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 $\langle k^2 \rangle$ studied in e.g. S. N. Dorogovtsev {\it et al}, Rev. Mod. Phys. {\bf 80}, 1275 (2008). It is shown that the Ising model on these 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. Possible extensions are discussed.


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 where J xy are interaction intensities, h is an external field, and the spins σ x take values ±1. The sums run over the sets of edges E and vertices V of the graph G, which is supposed to be connected and countably infinite. In view of the latter fact, the Hamiltonian (1.1) has no direct mathematical meaning and is used as a formula for local Hamiltonians, H Λ , which one obtains by restricting the sums to finite subsets Λ ⊂ V and E Λ ⊂ E, where E Λ is 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 (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 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 on this topic can be found in the review articles [2,3,4].
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 In simple cases, the degrees k(x) are independent and identically distributed, which in particular means that the probabilities p k (x) are the same for all x. If p k = Ck −λ , at least for big enough k and an appropriate C > 0, the corresponding graph is said to be scale-free, c.f. [5]. In this case, the m-th moment k m = k≥1 k m p k , m ≥ 1, exists provided λ > m + 1. Notably, k 2 = +∞ for many real-world networks [4]. An example of a random graph with independent k(x) is a Galton-Watson tree, which is the genealogic tree of a Galton-Watson branching process [6,7]. In more complex models, the graph G is obtained, more or less explicitly, as the limit of an increasing sequence of finite random graphs, (V n , E n ), n ∈ N, see, e.g. [8,9,10,11]. In the physical literature, such constructions are rather informal that leaves certain possibilities for different interpretations, which we are going to use in this work. In 2002, almost simultaneously there appeared two papers [12,13] where the model (1.1) on a scale-free graph was studied. The starting point is a scale-free distribution {p k } 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 kp k / k , see the beginning of the first section in [12]. In [13], a similar assumption was made, see equation (3) thereof, and also (1) in [3]. In the latter article, such graphs are called complex networks in contrast to the classical models called simple networks, one of which is presumably the Galton-Watson tree mentioned above. In both papers [12,13], it was found that the model (1.1) with h = 0, ferromagnetic interactions J xy = J > 0, and k 2 = +∞ is in an ordered (ferromagnetic) state at all temperatures 1 T > 0. Recently, in [10,11] the mathematical construction of a random graph model was presented, in which the mentioned properties were realized. For this model, the conclusion of [12,13] can be mathematically proven, see [15] and Subsection 5.2 below. The aim of this work is to construct "uncorrelated complex networks" with k 2 = ∞, consistent with the assumptions made in [3,12,13], for which the model (1.1) with interaction intensities of arbitrary signs can be in a paramagnetic state at finite temperatures. In Section 2, we perform this construction and obtain a graph model, which is a combination of the one used in [10,11] and of the ordinary Galton-Watson tree. In Section 3, we prove Theorem 3.1 the main result of which is that the model (1.1) on the introduced graphs, with any signs of J xy , can be in a paramagnetic state at finite temperatures and h = 0, provided k 1+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 give also an extension of the results of [12,13] concerning the model (1.1) with interaction intensities of arbitrary signs. Except possibly for Section 4, the paper is reasonably self-contained and accessible to non-mathematicians.

The graph models
Since the Galton-Watson tree will serve us as a prototype, we beging by presenting this model. More on this topic can be found in [6,7,16].

The Galton-Watson tree
To exclude the possibility of obtaining a finite tree, or a tree of bounded degree, we shall suppose that the basic random variable X takes positive integer values k ∈ N with probability Prob(X = k) = p k+1 such that One observes that a > 2, (2.2) which readily follows form the properties of p assumed in (2.1). Set L 0 = 1, L 1 = X, and where all X n,j are independent copies of X, see Chapters II and III in [6] or Chapter 5 in [7]. Then L n is the number of individuals in generation n of a Galton-Watson branching process. The process starts with one 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 does not extinct and L n → +∞ with probability one. Hence, G is an infinite tree -an acyclic connected graph, which has no leaves, except possibly for the root o, the degree of which coincides with the number of the corresponding offsprings L 0 . The degree of any other vertex equals the number of its offsprings plus one. By construction, the vertex degrees are independent identically distributed random variables such that 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 {x 0 , x 1 , . . . , x n }, n ∈ N, such that: (a) x 0 = x, x n = y; (b) {x l , x l+1 } ∈ E; (c) neither of x 0 , . . . , x n−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 origin x 0 is an infinite sequence of distinct vertices {x 0 , x 1 , . . . , x n , . . . } with property (b). A graph ensemble G is the collection of all possible realizations, e.g., 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 probability. For the Galton-Watson tree, P is defined by the collection p = {p k }. For A ∈ F, P (A) is the probability of the event: "the graph chosen at random from G belongs to A". Some events can occur with probability either zero or one -the celebrated zero-one law. In the Galton-Watson tree, if A is the set of all realizations with k(o) = 1, then P (A) = p 1 . 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. Note that this A need not be the whole ensemble G. Usually, when one deals with models with independent vertex degrees, the triple (G, F, P ) does not appear explicitly. This will also be the case in the rest of this paper.

The configuration model
Let p := {p k } k≥2 be as in (2.1), and "Uncorrelated complex networks" were 'defined' in [3,12,13] by means of the following conditions. The main one is that "the neighbor of a vertex has degree k with probabilityp k ", which can be interpreted as the condition that in every path {x 0 , x 1 , . . . , x n }, the probability that k(x l ) takes value k isp k . For convenience, we shall call it a path property. Another important assumptions are: (a) all k(x) are "uncorrelated" (in fact, independent); (b) two paths {x, x 1 , . . . , y} and {x, x ′ 1 , . . . , z}, with x 1 = x ′ 1 and y = z, never intersect each other, i.e., the graph is a tree 2 . These assumptions are consistent with the following mathematical models. The first one is the ordinary Galton-Watson tree with p replaced byp, in [10] it is T(ρ, ∞), see page 567 threof. A slightly different version is the so called configuration model, c.f. page 1279 in [3]. This model is a rooted tree with independent vertex degrees such that This model 3 was studied in [10,11], in [10] 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 inconsistency 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 mark 4 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) = p k . 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 Ckp k with C = 1/a since the total probability should sum up to one. In the result, each root neighbor has degree k with probabilityp k . In the same way one picks the next-neighbors of o keeping in mind that the only common element of the neighborhoods of different vertices is the root itself. This procedure is continued ad infinitum until the whole tree is constructed. An important observation here is that, except for the root degree, the distribution of k(x) is given byp, whereas p itself is the distribution of marks -auxiliary objects used in the graph construction.

The proposed model
Here the main object of our study in this article is introduced. This is a random tree model, also consistent with the assumptions of [3,12,13]. As in the case of the Galton-Watson tree, we beging with the description of the corresponding branching process. Let p,p, and a be as in (2.1)  where, for an integer κ, we write κ + = max{0, κ}, and X n,j and X n,j are independent copies of X and X, respectively. From the point of view of ordinary particles, those represented by the second term in the expression for L n are immigrants, and their number in generation n is Then the total number of individuals in generation n can be written in the form In the remaining part of the article we use the following nomenclature. By GW(s, p), s ∈ N 0 , we denote the model constructed in this subsection. Then GW(1, p) and GW(0, p) are is the size-biased and the ordinary Galton-Watson trees, respectively. and let E n be the set of all edges with both ends in V n . Let also E b n be the set of {x, y} such that x ∈ S n and y ∈ S n+1 .
By σ n we denote the configuration of spins in V n , that is, σ n = {σ x : x ∈ V n }. In the sequel, we suppose that The Gibbs probability distribution of configurations σ n at temperature T and h = 0 is the following probability measure where K xy := βJ xy , K := βJ, β := 1/k B T. Note that K xy can have arbitrary signs, whereas K is positive. In (3.2), the first (resp. second) summand in exp(· · · ) corresponds to the interaction of the spins in V n with each other (resp. with the spins ξ y fixed outside V n ). One observes that the latter interaction involves only ξ y with y ∈ S n+1 , which constitute the outer boundary of V n . For technical reasons, we add positive constants to the spin-spin interaction along each edge in E n , c.f. (1.1). The partition function is then where the summation is taken over all σ x = ±1, x ∈ V n . Given z ∈ V, let n z ∈ N 0 be such that z ∈ S nz , i.e., n z = ρ(o, z). For these z, n z , and for n > n z , is the magnetization at z ∈ V n in the state π n (·|ξ) (3.2). Here, for σ z = α, ̺ n,z (α|ξ) = σn: σz=α π n (σ n |ξ), where the summation is taken over all σ n with the fixed σ z = α. According to the theory of Gibbs states [1], the global thermodynamic state is unique if, for all z, the limits 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 (3.9) is a random event obeying the zero-one law. If the corresponding probability is one, we say that the model (1.1) on G is in a paramagnetic state with probability one.

The main statement
Let p andp be as in (2.1) and (2.5). For α ∈ (0, 1), we set kp k ln k, (3.10) and also 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 < K c . For s = 1, the same holds under the condition b < ∞. For s ≥ 2, the Ising model is in a paramagnetic state whenever b α < ∞ for some α ∈ (0, 1) and K < min{K c ; K c }.
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. Then for each z ∈ V, any n > n z , and arbitrary ξ and η, we have that In the next lemma, we describe the model GW(s, p) introduced in Subsection 2.3. with probability one exists and is finite, see (2.2) and Proposition 1.3 in [6, page 20]. If K < K c , then q(K) < q(K c ) = 1/(a − 1) < 1, and hence we can apply (3.13), which yields that with probability one [q(K)(a − 1)] n L n (a − 1) n → 0, n → +∞, (3.15) which in turn by (3.13) yields (3.9), and hence the proof for this case. For s = 1, by (2.10) we have L n = 1, and hence |S n | = L n = 1 + L n . Since b < ∞, with probability one we have For s ≥ 2, the random variables Y n with different n are no more identically distributed, and hence b < ∞ is not enough to get (3.16). Instead we use a more restrictive condition b α < ∞, under which we get, see Lemma 3.3, that c −n |S n | = c −n L n = c −n L n + c −n L n with probability one tends to a finite limit since c > s, cf. (2.10). Note that Hence, for K < K(c), we have q(K) < 1/c, and the proof follows as in (3.15). Let G = (V, E) be a general tree. For a path ϑ, by E ϑ we denote the set of edges {x l−1 , x l } of ϑ. Since G is a tree, there exists exactly one path ϑ(x, y) for any x and y. Recall that E n stands for the set of edges with both ends in V n . Thus, G n = (V n , E n ) is a finite graph; (σ z −σ z ) π n (σ n |ξ)π n (σ n |η) (4.1) For simplicity, we do not indicate the dependence of the latter functions on the spins. One observes that each Γ xy ≥ 0 due to our choice of the spinspin interactions in (3.2), cf. (3.1), and that Ψ n (ξ, η) depends only on σ x andσ x with x ∈ S n . Fix some E ′ in the last line in (4.1) and consider the subgraph G ′ with the edge set E ′ and the vertex set V n . If in G ′ the vertex z is disconnected from each x ∈ S n , then the sums over the spins σ z ,σ z and over σ x ,σ x with x ∈ S n get independent and hence the left-hand side of (4.1) vanishes as the term (σ z −σ z ) is antisymmetric with respect to the interchange σ ↔σ, whereas all Γ xy are symmetric and the only break of this symmetry is related with the fixed boundary spins ξ(y) and η(y), y ∈ S n+1 . Therefore, each non-vanishing term in (4.1) corresponds to a path ϑ(z, x) connecting z to some x ∈ S n . Let us take this into account and rewrite (4.1) as the sum over the subsets of E n containing the edges of at least one such path. Let Θ n (z) be the set of all paths connecting z to S n , and for ϑ ∈ Θ n (z), let E ϑ be the family of subsets of E n each of which contains E ϑ . That is, Note that, for distinct ϑ, ϑ ′ ∈ Θ n (z), the corresponding families E ϑ and E ϑ ′ are not disjoint -they include those E ′ which contain both E ϑ and E ϑ ′ . Finally, set that is, E contains all sets of edges E ′ ⊂ E n such that the corresponding graph G ′ contains at least one path connecting z to some x ∈ S n . Then (4.1) takes the form In view of the positivity of all Γ xy ≥ 0, this yields Note that the sum ϑ∈Θn(z) E ′ ∈E ϑ contains the same summands as E ′ ∈E , see (4.3), but a part of them are repeated, which yields the second ≤ in (4.5). From (4.2) and (3.12) we see that 0 ≤ Γ xy ≤ q(K), for any x, y ∈ V. Hence, which yields in (4.5) Here we have taken into account that, see (4.1), (1 + Γ xy )Ψ n (ξ, η) = Z n (ξ)Z n (η).
Note that a similar construction was used in deriving (2.16) and (2.17) in [17], see also Lemma 3.1 in [14]. Since n z is defined by the condition z ∈ S nz , the shortest path in Θ n (z) has length n − n z . Furthermore, for every x ∈ S n , there exists exactly one path ϑ ∈ Θ n (z). Then, for q(K) ≤ 1, the estimate (3.13) follows from (4.6).

The proof of Lemma 3.3
If we prove that, for c > s γ , with probability one see (2.8), then the proof that L n /c n → W ∈ [0, +∞), with probability one, follows as in the proof of Theorem 6.1 in [6, page 50]. The first summand of L n in (2.7) is under control in view of the assumed property c ≥ a − 1.

The graph model
Let us look at the structure of generation n in GW(s, p), cf. (2.7) -(2.10). For s = 1, we deal with a size-biased Galton-Watson tree, cf. [7] and [16], and b < ∞ is known as the X log X condition of the Kesten-Stigum theorem, see also [6, page 23]. If it holds, the influence of the distinguished ray on the structure of S n is asymptotically negligible, see Section 3 in [16]. For s ≥ 2, the number of distinguished vertices L n in generation n increases in such a way that the sequence {σ −n L n } n∈N with probability one tends to a random variable W such that Prob( W > 0) = 1 and W = 1. Here Note that σ ∈ (1, s], and σ = s if and only if p 2 = · · · = p s = 0, and hence the least value of X is s. In this case, a − 1 > s. If a − 1 > s γ , L n 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 ≤ s γ , the main contribution into L n comes from the immigration. If a − 1 is close to one and s γ ≫ 1, even is that they are uniformly bounded, cf. (3.1). As follows from the estimate (3.13), the model (1.1) on a general tree can be in a paramagnetic state if the number of vertices in the sphere S n admits the control |S n | ≤ c n for some c > 1. Note that in the example of [14], |S n | ∼ n!. For graphs other than trees 7 , one has to control Θ n (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 [14]. In a separate work, we plan to study the case of unbounded interaction intensities, which includes also random J xy , 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 It is naturally less precise as we cover the case of arbitrary signs of J xy . Regarding the model GW(0,p) studied in [12,13], by Theorem 1.1 we get that the Ising model with arbitrary signs of J xy obeying (3.1) is in a paramagnetic state if 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 (3.10). 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 ≥ 2 and p obeying (2.1) and b α < ∞. Then with probability one there is no giant component if θ < min{q(K c ); q( K c )}, where q(K) is given by (3.12), and K c and K c are as in Theorem 1.1.

Acknowledgment
The author benefited from the discussions on the matter of this work with Philippe Blanchard, Yurij Holovatch, and Yuri Kondratiev, for that 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.