinsystemswithunboundeddisorder AT pasurekt gibbsstatesoflatticespinsystemswithunboundeddisorder |
| first_indexed |
2025-07-03T12:39:44Z |
| last_indexed |
2025-07-03T12:39:44Z |
| _version_ |
1836629502491361280 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 4, 43601: 1–12
http://www.icmp.lviv.ua/journal
Gibbs states of lattice spin systems with unbounded
disorder ∗
Yu. Kondratiev1, Yu. Kozitsky2, T. Pasurek1
1 Fakultät für Mathematik, Universität Bielefeld, D–33615 Bielefeld, Germany
2 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20–031 Lublin, Poland
Received September 9, 2010, in final form September 28, 2010
The Gibbs states of a spin system on the lattice Z
d with pair interactions Jxyσ(x)σ(y) are studied. Here
〈x, y〉 ∈ E, i.e. x and y are neighbors in Z
d. The intensities Jxy and the spins σ(x), σ(y) are arbitrarily real.
To control their growth we introduce appropriate sets Jq ⊂ RE and Sp ⊂ RZ
d
and show that, for every
J = (Jxy) ∈ Jq : (a) the set of Gibbs states Gp(J) = {µ : solves DLR, µ(Sp) = 1} is non-void and weakly
compact; (b) each µ ∈ Gp(J) obeys an integrability estimate, the same for all µ. Next we study the case where
Jq is equipped with a norm, with the Borel σ-field B(Jq), and with a complete probability measure ν. We show
that the set-valued map Jq ∋ J 7→ Gp(J) has measurable selections Jq ∋ J 7→ µ(J) ∈ Gp(J), which are
random Gibbs measures. We demonstrate that the empirical distributions N−1
∑N
n=1 π∆n
(·|J, ξ), obtained
from the local conditional Gibbs measures π∆n
(·|J, ξ) and from exhausting sequences of ∆n ⊂ Zd, have
ν-a.s. weak limits as N → +∞, which are random Gibbs measures. Similarly, we show the existence of the
ν-a.s. weak limits of the empirical metastates N−1
∑N
n=1 δπ∆n
(·|J,ξ), which are Aizenman-Wehr metastates.
Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.
Key words: Aizenman-Wehr metastate, Newman-Stein empirical metastate, chaotic size dependence,
Komlós theorem, quenched pressure, spin glass
PACS: 61.43.Fs, 64.60.De, 64.70.kj
1. Introduction
In the present note, we announce a number of the results describing Gibbs states of a lattice
system with unbounded spins and unbounded random interactions. Their complete proof will be
published in a separate work. Our motivations, as well as the discussion of the connections of our
results with those known for spin models with random interactions are given in section 3.2 below.
Throughout the note, for a topological space S, by P(S) we denote the set of all probability
measures on (S,B(S)), where B(S) will always stand for the corresponding Borel σ-field.
Given a countable set X, a random field on X is a collection of random variables - spins, defined
on some probability space and taking values in the corresponding single-spin (Polish) spaces Sx,
x ∈ X. In a ‘canonical version’, the probability space is (S,B(S), µ), where S is the product space
of all Sx. Then the notion random field is attributed to the latter measure as well. A particular case
of such a field is the product measure of some single-spin probability measures χx, x ∈ X. Gibbs
random fields with pair interactions are constructed as perturbations of
∏
x∈X
χx by the ‘densities’
exp
∑
〈x,y〉
Wxy(σ(x), σ(y))
, (1.1)
where Wxy : Sx × Sy → R are measurable functions – interaction potentials, whereas the sum
is taken over a subset of X × X. Such a field defines the graph G = (X,E), where the set of
edges E consists of those pairs {x, y} where Wxy is not the zero function. The case of a special
∗Supported by the DFG through SFB 701: “Spektrale Strukturen und Topologische Methoden in der Mathe-
matik” and through the research project 436 POL 125/0–1.
c© Yu. Kondratiev, Yu. Kozitsky, T. Pasurek 43601-1
http://www.icmp.lviv.ua/journal
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
interest is where the potentials are random. Then one deals with another random field, this time
on E, represented by the triple (W ,F , P ). Here W is the space of interactions consisting of W =
(Wxy)〈x,y〉∈E, F is an appropriate σ-field, and P is a probability measure. A standard assumption
is that the degree of each vertex is finite and that the functions Wxy : Sx × Sy → R are P -almost
surely bounded, in which case the interactions are called regular, c.f. Definition 6.2.1 in [1], page 99.
The only irregular case studied in the literature is that of the long-range spin glasses, where the
single-spin spaces are finite, and thus the functions Wxy are bounded, but the vertex degrees are
infinite. In the case of regular W , a measurable map
W ∋ W 7→ µ(W ) ∈ P(S) (1.2)
is called a random Gibbs measure, or a random Gibbs state, if for P -almost all W , µ(W ) has a
Markov property, standard for Gibbs measures, c.f. Definition 6.2.5 in [1]. The measurability in
(1.2) is the key point since only in this case one can speak about averages with respect to the
disorder, that is, about the expectations EPΦ
(
Eµ(W )F
)
, where F : S → R and Φ : R → R
are appropriate functions, see the discussion in section 6.2 in [1]. In general, for models with the
interactions (1.1), there might exist multiple Gibbs measures1. Hence, the mapW 7→ {µ(W ) : µ(W )
is a Gibbs measure} can be set-valued and the existence of its measurable selections (1.2) is not
obvious. To the best of our knowledge, in a systematic way this aspect of the theory has never
been discussed so far. Thus, one of the aims of this work is to look at the problem of Gibbs fields
with random interactions from the point of view of the set-valued analysis [2]. Another aim is
to elaborate a method, which would allow us to study the models with unbounded interactions
– the other irregular case that has not been studied yet. In order to make the things as much
transparent as possible, we consider the simplest case where the graph is a lattice Zd with the edge
set E = {(x, y) : |x− y| = 1}, whereas the interaction potentials have the form
Wxy(u, v) = Jxyuv, Jxy, u, v ∈ R, (1.3)
that is, all Sx are the copies of R. In the physical terminology, this is a lattice spin model with
unbounded spins and a harmonic pair interaction. The existence and the properties of the corre-
sponding Gibbs fields for all Jxy being the same (or just uniformly bounded) and nonrandom have
been studied since the 1970th, see [3, 4] and the bibliographic notes in [5]. However, the case of
sup〈x,y〉∈E |Jxy| = +∞ has not been studied so far. To control the growth of J = (Jxy)〈x,y〉∈E and
σ = (σ(x))x∈X, we introduce two Banach spaces Jq ⊂ R
E and Sp ⊂ R
Z
d
. They are large enough
so that every ball in Jq contains J with arbitrarily big |Jxy|. Then the interaction randomness is
realized as the triple (Jq,B(Jq), ν), where ν is a general complete probability measure (need not
be product, etc). For every finite ∆ ⊂ Z
d, by means of the potentials (1.3) we introduce the local
conditional Gibbs measure π∆(·|J, ξ), J ∈ Jq and ξ ∈ Sp, which then allows us to define the set of
tempered Gibbs measures Gp(J) consisting of those µ ∈ P(RZ
d
) which solve the DLR equation and
are such that µ(Sp) = 1. We claim that:
(a) for every J ∈ Jq, the set Gp(J) is non-void and weakly compact, and that each
µ ∈ Gp(J) obeys an integrability estimate, the same for all such µ (Theorem 3.1);
(b) the map Jq ∋ J 7→ Gp(J) is measurable, as a set-valued map, and hence there exist
measurable selections Jq ∋ J 7→ µ(J) ∈ Gp(J)(Theorem 3.4).
The key element of the proof of (a) is an integrability estimate for the measures π∆(·|J, ξ) which
implies the existence of accumulation points of the family {π∆(·|J, ξ)}∆⊂Zd , that are elements of
Gp(J). Then, the corresponding estimate for µ ∈ Gp(J), which holds uniformly for all such µ and
all ‖J‖q 6 R, R > 0, are obtained therefrom. This allows us to prove that the map Jq ∋ J 7→ Gp(J)
is upper semi-continuous, which extends the result obtained (for bounded interactions) in item (d)
of Theorem 4.23 in [6], page 72. By Theorem 8.1.4 of [2], page 310, the mentioned upper semi-
continuity implies the measurability of Jq ∋ J 7→ Gp(J), which in turn yields the existence of
1The a.s. uniqueness of Gibbs measures of disordered spin systems, as well as the problem of phase transitions,
are highly nontrivial, see the discussion and the corresponding references in section 6.3 in [1].
43601-2
Gibbs states of spin systems with unbounded disorder
measurable selections, see Theorem 8.1.3 in [2]. In Corollary 3.2, we also establish the existence
of the averages EνΦ
(
Eµ(J)F
)
for appropriate functions F and Φ. Note that the constants in (3.3)
are explicitly expressed in terms of the model parameters.
As is commonly accepted, see chapter 7 in [6], the extreme elements of Gp(J) correspond to
thermodynamic phases of the physical system modeled by the family {π∆(·|J, ξ)}∆⊂Zd . These
elements are contained in the set of limiting Gibbs measures (Minlos states), see Corollary 7.30
on page 135 in [6], which are exactly the accumulation points of {π∆(·|J, ξ)}∆⊂Zd . The physical
meaning of such limiting Gibbs measures is that they approximate Gibbs measures of large finite
systems, c.f. the corresponding discussion in [7]. The random Gibbs measures obtained in (b)
as measurable selections need not be limiting Gibbs measures – thus, the result of Theorem 3.4
has rather got a theoretical value from the point of view of physics. The characteristic feature
of the spin models with random interactions is the so-called chaotic dependence of the measures
π∆(·|J, ξ) on ∆, see [8] and the references cited therein. This means that the limits of the sequences
{π∆n
(·|J, ξ)}n∈N need not be measurable (with respect to J) and hence cannot serve as limiting
random Gibbs measures. With the help of the Komlós theorem [9, 10] in Theorem 3.6 we obtain
that,
(c) for every ξ ∈ Sp, there exists a random Gibbs measure µξ(J) and an exhausting
sequence D = {∆n}n∈N such that µξ(J) is the ν-a.s. weak limit of the sequence
of ‘empirical distributions’
1
N
N
∑
n=1
π∆n
(·|J, ξ), N ∈ N. (1.4)
Under rather general assumptions, each Gibbs measure has an extreme decomposition, see Theo-
rem 7.26 in [6], page 133. Thus, every measurable selection can be written in the form
µ(J) =
∫
Gex
p (J)
µ w(J)(dµ). (1.5)
Here Gex
p (J) is the extreme boundary of Gp(J) and w(J) is a weight, uniquely determined by µ(J).
This decomposition holds for all J ∈ Jq, however, the weights w(J) need not be J-measurable.
Suppose now that a representation holds, which is similar to (1.5) with a measurable weight and
the integral taken over the whole set Gp(J). Then it yields a random Gibbs measure and the
corresponding weight is called an Aizenman-Wehr metastate [11], see also page 103 in [1] and
Definition 2.4 below. In Theorem 3.7, we show that
(d) for every ξ ∈ Sp, there exists an Aizenman-Wehr metastate mξ(J) and an exhausting
sequence D = {∆n}n∈N such that mξ(J) is the ν-a.s. weak limit of the sequence of
empirical metastates {N−1
∑N
n=1 δπ∆n (·|J,ξ)}N∈N.
The thermodynamic pressure, or the free energy density, is an important characteristic which one
obtains in the thermodynamic limit, see. e.g. the discussion in [1], p. 24–28. For non-random
(translation invariant) systems, the pressure exists and is independent of the way the limit has
been taken, see sections 2 and 3 in [3] or Theorem 3.10 in [12]. In Theorem 3.8, we show that under
an additional condition on the measure ν the pressure can be obtained as the almost sure limit of
the local pressures, ‘averaged’ over {∆n} similarly as in (1.4). In Theorem 3.9, we assume that ν is
a product measure with the zero first moment and prove that all the sequences of local pressures
averaged over the disorder have one and the same thermodynamic limit – the quenched pressure.
In proving Theorems 3.8 and 3.9, we employ a version of the first GKS inequality, known for such
models with Jxy > 0, which we obtain here by extending the approach of [13, 14] to the case of
unbounded interactions.
43601-3
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
2. Setup
2.1. General setting
In constructing Gibbs random fields, we follow the standard scheme [6]. Our Gibbs fields will
live on the set X = Z
d, d ∈ N, equipped with the adjacency relation x ∼ y defined by the condition
|x−y| = 1. By E we denote the set of edges of the corresponding graph. We also use the shorthand
∑
x
=
∑
x∈Zd
, sup
x
= sup
x∈Zd
,
∑
y∼x
=
∑
y∈Zd: y∼x
.
The set R
Z
d
is equipped with the product topology, which turns it into a Polish space – a sepa-
rable and completely metrizable topological space. Let Cb(R
Z
d
) be the Banach space of bounded
continuous functions f : RZ
d
→ R with the norm
‖f‖∞ = sup
σ∈RZd
|f(σ)|.
By means of Cb(R
Z
d
) we define the weak topology on the set of all probability measures P(RZ
d
),
which turns it into a Polish space, see e.g. page 39 in [15].
For any ∆ ⊂ Z
d, we let ∆c def
= Z
d \ ∆; by writing ∆ ⋐ X we mean that 0 < |∆| < ∞. A
sequence D = {∆n}n∈N, such that ∆n ⋐ Z
d for all n ∈ N, is said to be cofinal if it is: (a) ordered
by inclusion; (b) exhausting, i.e. such that each x ∈ Z
d belongs to a certain ∆n. For ∆ ⊂ Z
d, by
B∆ we denote the σ-sub-field of B(RZ
d
) generated by (σ(x))x∈∆. For ∆ ⋐ Z
d, a probability kernel
π∆(·|·) is a function on (B(RZ
d
),RZ
d
) such that for any ξ ∈ R
Z
d
, π∆(·|ξ) is in P(RZ
d
), and for
any A ∈ B(RZ
d
), π∆(A|·) is B∆c-measurable. Such a kernel is said to be proper if π∆(A|·) = IA(·)
for any A ∈ B∆c . Here IA(ξ) = 1 if ξ ∈ A, and IA(ξ) = 0 otherwise. Given a family {π∆}∆⋐Zd ,
suppose that there exists µ ∈ P(RZ
d
) such that
µ(A|B∆c) = π∆(A|·), (2.1)
which holds µ-almost surely for all A ∈ B(RZ
d
) and ∆ ⋐ Z
d. Then this measure µ is said to
be specified by the family {π∆}∆⋐Zd. In this case, all the kernels π∆ are µ-almost surely proper,
and their family is µ-almost surely consistent. The latter means that for µ-almost all ξ and all
A ∈ B(RZ
d
),
∫
RZd
πΛ(A|η)π∆(dη|ξ) = π∆(A|ξ), (2.2)
which holds for any pair of subsets such that Λ ⊂ ∆. It should be pointed out that (2.1) is
equivalent to
∫
RZd
π∆(A|ξ)µ(dξ) = µ(A), (2.3)
which holds for all A ∈ B(RZ
d
) and ∆ ⋐ Z
d. The condition (2.3) is called the Dobrushin-Lanford-
Ruelle (DLR) equation. It is equivalent to
∫
RZd
π∆(f |ξ)µ(dξ) = µ(f), (2.4)
for all f ∈ Cb(R
Z
d
) and all ∆ ⋐ Z
d. Here we use the notation
µ(f) =
∫
RZd
f(σ)µ(dσ). (2.5)
43601-4
Gibbs states of spin systems with unbounded disorder
2.2. The Gibbs fields
The Gibbs fields we are going to construct are specified by the kernels obtained as perturbations
of the products of single–spin measures by the factors (1.1) with the functions Wxy as in (1.3). For
∆ ⋐ Z
d and ξ ∈ R
Z
d
, we set
−H∆(σ∆|J, ξ) =
∑
〈x,y〉∈E∆
Jxyσ(x)σ(y) +
∑
x∈∆, y∈∆c, x∼y
Jxyσ(x)ξ(y), (2.6)
where E∆ consists of the edges with both endpoints in∆. In the mentioned terminology,H∆(σ∆|J, ξ)
is the energy of interaction of the spins located in∆ with each other and with the fixed spins outside
∆. For a family χ = (χx)x∈Zd , χx ∈ P(R), we put
χ∆(dσ∆) =
∏
x∈∆
χx(dσ(x)), σ∆ = (σ(x))x∈∆ , (2.7)
which is an element of P(R|∆|). Thereafter, for A ∈ B(RZ
d
), we define
π∆(A|J, ξ) =
1
Z∆(J, ξ)
∫
R|∆|
IA(σ∆ × ξ∆c) exp[−H∆(σ∆|J, ξ)]χ∆(dσ∆). (2.8)
Here Z∆(J, ξ) is a normalizing factor, that is,
Z∆(J, ξ) =
∫
R|∆|
exp[−H∆(σ∆|J, ξ)]χ∆(dσ∆), (2.9)
and the juxtaposition stands for the element of RZ
d
such that
(σ∆ × ξ∆c)(x) = σ(x), for x ∈ ∆; (σ∆ × ξ∆c)(x) = ξ(x), for x ∈ ∆c.
The family {π∆}∆⋐Zd is clearly consistent. It is the local Gibbs specification for our model.
As is typical of Gibbs measures of models with unbounded spins, the description of the prop-
erties possessed by all such measures is rather unrealistic. Usually, the study is restricted to those
measures which have a prescribed support property. Such measures are called tempered. To define
the mentioned property we use a weight function w : Zd → (0, 1], which by definition has the
following properties:
(a) |w|
def
=
∑
x
w(x) < ∞, (2.10)
(b) ∃w0 > 0 w(x) 6 w0w(y), for all x ∼ y. (2.11)
Note that w0 > 1, otherwise one would get w(x) ≡ 0. A typical example can be
w(x) = exp(−α|x|), α > 0. (2.12)
For w obeying (2.10) and (2.11) and for a p > 1, we set
‖σ‖p =
(
∑
x
|σ(x)|pw(x)
)1/p
, (2.13)
and
Sp = Lp(Zd, w) = {σ ∈ R
Z
d
: ‖σ‖p < ∞}. (2.14)
43601-5
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
This will be the space of tempered spin configurations. Next, for q > 1, we introduce the space of
tempered interaction intensities
‖J‖q =
∑
〈x,y〉∈E
|Jxy|
q[w(x) + w(y)]
1/q
, (2.15)
Jq = Lq(E, w) = {J ∈ R
E : ‖J‖q < ∞}.
Clearly, Lp(Zd, w) and Lq(E, w) are measurable subsets of the Polish spaces RZ
d
and R
E, respec-
tively. We equip these sets with the corresponding norm topologies, which turns them into separable
Banach spaces. It can easily be shown (see also the Kuratowski theorem, page 15 in [15]), that
B(Sp) = {Sp ∩A : A ∈ B(RZ
d
)}. (2.16)
Thus, one can consider the set
Ptemp = {µ ∈ P(RZ
d
) : µ (Sp) = 1}. (2.17)
The elements of Ptemp are called tempered measures. Now we impose conditions on the family of
single-spin measures χ = (χx)x∈Zd . For λ > 0 and q > 1, we set
sup
x
∫
R
exp
(
λ|u|2q/(q−1)
)
χx(du) = C+(λ),
inf
x
∫
R
exp
(
−λ|u|2q/(q−1)
)
χx(du) = C−(λ). (2.18)
And then
Kq
def
= {χ = (χx)x∈Zd : ∀λ > 0 C+(λ) < ∞, C−(λ) > 0}. (2.19)
As an example of χ ∈ Kq one can take the copies of the measure χ0(du) ∼ exp(−V (u))du, where
V is an even semi-bounded polynomial of degV > 2q/(q − 1), c.f. [3, 5]. This corresponds to the
physical model called an anharmonic crystal where the spins are the displacements of the oscillators
from their equilibrium positions.
In the sequel, we shall always choose J in Jq and χ in Kq with one and the same q > 1. We
also assume that this q and p in (2.14) and (2.17) satisfy
p =
2q
q − 1
, (2.20)
i.e. p > 2. As our main concern is the dependence on J , the dependence on χ will always be
suppressed from the notations.
Definition 2.1 Given J = (Jxy)〈x,y〉∈E ∈ Jq and p as in (2.20), by Gp(J) we denote the set of
all µ ∈ Ptemp which solve the DLR equation (2.3) with the kernels defined in (2.6) and (2.8). The
elements of Gp(J) are called (tempered) Gibbs measures.
We recall that a probability space (Ω,O, P ) is said to be complete if for every A such that P (A) = 0,
each subset of A is in O. We also recall that Jq is a separable Banach space.
Definition 2.2 By the lattice model with unbounded spins and unbounded random interactions we
mean the pair
(Jq,B(Jq), ν) and {π∆(·|J, ξ) : ∆ ⋐ Z
d, J ∈ Jq, ξ ∈ Sp},
where the probability space is complete and the kernels π∆ are defined in (2.6) and (2.8).
43601-6
Gibbs states of spin systems with unbounded disorder
Definition 2.3 A B(Jq)/B(P(RZ
d
))-measurable map Jq ∋ J 7→ µ(J) ∈ P(RZ
d
) is said to be a
random Gibbs measure if µ(J) ∈ Gp(J) for ν-almost all J ∈ Jq.
Note that when we speak about a Gibbs measure µ we mean merely an element of a given Gp(J).
However, a random Gibbs measure µ(J) will stand for a measure-valued function of J ∈ Jq.
Let P denote the space of all probability measures on (P(RZ
d
),B(P(RZ
d
))). We equip it with
the weak topology and thereby with the Borel σ-field B. For every f ∈ Cb(R
Z
d
), the evaluation
map P(RZ
d
) ∋ µ 7→ µ(f) is continuous and bounded.
Definition 2.4 A B(Jq)/B-measurable map Jq ∋ J 7→ m(J) ∈ P is said to be an Aizenman-Wehr
metastate if
(a) m(J) (Gp(J)) = 1 for ν-almost all J ∈ Jq;
(b) the map
Jq ∋ J 7→
∫
P(RZd )
µ m(J)(dµ) ∈ P(RZ
d
) (2.21)
is a random Gibbs measure.
Note that the integral in (2.21) is understood in terms of the pairing with f ∈ Cb(R
Z
d
).
3. The results
3.1. Theorems
For R > 0, we set Bq(R) = {J ∈ Jq : ‖J‖q 6 R}. From (2.15) it follows that
sup
J∈Bq(R)
sup
〈x,y〉∈E
|Jxy| = +∞, (3.1)
for any R > 0. Recall that the set of tempered measures Ptemp was defined in (2.17). In the sequel,
when we discuss topological properties of Gp(J) we always mean the topology induced by the weak
topology of the space P(RZ
d
), defined by means of Cb(R
d).
Theorem 3.1 For every J ∈ Jq, q > 1, the set Gp(J) (p as in (2.20)) is non-void and compact.
For any λ > 0, there exist positive constants Υi(λ), i = 1, 2, such that for every µ ∈ Gp(J), the
following estimate holds
∫
RZd
exp
(
λ‖σ‖pp
)
µ(dσ) 6 exp
(
Υ1(λ) +Υ2(λ)‖J‖
q
q
)
. (3.2)
By Jensen’s inequality, one readily gets from (3.2) the following.
Corollary 3.2 There exist positive constants A and B such that for any random Gibbs measure
µ(J), the following estimate
∫
Jq
Φ
∫
RZd
‖σ‖ppµ(J)(dσ)
ν(dJ) 6
∫
Jq
Φ
(
A+B‖J‖qq
)
ν(dJ) (3.3)
holds for any increasing function Φ : R+ → R+.
Random Gibbs measures can be obtained as measurable selections.
43601-7
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
Definition 3.3 A measurable map Jq ∋ J 7→ µ(J) ∈ P(RZ
d
) such that
∀J ∈ Jq : µ(J) ∈ Gp(J)
is called a measurable selection of the set-valued map Jq ∋ J 7→ Gp(J) ⊂ P(RZ
d
).
Theorem 3.4 Let p and q be as in Theorem 3.1 and the probability space (Jq,B(Jq), ν) be as in
Definition 2.2. Then the map Jq ∋ J 7→ Gp(J) has measurable selections.
It turns out that measurable selections constitute quite a big subset of the set of Gibbs measures,
c.f. item (vi) of Theorem 8.1.4 in [2], page 310.
Remark 3.5 There exists an at most countable family {µn}n∈N of measurable selections mentioned
in Theorem 3.4 such that, for every J ∈ Jq, the set {µn(J)}n∈N ⊂ Gp(J) is dense in Gp(J). Thus,
Gp(J) is a singleton for ν-almost all J if there is only one measurable selection.
As was already mentioned in Introduction, only limiting Gibbs measures can serve as the approxi-
mations of the Gibbs measures of large finite systems, see [7]. In the next theorem, we obtain random
Gibbs measures as weak limits of the averaged kernels πD,N . For a cofinal sequence D = {∆n}n∈N
and N ∈ N, we set
πD,N(·|J, ξ) =
1
N
N
∑
n=1
π∆n
(·|J, ξ). (3.4)
Theorem 3.6 For every ξ ∈ Sp, there exists a random Gibbs measure µξ and a cofinal sequence
D such that, in the topology of P(RZ
d
), one has µξ(J) = limN→+∞ πD,N(·|J, ξ) for ν-almost all
J ∈ Jq.
The fact that we use the sequences of averaged kernels to approximate the finite volume Gibbs mea-
sures rather than the sequences of kernels themselves can be explained by the chaotic dependence
of the kernels π∆(·|J, ξ) on ∆, which is smoothed up in (3.4).
For ∆ ⋐ Z
d, let d
ξ
∆
(J) denote the δ-measure centered at π∆(·|J, ξ), that is d
ξ
∆
(J)(A) =
IA (π∆(·|J, ξ)) for all A ∈ B. Then for a cofinal sequence D and N ∈ N, we set, c.f. (3.4),
d
ξ
D,N(J) =
1
N
N
∑
n=1
d
ξ
∆n
(J), (3.5)
which is the Newman-Stein empirical metastate, see equation (B19) in page 77 in [16] or equa-
tion (A18) in page 281 in [8]. Recall that the Aizenman-Wehr metastates were introduced in
Definition 2.4.
Theorem 3.7 For every ξ ∈ Sp, there exists an Aizenman-Wehr metastate mξ and a cofinal
sequence D such that, in the topology of P, mξ(J) = limN→+∞ d
ξ
D,N (J) for ν-almost all J ∈ Jq.
For ∆ ⋐ Z
d, J ∈ Jq, and ξ ∈ Sp, the (local) pressure in ∆ is
p∆(J, ξ) =
1
|∆|
logZ∆(J, ξ), (3.6)
where Z∆(J, ξ) is the same as in (2.9). Like in (3.4), for a cofinal sequence D = {∆n}n∈N and
N ∈ N, we consider
pD,N(J, ξ) =
1
N
N
∑
n=1
p∆n
(J, ξ). (3.7)
43601-8
Gibbs states of spin systems with unbounded disorder
Let now µ be a random Gibbs measure, see Definition 2.3. Then
p̄µ
∆
(J) =
∫
RZd
p∆(J, ξ)µ(J)(dξ),
p̄µD,N(J) =
∫
RZd
pD,N(J, ξ)µ(J)(dξ), (3.8)
are measurable functions of J ∈ Jq, and
ϑ(dσ, dJ) = µ(J)(dσ)ν(dJ) (3.9)
is a probability measure on the product space R
Z
d
× Jq.
Theorem 3.8 Suppose that ν has the property
sup
〈x,y〉∈E
∫
Jq
|Jxy|
qν(dJ) = aν < +∞. (3.10)
Then, for any random Gibbs measure µ, there exists a cofinal sequence D such that the sequence
{p̄µD,N(J)}N∈N converges, for ν-almost all J ∈ Jq, to a certain pµ ∈ L1(Jq, ν). Furthermore, for ν
obeying (3.10), let ϑ be as in (3.9). Then there exists a cofinal sequence D such that the sequence
{pD,N(J, ξ)}N∈N converges, for ϑ-almost all (ξ, J) ∈ R
Z
d
× Jq, to a certain p ∈ L1(RZ
d
× Jq, ϑ).
Imposing an additional condition on the measure ν we can strengthen the above result as follows.
A cofinal sequence D = {∆n}n∈N is called a van Hove sequence if
inf
n∈N
|∂∆n|
|∆n|
= lim
n→+∞
|∂∆n|
|∆n|
= 0,
see e.g. page 193 in [17]. Here ∂∆ = {y ∈ ∆c : ∃x ∈ ∆ x ∼ y}.
Theorem 3.9 In addition to (3.10), assume that ν is a product measure such that
∫
Jq
Jxyν(dJ) = 0, (3.11)
for all 〈x, y〉 ∈ E. Then, for any cofinal sequence D = {∆n}n∈N, there exists the quenched pressure
pquen = lim
n→+∞
∫
Jq
p∆n
(J, 0)ν(dJ) = sup
∆⋐Zd
∫
Jq
p∆(J, 0)ν(dJ), (3.12)
which thereby is independent of D. Furthermore, for any random Gibbs measure µ and any van
Hove sequence D = {∆n}n∈N, we have that
pquen = lim
n→+∞
∫
Jq
p̄µ
∆n
(J)ν(dJ). (3.13)
3.2. Comments
As was already mentioned, the model with the interaction as in (2.6), and with the single-
spin measures χ0(du) ∼ exp(−V (u))du, describes an anharmonic crystal with unbounded random
interactions. On the other hand, this is an irregular model, which has not been studied yet and it
seems a challenging task to develop its mathematical theory. We plan to include a random external
43601-9
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
field into consideration in a separate work, c.f. [11]. All the results presented above can be readily
extended to any bounded degree graph, and, after some modifications, to unbounded degree graphs
of a certain kind [18]. They can also be extended to more general pair interaction potentials Wxy,
c.f. (1.3). If every single-spin measure χx is supported on a bounded [a, b], then all the results
formulated above hold true with any q and p = 2q/(q − 1), including q = 1 and p = ∞. In this
case, we deal with regular random interactions2. An important particular model of this kind is the
Edwards-Anderson spin glass, see section 2 in [8]. In this model, the spins take values ±1 with equal
probabilities and the interaction intensities Jxy are symmetric, typically Gaussian, i.i.d.. Note that
such a model meets the conditions of Theorem 3.9.
More specific remarks to the above results are as follows:
• Theorem 3.1. The main point of this theorem is the lack of uniform boundedness of the
intensities Jxy, c.f. (3.1). Clearly, the growth of Jxy should be controlled in one or another
way. We do this by imposing the temperedness condition ‖J‖q < ∞, which appears in
the right-hand side of (3.2) and in similar estimates. The same results can be obtained for
the Euclidean Gibbs measures which describe equilibrium thermodynamic states of lattice
systems of interacting quantum anharmonic oscillators with random interactions. In this case,
our Theorem 3.1 would be an extension of Theorems 3.1 and 3.2 of [12] and of Theorems 3.3.1
and 3.3.6, p. 214–216 in [17]. For the Euclidean Gibbs measures, the single-spin spaces Sx
are the copies of the space of periodic continuous functions σx : [0, β] → R, where β > 0 is
the inverse temperature. In view of this, one needs to apply more sophisticated methods of
the path integral approach [17].
• Theorem 3.4. If J is random and fixed, the set Gp(J) describes the equilibrium thermody-
namic states of the spin system with quenched disorder. In order to average over the disorder,
one has to have the measurability as in Definition 2.3. In Theorem 3.1, we prove that Gp(J) is
non-void by showing that the family {π∆(·|J, ξ)}∆⋐Zd possesses accumulation points, which
are tempered Gibbs measures. Each measure of this kind is, therefore, obtained as the limit
of {π∆n
(·|J, ξ)}n∈N for the corresponding sequence {∆n}n∈N which, however, can depend on
J in an uncontrollable way (the so-called chaotic size dependence3). In view of this fact, it is
unclear whether these limiting points provide the measurability of J 7→ µ(J). In Theorem 3.4,
this measurability is obtained by means of general methods of the set-valued analysis. To the
best of our knowledge, this is the first instance of the use of such methods in the theory of
lattice models with random interactions.
• Theorems 3.6 and 3.7. These theorems give a constructive procedure of obtaining random
Gibbs measures as the infinite volume limits. Even for p = ∞ and q = 1, i.e. in the regular
case of bounded interactions, Theorems 3.6 and 3.7 are the corresponding extensions of
Theorems 6.2.6 and 6.2.8 in [1], p. 101–104. The novelty of these our theorems is that the
chaotic size dependence is harnessed with the help of the Komlós theorem [10] – a renowned
tool in the probability theory. This provides a new look at the approach put forward by
C. M. Newman and D. L. Stein, see [1, 7, 8, 16] and the references therein.
• Theorems 3.8 and 3.9. For the translation invariant lattice systems with nonrandom
interactions, the thermodynamic pressure exists and is unique even if the Gibbs measures
are multiple, see Theorem 3.10 and Corollary 3.11 in [12], and/or Theorems 5.1.2 and 5.1.3
in [17]. Thus, this is an important thermodynamic function by means of which one can
establish, e.g., the absence/existence of phase transitions, see [19] and/or chapter 6 in [17].
For disordered systems, the pressure in ∆ ⋐ Z
d clearly manifests the chaotic size dependence.
For the model considered here, we propose to eliminate this effect by passing to the averages
(3.7), as we did in Theorems 3.6 and 3.7. The existence of the limiting quenched pressure
obtained in (3.12) is a generalization to unbounded spins of the relevant result of [13, 14]. The
important point in Theorem 3.9 is that the pressure averaged over the disorder is the same
2See Definition 6.2.1 in [1].
3See the discussion in [8] and in [16], p. 55, 56, 64.
43601-10
Gibbs states of spin systems with unbounded disorder
in all states, which resembles the corresponding fact known for nonrandom interactions, see
Theorem 3.10 in [12] and Theorem 5.1.3 in [17], page 268. One observes that this result holds
true for the Edwards-Anderson spin glass as well. For the systems of quantum anharmonic
oscillators with the corresponding random interactions, the analogous statements can readily
be proven by means of a combination of the methods of [12, 17] and those of the present
work. This would be the extension of the results of [20].
Acknowledgements
The authors are grateful to Stas Molchanov and Michael Röckner for valuable discussions.
References
1. Bovier A., Statistical Mechanics of Disordered Systems. A Mathematical Perspective. Cambridge Series
in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006.
2. Aubin J.-P., Frankowska H., Set-valued Analysis. Reprint of the 1990 edition. Modern Birkhäuser
Classics. Birkhäuser Boston, Inc., Boston, MA, 2009.
3. Lebowitz J.L., Presutti E., Comm. Math. Phys., 1976, 50, 195.
4. Ruelle D., Comm. Math. Phys., 1976, 50, 189.
5. Pasurek T., Theory of Gibbs Measures with Unbounded Spins: Probabilistic and Analytic Aspects.
Habilitation Thesis, Universität Bielefeld, available as SFB 701 Preprint 08–101, 2008.
6. Georgii H.-O., Gibbs Measures and Phase Transitions. de Gruyter, New York, 1988.
7. Newman C.M., Stein D.L. Local vs. global variables for spin glasses. – In: Spin glasses, eds.
E. Bolthausen and A. Bovier, p. 145–158, Lecture Notes in Math., 1900, Springer, Berlin, 2007.
8. Newman C.M., Stein D.L., Thermodynamic chaos and the structure of short-range spin glasses. In –
Mathematical aspects of spin glasses and neural networks, eds. A. Bovier and P. Picco, p. 243–287,
Progr. Probab., 41, Birkhäuser Boston, Boston, MA, 1998.
9. Balder E.J., Probab. Th. Rel. Fields, 1989, 81, 185.
10. Komlós J. Acta Math. Acad. Sci. Hungar., 1967, 18, 217.
11. Aizenman M., Wehr J. Comm. Math. Phys., 1990, 130, 489.
12. Kozitsky Y., Pasurek T., J. Stat. Phys., 2007, 127, 985.
13. Contucci P., Lebowitz J., Ann. Henri Poincaré, 2007, 8, 1461.
14. Contucci P., Starr S., J. Stat. Phys., 2009, 135, 1159.
15. Parthasarathy K.R., Probability Measures on Metric Spaces. Probability and Mathematical Statistics,
No. 3. Academic Press Inc., New York-London, 1967.
16. Newman C.M., Topics in Disordered Systems. Lectures in Mathematics ETH Zürich. Birkhäuser Ver-
lag, Basel, 1997.
17. Albeverio S., Kondratiev Y., Kozitsky Y., Röckner M. The Statistical Mechanics of Quantum Lattice
Systems. A Path Integral Approach. EMS Tracts in Mathematics, 8. European Mathematical Society
(EMS), Zürich, 2009.
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43601-11
Yu. Kondratiev, Yu. Kozitsky, T. Pasurek
Стани Ґiббса ґраткових спiнових систем з необмеженим
безладом
Ю. Кондратьєв1, Ю. Козицький2, Т. Пазурек1
1 Факультет математики, Унiверситет Бiлефельд, D–33615, Нiмеччина
2 Iнститут математики, Унiверситет Марiї Кюрi-Склодовської, 20–031 Люблiн, Польща
Дослiджуються стани Ґiббса спiнової системи на ґратцi Zd з парною взаємодiєю Jxyσ(x)σ(y). Тут
〈x, y〉 ∈ E, тобто x та y є сусiдами в Zd. Iнтенсивностi Jxy i спiни σ(x), σ(y) приймають довiльнi
дiйснi значення. Щоб контролювати їх рiст, ми вводимо вiдповiднi множини Jq ⊂ RE i Sp ⊂ RZ
d
та
показуємо, що для кожного J = (Jxy) ∈ Jq : (a) множина cтанiв Ґiббса Gp(J) = {µ : ′ , µ(Sp) = 1}
є непорожньою i компактною в слабкому сенсi; (b) кожна µ ∈ Gp(J) задовiльняє iнтеґральнiй оцiнцi,
однаковiй для всiх µ. Далi ми дослiджуємо випадок, коли на Jq вводиться норма, борелевське σ-
тiло B(Jq) та повна ймовiрнiсна мiра ν. Ми показуємо, що мультивiдображення Jq ∋ J 7→ Gp(J)
має вимiрнi селектори Jq ∋ J 7→ µ(J) ∈ Gp(J) – суть випадкової мiри Ґiббса. Ми показуємо,
що емпiричнi розклади N−1
∑N
n=1 π∆n
(·|J, ξ), одержанi з локальних умовних мiр Ґiббса π∆n
(·|J, ξ)
та з вичерпуючих послiдовностей ∆n ⊂ Z
d, мають ν- майже певнi границi при N → +∞ –
суть випадкової мiри Ґiббса. Подiбним чином ми показуємо iснування ν- майже певних границь
емпiричних метастанiв N−1
∑N
n=1 δπ∆n
(·|J,ξ), якi є метастанами Айзенмана-Вера. Накiнець, ми
показуємо iснування граничного термодинамiчного тиску за певних додаткових умов на ν.
Ключовi слова: метастан Айзенмана-Вера, емпiричний метастан Ньюмена-Штайна, хаотична
просторова залежнiсть, теорема Комлоша, спiнове скло
43601-12
Introduction
Setup
General setting
The Gibbs fields
The results
Theorems
Comments
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