On thermodynamic states of the Ising model on scale-free graphs
There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent < k²> studied in, e.g., S. N. Dorogovtsev et al, Rev. Mod. Phys., 80, 1275 (2008). It is shown that the Ising model on the proposed graphs with interaction i...
Gespeichert in:
| Datum: | 2013 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
2013
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/120801 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On thermodynamic states of the Ising model on scale-free graphs / Yu. Kozitsky // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23001:1-12. — Бібліогр.: 23 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120801 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1208012017-06-14T03:02:39Z On thermodynamic states of the Ising model on scale-free graphs Kozitsky, Yu. There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent < k²> studied in, e.g., S. N. Dorogovtsev et al, Rev. Mod. Phys., 80, 1275 (2008). It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the percolation probability. These results and their possible extensions are also discussed. Запропоновано модель безмасштабних випадкових ґрафiв, яка є локально близькою до нескорельованих випадкових мереж з розбiжними 〈k²〉, що вивчалася, наприклад, в статтi Dorogovtsev S.N. et al., Rev. Mod. Phys., 2008, 80, 1275. Показано, що модель Iзинґа на запропонованих ґрафах з iнтенсивностями взаємодiї довiльного знаку iз ймовiрнiстю один є в парамагнiтному станi для достатньо високих скiнчених значень температури. Для таких ґрафiв модель перколяцiї по ребрах з ймовiрнiстю один є в неперколятивному станi для позитивних значень перколяцiйної ймовiрностi. Обговорюються цi результати i можливостi їх розширення. 2013 Article On thermodynamic states of the Ising model on scale-free graphs / Yu. Kozitsky // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23001:1-12. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 05.70.Fh, 05.50.+q, 02.50.Ga DOI:10.5488/CMP.16.23001 http://dspace.nbuv.gov.ua/handle/123456789/120801 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent < k²> studied in, e.g., S. N. Dorogovtsev et al, Rev. Mod. Phys., 80, 1275 (2008). It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the percolation probability. These results and their possible extensions are also discussed. |
| format |
Article |
| author |
Kozitsky, Yu. |
| spellingShingle |
Kozitsky, Yu. On thermodynamic states of the Ising model on scale-free graphs Condensed Matter Physics |
| author_facet |
Kozitsky, Yu. |
| author_sort |
Kozitsky, Yu. |
| title |
On thermodynamic states of the Ising model on scale-free graphs |
| title_short |
On thermodynamic states of the Ising model on scale-free graphs |
| title_full |
On thermodynamic states of the Ising model on scale-free graphs |
| title_fullStr |
On thermodynamic states of the Ising model on scale-free graphs |
| title_full_unstemmed |
On thermodynamic states of the Ising model on scale-free graphs |
| title_sort |
on thermodynamic states of the ising model on scale-free graphs |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120801 |
| citation_txt |
On thermodynamic states of the Ising model on scale-free graphs / Yu. Kozitsky // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23001:1-12. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kozitskyyu onthermodynamicstatesoftheisingmodelonscalefreegraphs |
| first_indexed |
2025-07-08T18:35:56Z |
| last_indexed |
2025-07-08T18:35:56Z |
| _version_ |
1837104898170159104 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 2, 23001: 1–12
DOI: 10.5488/CMP.16.23001
http://www.icmp.lviv.ua/journal
On thermodynamic states of the Ising model
on scale-free graphs∗
Yu. Kozitsky
Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland
Received October 11, 2012, in final form November 2, 2012
There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex
random networks with divergent 〈k2〉 studied in, e.g., Dorogovtsev S.N. et al., Rev. Mod. Phys., 2008, 80, 1275.
It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with
probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same
graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the
percolation probability. These results and their possible extensions are also discussed.
Key words: random graphs, paramagnetic phase, percolation, branching process
PACS: 05.70.Fh, 05.50.+q, 02.50.Ga
1. Introduction
In statistical physics, a paradigm model for cooperative phenomena is the Ising model of interacting
spins attached to the vertices of a graph G= (V,E). This model is described by the Hamiltonian
H =−
∑
{x,y}∈E
Jx yσxσy −h
∑
x∈V
σx , (1.1)
where Jx y are interaction intensities, h is an external field, and the spins σx take values ±1. The sums
run over the sets of edges E and vertices V of the graph G, respectively. The graph is supposed to be
connected and countably infinite. In view of the latter fact, the Hamiltonian (1.1) has no direct mathe-
matical meaning and is used as a formula for local Hamiltonians, HΛ, which one obtains by restricting
the sums to finite subsets Λ⊂V and EΛ ⊂E (EΛ being the collection of edges with both ends in Λ). Then,
the thermodynamic properties of (1.1) are studied in the limit Λ→ V. In a more sophisticated theory [1],
a thermodynamic state is a Gibbs measure, and a phase transition is associated with the possibility for
multiple thermodynamic states to exist at the same values of the temperature and h.
Obviously, the properties of the model in (1.1) are closely related to those of the underlying graph.
The case of a special interest is where this graph is random, with vertices having an arbitrary number
of neighbors. Then, the graph itself can be characterized by the properties of the corresponding Ising
model. Perhaps, In view of this fact methods of statistical physics are widely used also in the study of the
underlying random graphs. A long list of publications regarding this topic can be found in the review
articles [2, 3].
For a vertex x, the degree k(x) is defined as the number of neighbors of x, i.e., the number of y ∈ V
such that {x, y} ∈ E. If G is random, the degree of each x ∈ V is an integer valued random variable and
hence is characterized by the probability distribution
Prob{k(x) = k} = pk (x), k Ê 1, x ∈V. (1.2)
∗The paper is dedicated to Mykhailo Kozlovskii on the occasion of his 60-th birthday.
© Yu. Kozitsky, 2013 23001-1
http://dx.doi.org/10.5488/CMP.16.23001
http://www.icmp.lviv.ua/journal
Yu. Kozitsky
In simple cases, the degrees k(x) are independent and identically distributed, which particularly means
that the probabilities pk (x) are the same for all x. If pk =Ck−λ, at least for big enough k , the correspond-
ing graph is said to be scale-free, cf. [4]. In this case, the m-th moment
〈km
〉 =
∑
kÊ1
km pk , m Ê 1
exists provided λ> m +1. Notably, 〈k2〉 =+∞ for many real-world networks [3].
An example of a random graph with independent k(x) is a Galton-Watson tree, which is a genealogic
tree of a Galton-Watson branching process [5, 6]. In more complex models, the graph G is obtained, more
or less explicitly, as the limit of an increasing sequence of finite random graphs, (Vn ,En), n ∈N, see, e.g.,
[7–10]. In the physical literature, such constructions are rather informal which leaves certain possibilities
for different interpretations. In a sense, the aim of this paper is to show that the lack of precision in
defining the underlying random graph may yield different conclusions on the thermodynamic states of
the corresponding Ising model. In 2002, almost simultaneously there appeared two papers [11, 12] where
the model (1.1) on a scale-free graph was studied. The starting point is a scale-free distribution {pk }k∈N,
which “characterizes the entire net”, whereas the “local structure of connections” is characterized by
“the distribution of the number of connections of the nearest neighbor of a vertex”, which is taken to be
kpk /〈k〉, see the beginning of the first section in [11]. In [12], a similar assumptionwasmade, see equation
(3) therein, as well as (1) in [2]. In the latter article, such graphs are referred to as complex networks in
contrast to the classical models — simple networks, one of which is presumably the Galton-Watson tree
mentioned above. In both papers [11, 12], it was found that the model (1.1) with h = 0, ferromagnetic
interactions Jx y = J > 0, and 〈k2〉 = +∞ is in an ordered (ferromagnetic) state at all temperatures1 T > 0
. Recently, in [9, 10] a mathematical construction of a random graph model was presented, in which the
mentioned properties were realized. Then, it has become clear that the conclusions of [11, 12] repeat, in
a sense, the results of [14], see subsection 5.2 below. In this work, we construct “uncorrelated complex
networks” with 〈k2〉 =∞, consistent with the assumptions made in [2, 11, 12] and mentioned above, for
which the model (1.1) with interaction intensities of arbitrary signs can be in a paramagnetic state at
finite temperatures. Note that the graph model which we propose here is only locally mathematically
similar to that of [11, 12]. This means that our result does not contradict the conclusions of the latter
papers because it applies to a different model. A more detailed discussion regarding this subject can be
found in section 5 below.
In section 2, we construct the graphmodel which is a combination of the one used in [9, 10] and of the
ordinaryGalton-Watson tree. In section 3, we prove theorem 3.1 themain result of which is that themodel
(1.1) on the introduced graphs, with any signs of Jx y , can be in a paramagnetic state at finite temperatures
and h = 0, provided 〈k1+α〉 <∞ for some α > 0. This covers the case of scale-free distributions with λ ∈
(2,3). The proof of theorem 3.1 relies upon two lemmas, proven in section 4. The result of theorem 3.1 and
the graph model introduced in section 2 are discussed in the concluding section 5, where we also present
an extension of the results of [11, 12] concerning the model (1.1) with interaction intensities of arbitrary
signs. Except possibly for section 4, the paper is self-contained and accessible to non-mathematicians.
2. The graph models
Since the Galton-Watson tree will serve us as a prototype, we begin by presenting this model. More
on this topic can be found in [5, 6, 15].
2.1. The Galton-Watson tree
To exclude the possibility of obtaining a finite tree or a tree of a bounded degree, we suppose that the
underlying random variable X takes positive integer values k ∈Nwith probability
Prob(X = k) = pk+1 ∈ [0,1)
1In contrast to regular graphs, e.g., crystal lattices, the Isingmodel on sufficiently ‘dense’non-randomgraphs can be in an ordered
state at all temperatures, see, e.g., the example in [13].
23001-2
On thermodynamic states of the Ising model
such that
∑
kÊ2
pk = 1, a :=
∑
kÊ2
kpk <∞,
n∑
k=2
pk < 1, for any n ∈N. (2.1)
One observes that
a > 2, (2.2)
which readily follows from the properties assumed in (2.1). Set L0 = 1, L1 = X , and
Ln =
Ln−1∑
j=1
Xn, j , (2.3)
where all Xn, j are independent copies of X , see chapters II and III in [5] or chapter 5 in [6]. Then, Ln is
the number of individuals in generation n of a Galton-Watson branching process. The process starts with
a single individual, and each individual in generation n produces offsprings at random, independently
and with the same distribution as X . The individuals existing in all generations form the vertex set V
of the corresponding (genealogic) Galton-Watson graph G, with undirected edges {x, y} ∈ E connecting
each individual and its offsprings. Under the assumptions made in (2.1), the corresponding process is not
extinct and Ln → +∞ with probability one. Hence, G is an infinite tree — an acyclic connected graph,
which has no leaves, except possibly for the root o. By construction, the vertex degrees are independent
identically distributed random variables such that
Prob{k(o) = k} = pk+1, Prob{k(x) = k} = pk , x ∈V \ {o}. (2.4)
Now let us fix the graph-theoretical and probabilistic terminology used in this article. For a graph G =
(V,E) and vertices x, y ∈ V, a path ϑ(x, y) is a sequence {x0, x1, . . . , xn }, n ∈N, such that: (a) x0 = x, xn = y ;
(b) {xl , xl+1} ∈ E; (c) neither of x0, . . . , xn−1 can be repeated. The length of the path is ‖ϑ(x, y)‖ = n;
x is its origin and y is terminus. A ray with the origin x0 is an infinite sequence of distinct vertices
{x0, x1, . . . , xn , . . . } with the property (b) as above. A graph ensemble G is the collection of all possible
realizations as just described. A random graph model is a triple (G ,F ,P ), where F is a certain family of
subsets of G , and P : F → [0,1] is the probability. For the Galton-Watson tree, P is defined by the collec-
tion p = {pk }. For instance, if A is the set of all realizations with k(o) = 1, then P (A ) = p1. In general,
for A ∈ F , P (A ) is the probability of the event: “the graph chosen at random from G belongs to A ”.
Events of a certain sort can occur with probability either zero or one — the celebrated zero-one law. We
say that a property of a random graph holds with probability one if this property is possessed by all of
the elements of some A such that P (A ) = 1. Usually, when one deals with the models with independent
vertex degrees, the triple (G ,F ,P ) does not appear explicitly. This will also be the case in the rest of the
paper.
2.2. The configuration model
Let p := {pk }kÊ2 be as in (2.1), and
p̂k := kpk /a, k Ê 2. (2.5)
“Uncorrelated complex networks” were ‘defined’ in [2, 11, 12] by means of the following conditions. The
main one is that “the neighbor of a vertex has a degree k with probability p̂k”, which can be interpreted as
the condition that in every path {x0, x1, . . . , xn }, the probability that k(xl ) takes value k is p̂k . We shall call
it a path property. Other important assumptions are: (a) all k(x) are “uncorrelated” (in fact, independent);
(b) two paths {x, x1, . . . , y} and {x, x′
1 , . . . , z}, with x1 , x′
1 never intersect each other, i.e., the graph is a
tree2. These assumptions are consistent with the following mathematical models. The first one is the
ordinary Galton-Watson tree with p replaced by p̂, in [9] it is T(ρ,∞), see page 567 thereof. A slightly
2In [2, 11], the graphs were called “tree-like” if the intersections are negligible. Mathematical definitions of tree-like graphs can
be found, e.g., in [9, 10, 14].
23001-3
Yu. Kozitsky
different version is the so-called configuration model, cf. page 1279 in [2]. This model is a rooted tree with
independent vertex degrees such that
Prob(k(o) = k) = pk , Prob(k(x) = k) = p̂k . (2.6)
This model3 was studied in [9, 10], in [9] it appears as T(P,ρ,∞). The main characteristic feature of both
T(P,ρ,∞) and T(ρ,∞) is that each path ϑ(x, y), x , o, has the path property described above. The in-
consistency with the root degrees does not change the global behavior of the graph, and hence of the
corresponding model (1.1).
The construction of the configuration model can be visualized as follows. Consider a countable set
of ‘points’ (a configuration), each of which is given a random mark4 k ∈ N with probability distribution
p as in (2.1). The marks of different points are independent. Then, the graph is obtained as follows. The
root degree is drawn from N according to Prob(k(o) = k) = pk . If the result is m, the root receives m
neighbors— ‘particles’ independently drawn from the configuration with probability proportional to the
particle mark, i.e., Ck. Then, the (unconditional) probability of drawing some particle, which bears mark
k , is Ckpk with C = 1/a since the total probability should sum up to one. As a result, each root neighbor
has a degree k with probability p̂k . In the same way one picks the next-neighbors of o. This procedure
is continued ad infinitum until the whole tree is constructed. Notably, except for the root degree, the
distribution of k(x) is given by p̂, whereas p itself is the distribution of marks, i.e., auxiliary objects used
in the graph construction.
2.3. The proposed model
Our model is a random tree, also consistent with the assumptions of [2, 11, 12]. As in the case of the
Galton-Watson tree, we begin with the description of the corresponding branching process. Let p , p̂ , and
a be as in (2.1) and (2.5). An additional parameter is s ∈N. The whole population falls into two types —
‘distinguished’ and ‘ordinary’ individuals. At the beginning, there are no ordinary individuals and there is
one distinguished individual, who independently produces m offsprings with probability p̂m . If m É s, all
of them are set to be distinguished. For m > s, s the offsprings are distinguished, and the remaining m− s
ones are set to be ordinary. Each ordinary individual independently produces k ordinary individuals
with probability pk+1. Each distinguished individual, other than the initial one, independently produces
m offsprings with probability p̂m+1. As above, if m É s, then all of the offsprings are distinguished. For
m > s, s offsprings are distinguished and the remaining m− s ones are ordinary. This process is repeated
ad infinitum. For n ∈N0, by L̂n and Ln we denote the number of distinguished and ordinary individuals
in generation n, respectively. Let also X̂ and X be independent and such that Prob(X̂ = m) = p̂m+1 and
Prob(X = m) = pm+1, m ∈N. Then, we have, cf. (2.3),
L̂n =
L̂n−1∑
j=1
[
X̂n, j −
(
X̂n, j − s
)
+
]
, Ln = Yn +
Ln−1∑
j=1
Xn, j , (2.7)
where
Yn =
L̂n−1∑
j=1
(
X̂n, j − s
)
+
, (2.8)
and for an integer κ, we write κ+ = max{0;κ}, whereas X̂n, j and Xn, j are independent copies of X̂ and X ,
respectively. From the point of view of ordinary particles, those represented by the first term in the ex-
pression for Ln are immigrants. Then, the total number of individuals in the generation n can be written
in the form
L̃n := L̂n +Ln =
L̂n−1∑
j=1
X̂n, j +
Ln−1∑
j=1
Xn, j . (2.9)
Since each distinguished individual can have at most s distinguished offsprings, we have that
L̂n É sn . (2.10)
3Also models which locally converge to T(P,ρ,∞), cf. Definition 2.1 in [9].
4In [2], a mark is the number of stubs.
23001-4
On thermodynamic states of the Ising model
The graph in question is the genealogic tree of the process just described. Its construction can be visual-
ized as follows. The graph is a rooted tree with Prob(k(o) = k) = p̂k+1. Let k(o) = m É s. Then, we draw
m neighbors of o as in the configuration model. Denote them x1, . . . , xm ; by construction Prob(k(xl ) =
k) = p̂k for all l = 1, . . . m. If m > s, we draw s neighbors x1, . . . , xs of o as in the configuration model. The
remaining neighbors ys+1, . . . , ym are set to be the roots of independent Galton-Watson trees with proba-
bility distribution p , which in particular means that Prob(k(yl ) = k) = pk , l = s +1, . . . ,m. This procedure
is continued ad infinitum. For s > 1, the obtained tree contains a subtree, comprised by an infinite num-
ber of distinguished rays {o, x1, x2, . . . , xn , . . . } with the mentioned path property “the neighbor of xn−1 has
a degree k with probability p̂k”. The set of vertices of this subtree is contained in that of a Cayley tree
with root o and branching number s. For s = 1, our graph is a size-biased Galton-Watson tree, see [6, pp.
407–412] or [15], in which there is only one distinguished ray.
In the remaining part of the article we use the following nomenclature. By GW(s, p), s ∈ N0, we de-
note the model constructed in this subsection. Then, GW(1, p) and GW(0, p) are the size-biased and the
ordinary Galton-Watson trees, respectively.
3. The thermodynamic states of the Ising model
3.1. The thermodynamic states
Let G = (V,E) be a general tree with root o. Given x ∈ V, the distance ρ(o, x) is the length of the path
ϑ(o, x). For n ∈N0, by Sn we denote the set of vertices x such that ρ(o, x) = n. Put
Vn =
n⋃
m=0
Sm ,
and let En be the set of all edges with both ends in Vn. Let also Eb
n be the set of {x, y} such that x ∈Sn and
y ∈Sn+1.
In the sequel, we assume that
J := sup
{x,y}∈E
|Jx y | <∞. (3.1)
By σn we denote the configuration of spins in Vn , that is, σn = {σx : x ∈Vn}. The Gibbs probability distri-
bution of configurations σn at temperature T and h = 0 is the following probability measure
πn(σn |ξ) =
1
Zn(ξ)
exp
∑
{x,y}∈En
(Kx yσxσy +K )+
∑
{x,y}∈Eb
n
Kx yσxξy
, (3.2)
where
Kx y :=βJx y , K :=βJ , β := 1/kBT . (3.3)
Note that Kx y can have arbitrary signs, whereas K is positive. In (3.2), the first (respectively the second)
summand in exp(· · · ) corresponds to the interaction of the spins in Vn with each other (respectively with
the spins ξy fixed outside Vn). One observes that the latter interaction involves only ξy with y ∈Sn+1. For
technical reasons, we add positive constants to the spin-spin interaction along each edge in En , cf. (1.1).
The partition function is then
Zn(ξ) =
∑
σn
exp
∑
{x,y}∈En
(Kx yσxσy +K )+
∑
{x,y}∈Eb
n
Kx yσxξy
, (3.4)
where the summation is taken over all σx =±1, x ∈Vn .
Given z ∈V, let nz ∈N0 be such that z ∈Snz , i.e., nz = ρ(o, z). For these z, nz , and for n > nz ,
Mn,z (ξ)=
∑
σn
σzπn(σn |ξ) =
∑
σz=±1
σz̺n,z (σz |ξ) = ̺n,z (1|ξ)−̺n,z (−1|ξ) (3.5)
23001-5
Yu. Kozitsky
is the magnetization at z ∈Vn in state πn(·|ξ) (3.2). Here, for σz =α,
̺n,z (α|ξ) =
∑
σn : σz=α
πn(σn |ξ), α=±1, (3.6)
where the summation is taken over all σn with the fixed σz =α. According to the theory of Gibbs states
[1], the global thermodynamic state is unique if, for all z, the limits
̺z (α) := lim
n→+∞
̺n,z (α|ξ) (3.7)
exist and are independent of ξ. In this case, by (3.5) the global magnetization Mz exists and
Mz = lim
n→+∞
Mn,z (ξ) = ̺z (1)−̺z (−1).
As h = 0, we have Mn,z (ξ) = −Mn,z (−ξ), and hence Mz = −Mz . Therefore, the uniqueness of thermody-
namic states occurs if and only if, for all z ∈V and ξ,
Mz = lim
n→+∞
Mn,z (ξ) = 0. (3.8)
In this case, we say that the model is in a paramagnetic state.
If the underlying tree G is random, then the measure (3.2) is also random, and hence (3.8) is a random
event obeying the zero-one law. If Prob(Mz = 0) = 1, we say that the model (1.1) on G is in a paramagnetic
state with probability one.
3.2. The main statement
Let p and p̂ be as in (2.1) and (2.5). For α ∈ (0,1), we set
bα =
∑
kÊs+1
(k − s)αp̂k , b =
∑
kÊ2
kpk ln k, (3.9)
and also
Kc =
1
4
ln
a
a −1
, K̂c =
1
4
ln
sγ+1
sγ
, γ= 1/α. (3.10)
In the statement below we describe the Ising model (1.1) on GW(s, p) graphs with p obeying (2.1).
Theorem 3.1 For s = 0, the Ising model is in a paramagnetic state whenever K < Kc. For s = 1, the same
holds under the condition b <∞. For s Ê 2, the Ising model is in a paramagnetic state whenever bα <∞ for
some α ∈ (0,1) and K < min{Kc; K̂c }.
In section 5, we discuss the above statement in detail. Its proof is based on two lemmas proven in the next
section. In the first lemma, we describe the model (1.1) on a general tree, for which the corresponding
quantities were introduced in subsection 3.1.
Lemma 3.2 Let K > 0 be such that
q(K ) := exp(4K )−1< 1. (3.11)
Then, for each z ∈V, any n > nz , and arbitrary ξ and η, we have that
∣∣Mn,z (ξ)−Mn,z (η)
∣∣É 2[q(K )]n−nz |Sn |. (3.12)
In the next lemma, we describe the model GW(s, p) introduced in subsection 2.3.
Lemma 3.3 Let Ln be as in (2.7), s Ê 2, and bα < ∞ for some α ∈ (0,1). Then, for every c obeying the
conditions c > sγ and c Ê a −1, with probability one
c−nLn →W ∈ [0,∞), as n →∞.
23001-6
On thermodynamic states of the Ising model
Proof of theorem 3.1: For s = 0, by construction |Sn | = Ln is the number of individuals in generation n, see
(2.3), and a −1 is the mean number of offsprings in the corresponding Galton-Watson process. Then, the
limit
lim
n→+∞
Ln
(a −1)n
(3.13)
with probability one exists and is finite, see (2.2) and proposition 1.3 in [5, page 20]. If K < Kc, then
q(K )< q(Kc) = 1/(a −1) < 1, and hence we can apply (3.12), which yields that with probability one
[
q(K )(a −1)
]n Ln
(a −1)n
→ 0, n →+∞, (3.14)
which in turn by (3.12) yields (3.8), and hence the proof for this case.
For s = 1, by (2.10) we have L̂n = 1, and hence |Sn | = L̃n = 1+Ln . Since b <∞, with probability one
we have ∑
nÊ1
(a −1)−nYn <∞, (3.15)
see proposition 6.2 in [5, page 50]. Then, by theorem 6.1 in [5, page 50], L̃n /(a −1)n with probability one
tends to a finite limit. Then, the proof follows as in the case of s = 0.
For s Ê 2, the random variables Yn with different n are no more identically distributed, and hence
b < ∞ is not sufficient to get (3.15). Instead, we use a more restrictive condition bα < ∞, under which
we get, see lemma 3.3, that c−n |Sn | = c−n L̃n = c−n L̂n + c−nLn with probability one tends to a finite limit
since c > s, cf. (2.10). Note that
K (c) :=
1
4
ln
c +1
c
É min{Kc; K̂c }.
Hence, for K < K (c), we have q(K )< 1/c, and the proof follows as in (3.14).
4. The proof of the lemmas
4.1. The proof of lemma 3.2
Let G= (V,E) be a general tree. For a path ϑ, by Eϑ we denote the set of edges {xl−1, xl } of ϑ. Since G is
a tree, there exists exactly one path ϑ(x, y) for any x and y . Recall that En stands for the set of edges with
both ends in Vn . Thus, Gn = (Vn ,En) is a finite graph; G′ = (V′,E′) is called a subgraph of Gn if V′ ⊂ Vn
and E′ ⊂En . A path ϑ is said to be in G′ if Eϑ ⊂E′. Two vertices z, x ∈V′ are said to be disconnected in G′
if there is no path ϑ(z, x) in G′.
By (3.5) and (3.2), employing replica spins σ̃we obtain the following Meyer-like expansion
Mn,z (ξ)−Mn,z (η) =
∑
σn ,σ̃n
(σz − σ̃z )πn(σn |ξ)πn(σ̃n |η)
=
1
Zn(ξ)Zn(η)
∑
σn ,σ̃n
(σz − σ̃z )
∏
{x,y}∈En
(1+Γx y )Ψn(ξ,η)
=
1
Zn(ξ)Zn(η)
∑
E′⊂En
∑
σn ,σ̃n
(σz − σ̃z )Γ(E′)Ψn(ξ,η), (4.1)
where
Γ(E′) =
∏
{x,y}∈E′
Γx y ,
Γx y = exp
[(
Kx yσxσy +K
)
+
(
Kx y σ̃x σ̃y +K
)]
−1,
Ψn(ξ,η) =
∏
{x,y}∈Eb
n
exp
[
Kx y
(
σxξy + σ̃xηy
)]
. (4.2)
23001-7
Yu. Kozitsky
For simplicity, we do not indicate the dependence of the latter functions on the spins. One observes that
each Γx y Ê 0 due to our choice of the spin-spin interactions in (3.2), cf. (3.1), and that Ψn(ξ,η) depends
only on σx and σ̃x with x ∈Sn . Fix some E′ in the last line in (4.1) and consider the subgraph G′ with the
edge set E′ and the vertex set Vn . If in G′, the vertex z is disconnected from each x ∈ Sn , then the sums
over the spins σz ,σ̃z and over σx ,σ̃x with x ∈ Sn get independent and hence the left-hand side of (4.1)
vanishes as the term (σz − σ̃z ) is antisymmetric with respect to the interchange σ↔ σ̃, whereas all Γx y
are symmetric and the only break of this symmetry is related to the fixed boundary spins ξ(y) and η(y),
y ∈Sn+1. Therefore, each non-vanishing term in (4.1) corresponds to a path ϑ(z, x) connecting z to some
x ∈ Sn . Let us take this into account and rewrite (4.1) as the sum over the subsets of En containing the
edges of at least one such path. Let Θn(z) be the set of all paths connecting z to Sn , and for ϑ ∈Θn(z), let
Eϑ be the family of subsets of En each of which contains Eϑ. That is, Eϑ = {E′ ⊂ En : Eϑ ⊂ E′}. Note that,
for distinct ϑ,ϑ′ ∈Θn(z), the corresponding families Eϑ and Eϑ′ are not disjoint — they include those E′
which contain both Eϑ and Eϑ′ . Finally, set
E =
⋃
ϑ∈Θn (z)
Eϑ , (4.3)
that is, E contains all sets of edges E′ ⊂ En such that the corresponding graph G′ contains at least one
path connecting z to some x ∈Sn. Then, (4.1) takes the form
Mn,z (ξ)−Mn,z (η) =
1
Zn(ξ)Zn(η)
∑
E′∈E
∑
σn ,σ̃n
(σz − σ̃z )Γ(E′)Ψn(ξ,η). (4.4)
In view of the positivity of all Γx y Ê 0, this yields
∣∣Mn,z (ξ)−Mn,z (η)
∣∣ É
2
Zn(ξ)Zn(η)
∑
E′∈E
∑
σn ,σ̃n
Γ(E′)Ψn(ξ,η)
É
2
Zn(ξ)Zn(η)
∑
σn ,σ̃n
∑
ϑ∈Θn (z)
Γ(Eϑ)
∑
E′∈Eϑ
Γ(E′ \ Eϑ)Ψn(ξ,η). (4.5)
Note that the sum
∑
ϑ∈Θn (z)
∑
E′∈Eϑ
contains the same summands as
∑
E′∈E , see (4.3), but a part of them
are repeated, which yields the second É in (4.5). From (4.2) and (3.11) we see that 0 É Γx y É q(K ), for any
x, y ∈V. Hence,
Γ(Eϑ) É [q(K )]‖ϑ‖,
which yields in (4.5)
∣∣Mn,z (ξ)−Mn,z (η)
∣∣ É
2
Zn(ξ)Zn(η)
∑
ϑ∈Θn (z)
[q(K )]‖ϑ‖
×
∑
σn ,σ̃n
∏
{x,y}∈Eϑ
(1+Γx y )
∑
E′′⊂En\Eϑ
Γ(E′′)Ψn(ξ,η)
= 2
∑
ϑ∈Θn (z)
[q(K )]‖ϑ‖. (4.6)
Here, we have taken into account that, see (4.1),
∑
σn ,σ̃n
∏
{x,y}∈Eϑ
(1+Γx y )
∑
E′′⊂En\Eϑ
Γ(E′′)Ψn(ξ,η) =
∑
σn ,σ̃n
∏
{x,y}∈En
(1+Γx y )Ψn(ξ,η) = Zn(ξ)Zn(η).
Note that a similar construction was used in deriving (2.16) and (2.17) in [16], see also lemma 3.1 in [13].
Since nz is defined by the condition z ∈ Snz , the shortest path in Θn(z) has length n −nz . Furthermore,
for every x ∈ Sn , there exists exactly one path ϑ ∈ Θn(z). Then, for q(K ) É 1, the estimate (3.12) follows
from (4.6).
23001-8
On thermodynamic states of the Ising model
4.2. The proof of lemma 3.3
If we prove that, for c > sγ, with probability one
∑
nÊ1
c−n Yn <∞, (4.7)
see (2.8), then the proof that Ln /cn → W ∈ [0,+∞), with probability one, follows as in the proof of theo-
rem 6.1 in [5, page 50]. The first summand of Ln in (2.7) is under control in view of the assumed property
c Ê a −1.
As α ∈ (0,1), by the standard Minkowski inequality, cf. [17, theorem 8, page 319], as well as by (2.10)
and (3.9), we have
〈Y α
n 〉 É
sn−1∑
j=1
〈(
X̂ j − s
)α
+
〉
= sn−1bα .
Then, for any q > cγ, by Markov’s inequality, cf. [17, page 311],
Pn := Prob
(
q−nYn > 1
)
= Prob
(
Y α
n > qαn
)
É
bα
s
(
s
qα
)n
. (4.8)
On the other hand, for c > sγ, we take q ∈ (sγ,c) and rewrite
∑
nÊ1
c−n Yn =
∑
nÊ1
( q
c
)n (
q−nYn
)
. (4.9)
For such q , by (4.8) we have that
∑
nÊ1 Pn <∞. Thus, the Borel-Cantelli lemma, cf. [17, page 320], yields
that with probability one, only finitely many of the events q−nYn > 1 may occur, which by (4.9) readily
yields (4.7).
5. Concluding comments and remarks
5.1. The graph model
Let us look at the structure of the generation n in GW(s, p), cf. (2.7) – (2.10). For s = 1, we deal with a
size-biased Galton-Watson tree, cf. [6] and [15], and b <∞ is known as the X log X condition of the Kesten-
Stigum theorem, see also [5, page 23]. If it holds, the effect of the distinguished ray on the structure of Sn
is asymptotically negligible, see section 3 in [15]. For s Ê 2, the number of distinguished vertices L̂n in
the generation n increases in such a way that the sequence {σ−n L̂n}n∈N with probability one tends to a
random variable Ŵ such that Prob(Ŵ > 0) = 1 and 〈Ŵ 〉 = 1. Here,
σ= s − (s −1)p̂2 − (s −2)p̂3 −·· ·− p̂s .
Note thatσ ∈ (1, s], andσ= s if and only if p2 = ·· · = ps = 0, and hence the least value of X is s. In this case,
a −1 > s. If a −1 > sγ, Ln asymptotically ‘behaves’ like in the ordinary Galton-Watson tree, which means
that the production of offsprings with probability p ‘dominates’ the immigration from the distinguished
part of the population, which is always the case for GW(1, p), cf. (2.2). However, for a −1 É sγ, the main
contribution into Ln comes from the immigration. If a −1 is close to one and sγ ≫ 1, even for big n the
structure of (Vn ,En) is quite close to that of the corresponding subgraph of GW(0, p̂), i.e., as in the case of
the configuration model. Note that sγ is big whenever (a) s is big, and hence the distinguished subgraph
is ‘big’; (b) α is small, and hence the offspring production in the distinguished subgraph is very intensive.
Noteworthy, the procedure of formation of edges in GW(s, p) resembles that of [7, 8]. The ‘value’ spec-
ified in [8] by a positiveω, in our case appears as a mark k , distributed according to p with pk ‘decreasing
at infinity’, cf. “inverse mass-action principle” of [8]. Distinguished individuals have the right to choose
neighbors. Each chooses according to its own mark at most s distinguished neighbors, preferring those
with big marks. The restriction of the number of distinguished neighbors to s corresponds to the restric-
tion of the number of outgoing edges in the Cameo principle [8].
23001-9
Yu. Kozitsky
One observes that GW(s, p) is a natural generalization of the size-biased Galton-Watson tree GW(1, p).
In a separate work, we plan to study this model in more detail. Another generalization of GW(1, p) can
be the model in which the distinguished part of X̂ is not X̂ − (X̂ − s)+ like in GW(s, p), cf. (2.7), but rather
it is a more general increasing function of X̂ , or it is random. Further generalizations of this kind can be
obtained in the approach of [18].
5.2. Thermodynamic states
In 1989, R. Lyons [14] proved that, for any tree, the critical temperature of the model (1.1) with h = 0
and Jx y = J > 0 is
Tc =
J
kB coth−1̺
, (5.1)
where ̺ is the branching number of the underlying tree. For an infinite (almost surely non-extinct) Galton-
Watson tree, with probability one ̺ equals the mean number of offsprings, see proposition 6.4 in [19].
Hence, for GW(0, p̂), we have ̺= 〈k2〉/〈k〉−1, which yields in (5.1)
Tc = 2J
/
kB ln
〈k2〉
〈k2〉−2〈k〉
. (5.2)
The very same expression for Tc was ‘rediscovered’ in 2002 in [11, 12]. It is significant that Lyons’ paper
[14] was quoted5 in [2] and even the same formula (5.1) was discussed, see the text between equations
(89) and (90) on page 1304 in [2] or the text between equations (89) and (90) on page 33 in [20].
Now let us turn to the discussion of theorem 3.1, which describes the model (1.1) with arbitrary signs6
of the intensities Jx y . The only condition is that they are uniformly bounded, cf. (3.1). As follows from the
estimate (3.12), the model (1.1) on a general tree can be in a paramagnetic state if the number of vertices
in the sphere Sn admits the control |Sn | É cn for some c > 1. Note that in the example of [13], |Sn | ∼ n!.
For graphs other than trees7, one has to controlΘn(z), cf. (4.6). The results of theorem 3.1 can naturally be
extended to models with continuous spins and bounded interaction, as, e.g., in [13]. In a separate work,
we plan to study the case of unbounded interaction intensities, which also includes random Jx y , as, e.g.,
in the Edwards-Anderson model. For a ferromagnetic Ising model on a GW(0, p), the exact value of the
critical temperature is given in (5.1) with ̺ = a −1. The corresponding result of theorem 3.1 is that the
model is in a paramagnetic state for
T > 4J
/
kB ln
a
a −1
> Tc .
It is naturally less precise as we cover the case of arbitrary signs of Jx y .
Regarding the model GW(0, p̂) studied in [11, 12], by theorem 1.1 we get that the Ising model with
arbitrary signs of Jx y obeying (3.1) is in a paramagnetic state if
T > 4J
/
kB ln
〈k2〉
〈k2〉−1
.
It is believed, cf. [22], that the Internet can well be modeled as a scale-free graph with λ = 5/2, which
corresponds to the choice of α< 1/2, see (3.9). As a byproduct of the result of theorem 3.1, we obtain the
following, cf. [22, 23],
Proposition 5.1 Let θ ∈ (0,1) be the Bernoulli bond percolation probability on GW(s, p) graphs with s Ê 2
and p obeying (2.1) and bα <∞. Then, with probability one there is no giant component if
θ < min{q(Kc); q(K̂c)},
where q(K ) is given by (3.11), and Kc and K̂c are as in theorem 1.1.
5In the preprint version [20] of [2], both papers [14, 19] were quoted.
6The antiferromagnetic case Jx y =−J , J > 0, can be reduced to the ferromagnetic one by changing signs of σx , x ∈S2n .
7For more on phase transitions on graphs see [21].
23001-10
On thermodynamic states of the Ising model
Acknowledgements
The author benefited from the discussions on the matter of this work with Philippe Blanchard, Yurij
Holovatch, and Yuri Kondratiev, for which he is cordially indebted. The work was supported in part by
the DFG through SFB 701: “Spektrale Strukturen und Topologische Methoden in der Mathematik” and
through the research project 436 POL 125/113/0-1.
References
1. Georgii H.-O., Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, Vol. 9, Walter de
Gruyter, Berlin, New York, 1988.
2. Dorogovtsev S.N., Goltsev A.V., Mendes J.F.F., Rev. Mod. Phys., 2008, 80, 1275; doi:10.1103/RevModPhys.80.1275.
3. Newman M.E.J., SIAM Rev., 2003, 45, 167; doi:10.1137/S003614450342480.
4. Barabási A.-L., Science, 2009, 325, 412l; doi:10.1126/science.1173299.
5. Asmussen S., Hering H., Branching Processes, Progress in Probability and Statistics, Vol. 3, Birkhäuser Boston,
Boston, 1983.
6. Lyons R., Peres Y., Probability on Trees and Networks, Cambridge University Press (in press,
http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html).
7. Blanchard Ph., Chang C.-H., Krüger T., Ann. Henri Poincaré, 2003, 4, No. 2 Suppl., S954;
doi:10.1007/s00023-003-0975-1.
8. Blanchard Ph., Krüger T., J. Stat. Phys., 2004, 114, 1399; doi:10.1023/B:JOSS.0000013969.87579.39.
9. Dembo A., Montanari A., Ann. Appl. Probab., 2010, 20, 565; doi:10.1214/09-AAP627.
10. Dommers S., Giardinà C., van der Hofstad R., J. Stat. Phys., 2010, 141, 638; doi:10.1007/s10955-010-0067-9.
11. Dorogovtsev S.N., Goltsev A.V., Mendes J.F.F., Phys. Rev. E, 2002, 66, 016104; doi:10.1103/PhysRevE.66.016104.
12. Leone M., Vázquez A., Vespignani A., Zecchina R., Eur. Phys. J. B, 2002, 28, 191; doi:10.1140/epjb/e2002-00220-0.
13. Kepa D., Kozitsky Y., Condens. Matter Phys., 2008, 11, 313; doi:10.5488/CMP.11.2.313.
14. Lyons R., Commun. Math. Phys., 1989, 125, 337; doi:10.1007/BF01217911.
15. Lyons R., Pemantle R., Peres Y., Ann. Probab., 1995, 23, 1125; doi:10.1214/aop/1176988176.
16. Von Dreifus H., Klein A., Perez F.J., Commun. Math. Phys., 1995, 170, 21; doi:10.1007/BF02099437.
17. Grimmett G., Stirzaker D., Probability and Random Processes, 3rd ed., Oxford University Press, New York, 2001.
18. Jagers P., Sagitov S., Bernoulli, 2008, 14, No. 4, 949; doi:10.3150/08-BEJ138.
19. Lyons R., Ann. Probab., 1990, 18, 931; doi:10.1214/aop/1176990730.
20. Dorogovtsev S.N., Goltsev A.V., Mendes J.F.F., Preprint arXiv:0705.0010v2, 2007.
21. Lyons R., J. Math. Phys., 2000, 41, 1099; doi:10.1063/1.533179.
22. Cohen R., Erez K., ben-Avraham D., Havlin S., Phys. Rev. Lett., 2000, 85, 4626; doi:10.1103/PhysRevLett.85.4626.
23. Cohen R., ben-Avraham D., Havlin S., Phys. Rev. E, 2002, 66, 036113; doi:10.1103/PhysRevE.66.036113.
23001-11
http://dx.doi.org/10.1103/RevModPhys.80.1275
http://dx.doi.org/10.1137/S003614450342480
http://dx.doi.org/10.1126/science.1173299
http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html
http://dx.doi.org/10.1007/s00023-003-0975-1
http://dx.doi.org/10.1023/B:JOSS.0000013969.87579.39
http://dx.doi.org/10.1214/09-AAP627
http://dx.doi.org/10.1007/s10955-010-0067-9
http://dx.doi.org/10.1103/PhysRevE.66.016104
http://dx.doi.org/10.1140/epjb/e2002-00220-0
http://dx.doi.org/10.5488/CMP.11.2.313
http://dx.doi.org/10.1007/BF01217911
http://dx.doi.org/10.1214/aop/1176988176
http://dx.doi.org/10.1007/BF02099437
http://dx.doi.org/10.3150/08-BEJ138
http://dx.doi.org/10.1214/aop/1176990730
http://arxiv.org/abs/0705.0010v2
http://dx.doi.org/10.1063/1.533179
http://dx.doi.org/10.1103/PhysRevLett.85.4626
http://dx.doi.org/10.1103/PhysRevE.66.036113
Yu. Kozitsky
До термодинамiчних станiв моделi Iзинґа
на безмасштабних ґрафах
Ю. Козицький
Iнститут математики, Унiверситет Марiї Кюрi-Складовської, 20-031 Люблiн, Польща
Запропоновано модель безмасштабних випадкових ґрафiв, яка є локально близькою до нескорельова-
них випадкових мереж з розбiжними 〈k2〉, що вивчалася, наприклад, в статтi Dorogovtsev S.N. et al., Rev.
Mod. Phys., 2008, 80, 1275. Показано, що модель Iзинґа на запропонованих ґрафах з iнтенсивностями
взаємодiї довiльного знаку iз ймовiрнiстю один є в парамагнiтному станi для достатньо високих скiнче-
них значень температури. Для таких ґрафiв модель перколяцiї по ребрах з ймовiрнiстю один є в непер-
колятивному станi для позитивних значень перколяцiйної ймовiрностi. Обговорюються цi результати i
можливостi їх розширення.
Ключовi слова: випадковi ґрафи, парамагнiтна фаза, перколяцiя, процес розгалуження
23001-12
Introduction
The graph models
The Galton-Watson tree
The configuration model
The proposed model
The thermodynamic states of the Ising model
The thermodynamic states
The main statement
The proof of the lemmas
The proof of lemma 3.2
The proof of lemma 3.3
Concluding comments and remarks
The graph model
Thermodynamic states
|