LACK OF MONOTONICITY IN FERROMAGNETIC

ISING MODEL PHASE DIAGRAMS

Roberto H. Schonmann (UCLA) and Nelson I. Tanaka (IME/USP)

Abstract. We study patterns of the phase diagram of ferromagnetic Ising models on graphs under an external magnetic field. We provide an example of a tree with only two types of vertices on which for a range of values of the external field there is a unique Gibbs distribution at low enough and at high enough temperatures, while at intermediate temperatures there is phase coexistence (in other words, a reentrance transition takes place).

1. Introduction.

The investigation reported in this paper started with the following thoughts. The ferromagnetic Ising model can be seen as the most basic model in mathematical statistical mechanics and its natural mathematical setting is a graph, with the vertices being the sites which we can think of as the locations of spins +1 or −1, and the edges indicating the pairs of spins which do interact. Two parameters appear in this model: an external field h ∈ ℝ and a temperature T ∈ ℝ_{+}. A fundamental question is then to locate on the phase diagram h × T the region where there is more than one Gibbs distribution (the so called “phase-coexistence” region). It is natural to ask what the general features of the phase-coexistence region are. Regarding its intersection with the T -axis (i.e., the reduced problem in which h is held fixed equal to 0), there is a well-known argument – to be reviewed later, when we have enough notation available – based on one of Griffiths’ inequalities, which shows that this intersection is always an interval (possibly degenerate), which has one of its end-points at (h, T) = (0, 0). This simple feature of the phase-coexistence region can, of course, be seen as a monotonicity property: if there is a unique Gibbs measure at the point (0, T_{1}) of the phase diagram, then the same is also true at all the points (0, T), with T > T_{1}. The main point of this paper is that if h ≠ 0, then the same sort of monotonicity in T is not true, even if we restrict ourselves to very simple graphs (our example will be an almost homogeneous tree, with only two types of vertices).

Before proceeding, we will have to introduce a certain amount of notation and terminology. We will also recall some well known facts about the statistical mechanics of lattice systems and the Ising model, and we refer the reader to [Geo] and [Lig] for their proofs.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 60K35, 82B27.

Key words and phrases. Ising model, ferromagnetism, graphs, phase diagram, monotonicity, reentrance transition.

We will consider graphs with countably many vertices and locally bounded degree (meaning that each vertex belongs to a finite number of edges). This class of graphs will be denoted by C. No assumption that the graph is connected or infinite is being made, but these are the typical cases in which one is interested. Given a graph G ∈ C, we will use V (G) to denote its set of vertices (also called sites) and E(G) to denote its set of edges. When there is no risk of confusion, G will be omitted in this notation. Two vertices are called neighbors if they belong to a common edge.

An isomorphism between two graphs, G_{1} and G_{2}, is a one-to-one mapping from V (G_{1}) onto V (G_{2}) which preserves the graph structure, i.e., such that the set of edges of G_{2} can be obtained as the set of pairs of images of vertices of G_{1} which form edges. An isomorphism between a graph G and itself is called an automorphism of G. Two vertices of G are said to be of the same type, if each one can be mapped into the other one by an automorphism of G. Graphs which have a single type of vertex will be called homogeneous graphs (sometimes the term transitive graphs is used in the literature). Graphs which have a finite number of types of vertices will be called almost-homogeneous graphs (sometimes the term almost-transitive graphs is used in the literature).

Configurations are elements of the set Ω = {−1, +1}^{V}, interpreted as the assignment of a spin −1 or +1 to each site in V. Given σ ∈ Ω and x ∈ V, we use σ(x) for the value of the spin at x. We will consider the formal ferromagnetic Ising Hamiltonian:

H_{h}(σ) = − ∑_{{x,y}∈E} σ(x)σ(y) − h ∑_{x} σ(x),  (1.1)

where h ∈ ℝ is the external field and σ ∈ Ω is a generic configuration. In order to give precise definitions, we consider finite subsets of V. The expression Λ ⊂⊂ V will mean that Λ is a finite subset of V. Given a set Λ ⊂⊂ V we define also

E_{Λ} = {{x, y} ∈ E : x, y ∈ Λ},
∂E_{Λ} = {{x, y} ∈ E : x ∈ Λ, y ∉ Λ}.

Given also a configuration η ∈ Ω, we define the following set of configurations:

Ω_{Λ,η} = {σ ∈ Ω : σ(x) = η(x) for all x ∉ Λ}.

For each set Λ ⊂⊂ V and each boundary condition η ∈ Ω, we define

H_{Λ,η,h}(σ) = − ∑_{\{x,y\}∈E_{Λ}} σ(x)σ(y) − ∑_{\begin{matrix}\{x,y\}∈∂E_{Λ} \\ y∉Λ \end{matrix}} σ(x)η(y) − h ∑_{x∈Λ} σ(x).  (1.2)

The Gibbs (probability) measure in Λ with boundary condition η, under external field h and at temperature T > 0 is now defined on Ω as

μ_{Λ,η,T,h}(σ) = \{ \begin{matrix}\frac{exp(−βH_{Λ,η,h}(σ))}{Z_{Λ,η,T,h}}, & if σ ∈ Ω_{Λ,η}, \\ 0, & otherwise, \end{matrix}

where β = 1/T and

Z_{Λ,η,T,h} = ∑_{σ∈Ω_{Λ,η}} exp(−βH_{Λ,η,T,h}(σ)).

A Gibbs measure for the system on the graph G (possibly infinite) is defined as any probability measure μ which satisfies the DLR equations in the sense that for every Λ ⊂⊂ V and μ-almost all η ∈ Ω

μ( · |Ω_{Λ,η}) = μ_{Λ,η,T,h}( · ).  (1.3)

It is easy to check and very important that this definition is self-consistent in case V is finite.

Alternatively and equivalently, Gibbs measures on a possibly infinite graph can be defined as limits of the corresponding Gibbs measures on finite subsets of the system, with arbitrary boundary conditions. For this purpose one says that a sequence of probability measures, (μ_{n})_{n=1,2,...}, converges weakly to the probability measure ν in case

lim_{n→∞} ∫ fdμ_{n} = ∫ fdν,  (1.4)

for each f : Ω → ℝ which depends only on the value of the spins on a finite set of sites. The set of Gibbs measures for the system on G coincides with the closed convex hull of the set of weak limit points of sequences of the form (μ_{Λ_{i},η_{i},T,h})_{i=1,2,...}, where each Λ_{i} is finite and Λ_{i} → V, as i → ∞, in the sense that ∪_{i=1}^{∞} ∩_{j=i}^{∞} Λ_{j} = V.

The set of Gibbs measures will be denoted by G_{T,h}. The subset of the h×T halfplane where G_{T,h} has a single element is called the phase-uniqueness region, and its complement is called the phase coexistence region. From Dobrushin’s uniqueness condition (see, e.g, Chapter 8 of [Geo]), it is easy to see that if our graph G has the degree of all the vertices bounded by a common constant κ, then there are finite positive constants h(κ) and T (κ) such that if |h| > h(κ), or T > T (κ), the system is in the phase-uniqueness region.

For the expected value corresponding to a Gibbs measure μ..., in finite or infinite volume, we will use the notation

⟨f⟩_{...} = ∫ fdμ...,

where ... stands for arbitrary subscripts. We will use a common and convenient form of abuse of notation: σ(x) will be used to denote the function which associates to each configuration the value of the spin at the site x in that configuration.

The Gibbs measures satisfy the following monotonicity relations to which we will refer as the FKG-Holley inequalities.

If η ≤ ζ and h_{1} ≤ h_{2}, then, for each Λ ⊂⊂ V, μ_{Λ,η,T,h_{1}} ≤ μ_{Λ,ζ,T,h_{2}}.

In what follows we will abbreviate by + (resp. −) the configuration with all spins +1 (resp. −1). We will also use the notation ± in a standard way, with each equation in which it appears representing the two equations obtained by replacing this symbol consistently with +, or consistently with −.

A consequence of the FKG-Holley inequalities in combination with the DLR equations is that

μ_{Λ,±,T,h} → μ_{±,T,h} weakly, as Λ → V.  (1.5)

The Gibbs distributions μ_{+,T,h} and μ_{−,T,h} so obtained are called respectively the (+)-phase and the (−)-phase of the Ising model on G. Moreover the following three statements are equivalent

U1) |G_{T,h}| = 1.

U2) μ_{−,T,h} = μ_{+,T,h}.

U3) ⟨σ(x)⟩_{−,T,h} = ⟨σ(x)⟩_{+,T,h}, for each site x ∈ V.

In case the graph G is not connected, the equality in (U3) above can be satisfied for some sites x while it fails for others. On the other hand if G is connected, that equality is either true for all sites x or false for all sites x. To prove this last statement one can proceed as follows. Suppose that the equality fails for one given site x. Consider next a site y which is neighbor to x. By conditioning on the configuration in the neighbors of y and using the Holley-FKG inequalities, one can readily see that the equality in (U3) also fails for y. By proceeding inductively, one concludes then that once the equality in (U3) fails for one site, it will fail for all sites which belong to the connected component of the graph to which this site belongs.

The Ising model has been mostly studied on the the cubic lattices, V = ℤ^{d}, with edges connecting sites which are separated by Euclidean distance 1. In this fundamental case, the phase-coexistence region is contained in the T -axis. In case d = 1 it is empty, while for d ≥ 2 it is a non-degenerate interval contained in this axis, and of the form (0, T_{c}]. The proofs that there is uniqueness of the Gibbs distribution for the Ising model on these graphs when h ≠ 0 depend only on the facts that these graphs are homogeneous and the number of sites at distance N from a fixed site grows slower than the number of sites at distances smaller than N from this site, as N → ∞.

Also greatly studied is the Ising model on a homogeneous tree, 𝕋_{b}. In this notation b is the branching number, so that each site has b + 1 neighbors. Note that 𝕋_{1} = ℤ^{1}. In the case b ≥ 2 the phase-coexistence region is not confined to the T -axis; it is given by the set

{(h, T) : 0 < T < T_{c}, −h_{c}(T) ≤ h ≤ h_{c}(T)},  (1.6)

where

h_{c}(T) = T max_{t≥0}(bφ_{β}(t) − t),  (1.7)

with

φ_{β}(t) = \frac{1}{2} log (\frac{cosh(t + β)}{cosh(t − β)}) = tanh^{−1}(tanh(β) tanh(t)),  (1.8)

and

T_{c} = sup\{T > 0 : h_{c}(T) > 0\} = \frac{1}{coth^{−1}(b)} = \frac{2}{log (\frac{b+1}{b−1})}.  (1.9)

The following properties of the function φ_{β} are important. It is a continuous odd function, which is strictly increasing and has range (−β, β). It is also strictly convex on (−∞, 0] and strictly concave on [0, ∞).

Before proceeding we recall the well-known argument to the effect that if there is a unique Gibbs measure at the point (0, T_{1}) of the phase diagram, then the same is also true at all the points (0, T), with T > T_{1}. For this purpose one uses the symmetry between +s and −s in the case h = 0, and writes

0 ≤⟨σ(x)⟩_{+,Λ,T,0} − ⟨σ(x)⟩_{−,Λ,T,0} = 2⟨σ(x)⟩_{+,Λ,T,0}
≤2⟨σ(x)⟩_{+,Λ,T_{1},0} = ⟨σ(x)⟩_{+,Λ,T_{1},0} − ⟨σ(x)⟩_{−,Λ,T_{1},0}.

Here the first inequality is an instance of the FKG-Holley inequalities, while the second inequality is an instance of one of Griffiths’ inequalities. Letting Λ → V, and using the equivalence between (U1) and (U3), one obtains then phase uniqueness at (T, 0) from phase uniqueness at (T_{1}, 0).

A similar well-known argument can be used to show that if we delete edges from a graph on which there is phase uniqueness at a certain point (0, T), then at the same point of the phase diagram, there will be phase uniqueness for the new graph.

Note nevertheless that the absence of symmetry between +s and −s when h ≠ 0, spoils the argument above, in that case.

We describe next a tree with only two types of sites (an almost-homogeneous tree), on which for certain values of h ≠ 0 phase uniqueness at certain values of T does not imply phase uniqueness at larger values of T.

Our tree can be constructed from 𝕋_{b}, by adding vertices and edges to it. In this procedure each vertex of 𝕋_{b} is connected to A new vertices, by means of A new edges. These new vertices are not connected to any other vertex, so that they are leaves of the graph. This completes the construction of the graph, which we denote by 𝕋_{b,A}. This graph has two types of vertices, those of first type are the ones with which we started; the ones we added are of the second type. Each vertex of first type is connected to b + 1 other vertices of the same type and to A vertices of the second type. Each vertex of the second type is connected to exactly one vertex of the first type and to no vertex of the second type.

Proposition 1. For the tree 𝕋_{b,A} with a proper choice of b and A (e.g., A = 2b and b large enough) for h in a non-degenerate interval which contains 1 the following happens. There are values of T for which there is phase coexistence, but for T either large enough or small enough there is phase uniqueness.

In the proof of this proposition we will see that when A > 0 the phase-coexistence region of 𝕋_{b,A} is a strict subset of that of 𝕋_{b}. It is clear that the graph that has the same set of vertices as 𝕋_{b,A} but only has edges connecting the sites corresponding to sites of the first type in 𝕋_{b,A} has the same phase diagram as 𝕋_{b} (the other sites are not connected to anything, and in particular the graph is not connected, but still it is in C). Hence the conclusion that the addition of edges to a graph may reduce the phase-coexistence region, contrary to what happens with the intersection of this region and the T -axis.

The lack of monotonicity in T exemplified in Proposition 1 contrasts with the monotonicity in |h| expressed in the next one.

Proposition 2. For all graphs in C, if there is a unique Gibbs measure at the point (h_{1}, T) of the phase diagram, then the same is also true at all the points (h, T), with |h| > |h_{1}|.

Proof. With no loss of generality we can take 0 ≤ h_{1} ≤ h. The following correlation inequalities are then available, where Λ ⊂⊂ V and x ∈ Λ,

0 ≤ ⟨σ(x)⟩_{+,Λ,T,h} − ⟨σ(x)⟩_{−,Λ,T,h} ≤ ⟨σ(x)⟩_{+,Λ,T,h_{1}} − ⟨σ(x)⟩_{−,Λ,T,h_{1}}.  (1.10)

Here the first inequality is an instance of the FKG-Holley inequalities, while the second inequality can be obtained from Lebowitz’s inequalities for duplicated spins in [Leb]. (The derivation appears in the 2nd step of the proof of Theorem 2 in [Hig2], p. 6, where only the case 0 = h_{1} ≤ h is considered, but the same argument works as well for 0 ≤ h_{1} ≤ h. The reader should also beware that Higuchi uses a slight generalization of the work in [Leb], in which now different external fields act on each copy of the spin system.) Letting Λ → V in (1.10), and using the equivalence between (U1) and (U3), one obtains phase uniqueness at (T, h) from phase uniqueness at (T, h_{1}).

Section 2 will be dedicated to the proof of Proposition 1. Here we give a heuristic argument, which makes this proposition at least plausible. The idea in this heuristic argument is to consider the spins at the sites of first type in 𝕋_{b,A} as being the sites of the smaller graph 𝕋_{b}, and the spins at the sites of second type (the leaves of 𝕋_{b,A}) as providing an extra effective external field which acts on the sites of first type, in addition to the external field h.

For the heuristic argument we will need two facts about the phase diagram of the Ising model on 𝕋_{b}. The first one is

lim_{T →0} h_{c}(T) = b − 1.  (1.11)

One way to obtain (1.11) is by computing h_{c}(T) explicitly from its definition (1.7). This is done in [Geo], where the result appears as equation (12.30). There J is our β, d is our b and h(J, d) is our βh_{c}(T). The behavior of h_{c}(T) as T → 0 is also found in [Geo] (first display after (12.30)), and in particular one has (1.11).

The second fact is that for fixed T

lim_{b→∞} \frac{h_{c}(T)}{b} = 1.  (1.12)

This is a straightforward consequence of (1.7) and the properties of φ_{β}.

Suppose that h is slightly larger than 1. At low temperature the spins at the sites of second type will, with overwhelming probability, be aligned with this external field and so be +1. This is so because in this state the contribution to the energy from each such spin is lower than in the opposite state, regardless of the state of the spin at the site of first type neighbor to the site that we are considering. But then the spins at the A sites of second type, neighbors to a site of first type x, produce an effective extra external field acting on the spin at x of magnitude close to A. This should be contrasted with (1.11), which tells us that h_{c}(T) is close to b − 1. The total effective external field close to A + 1 = 2b + 1 should therefore be enough to assure phase uniqueness.

We suppose that b is so large that we can take a temperature large compared with 1, while still small compared with b. When the temperature is much larger than 1, there should be a substantial entropy effect affecting the spins at the leaves of the tree and the extra effective field acting on the spins at sites of first type should be just a small fraction of A = 2b. Contrasting this with (1.12), we see that, if b is large, now the effective field is no longer strong enough to bring the system into the phase-uniqueness region.

At larger temperatures, of course, Dobrushin’s uniqueness condition tells us that phase uniqueness will again be restored.

2. Proof of Proposition 1.

When studying Gibbs measures on trees, it is natural to look for recursions. In such an approach success depends on making a good choice of the quantities for which the recursion is written. Our choice of L_{x,n,η} below as this quantity was motivated by the paper [Hig1].

For the moment the setting is an arbitrary connected tree G in C. Call one of the vertices of the tree its root, denoted by 0. Each vertex x in the tree has a generation index g(x), defined inductively by setting g(0) = 0, and giving the generation index n + 1 to the vertices which are neighbors of a vertex with index n unless they already have index n − 1. We write x →_{1} y in case x and y belong

to a common edge and g(y) = g(x) + 1. A vertex z is a descendent of a vertex x if there is a sequence of vertices x = x_{0}, x_{1}, ..., x_{i} = z such that x_{j} →_{1} x_{j+1}, for j = 0, ..., i − 1.

Given x ∈ V, we define the tree G_{x}, obtained from the original tree G that we are considering by only keeping the vertex x and its descendents, and keeping all the edges connecting any two of these vertices. Given also n ≥ g(x), we define V_{x,n} as the set of vertices containing x and its descendents with generation index not exceeding n. Note that V_{x,n} is a subset of the set of vertices of G_{x}.

In what follows T and h are fixed, and will be omitted from the new notation being introduced. First define

H_{x,n,η}(σ) = − ∑_{\begin{matrix}\{y,z\}∈E(G_{x}) \\ y∈V_{x,n} z∈V_{x,n} \end{matrix}} σ(y)σ(z) − ∑_{\begin{matrix}\{y,z\}∈E(G_{x}) \\ y∈V_{x,n} z∉V_{x,n} \end{matrix}} σ(y)η(z) − h ∑_{y∈V_{x,n}} σ(y). (2.1)

Now, given A ⊂ {−1, +1}^{V_{x,n}} define

Z_{x,n,η}(A) = ∑_{σ∈A} exp(−βH_{x,n,η}(σ)).

Next set

L_{x,n,η} = \frac{1}{2} log \frac{Z_{x,n,η}(σ(x) = +1)}{Z_{x,n,η}(σ(x) = −1)}.

We can now write the following recursion:

L_{x,n,η} =\frac{1}{2} log \frac{e^{βh} ∏_{y:x→_{1}y} \{e^{β}Z_{y,n,η}(σ(y) = +1) + e^{−β}Z_{y,n,η}(σ(y) = −1)\}}{e^{−βh} ∏_{y:x→_{1}y} \{e^{−β}Z_{y,n,η}(σ(y) = +1) + e^{β}Z_{y,n,η}(σ(y) = −1)\}}
=βh + ∑_{y:x→_{1}y} φ_{β}(L_{y,n,η}),  (2.2)

where φ_{β} is defined by (1.8).

We will write l_{n,η} = L_{0,n,η} and V_{0,n} = V_{n}. Note that

l_{n,η} = \frac{1}{2} log \frac{μ_{V_{n},η,T,h}(σ(0) = +1)}{μ_{V_{n},η,T,h}(σ(0) = −1)}.  (2.3)

In what follows we will suppose that η is either the configuration with all spins −1 or that with all spins +1. From (1.5) and (2.3) we obtain

l_{±} := lim_{n→∞} l_{n,±} = \frac{1}{2} log \frac{μ_{±,T,h}(σ(0) = +1)}{μ_{±,T,h}(σ(0) = −1)}.

A simple computation yields now

⟨σ(0)⟩_{±,T,h} = tanh(l_{±}).

Therefore the remark in the paragraph after the equivalent conditions (U1) – (U3) were introduced implies that for an arbitrary connected tree G ∈ C also

|G_{T,h}| = 1 ⟺ l_{−} = l_{+}.  (2.4)

We want to use the equivalence in (2.4) to find the coexistence region for the tree 𝕋_{b,A}. From this point on we will use the notation above, having in mind that it refers now to the tree G = 𝕋_{b,A}.

Before we can study 𝕋_{b,A}, we will have to consider a related but somewhat different tree G′ = 𝕋′_{b,A}, and for this tree we will use similar notation, but distinguished by a prime. The tree 𝕋′_{b,A} is obtained from 𝕋_{b,A} by removing one of the neighbors of its root, as well as all the descendents of this site. All the edges from 𝕋_{b,A} which connect sites which are not being removed are kept.

The tree G′ = 𝕋′_{b,A} has the particularly nice feature that for each of its vertices x, the corresponding tree G′_{x} (as defined above, before (2.1) was introduced) is isomorphic to 𝕋′_{b,A} itself. This makes the recursion (2.2) become particularly simple and yields

l′_{n,±} = βh + Aφ_{β}(βh) + bφ_{β}(l′_{n−1,±}),  (2.5)

where the second term comes from summing over the leaves among the neighbors of the root. We will introduce the notation

H_{eff}^{A}(β, h) = βh + Aφ_{β}(βh),  (2.6)

so that (2.5) turns into

l′_{n,±} = H_{eff}^{A}(β, h) + bφ_{β}(l′_{n−1,±}).  (2.7)

From the last display and the continuity of φ_{β}, it follows that l′_{−} = lim_{n→∞} l′_{n,−} and l′_{+} = lim_{n→∞} l′_{n,+} are solutions of the following equation in t

t = H_{eff}^{A}(β, h) + bφ_{β}(t).  (2.8)

This equation is precisely equation (12.22) of [Geo], where the case of the homogeneous trees is studied. One simply has to replace the variable J in [Geo] with our β, d in [Geo] with our b and the h in [Geo] with our H_{eff}^{A}(β, h). From the analysis in [Geo] of this equation we know that (2.8) has exactly 1 solution if T ≥ T_{c} or |H_{eff}^{A}(β, h)| > βh_{c}(T), where T_{c} is given by (1.9) and h_{c}(T) is given by (1.7). It has exactly 2 solutions if T < T_{c} and |H_{eff}^{A}(β, h)| = βh_{c}(T). And it has exactly 3 solutions in case T < T_{c} and |H_{eff}^{A}(β, h)| < βh_{c}(T). We will denote by t_{−} the smallest solution of (2.8) and by t_{+} the largest solution of that equation.

Next we want to argue that

l′_{±} = t_{±}.  (2.9)

For this purpose we extend the definition of φ_{β} by continuity, setting

φ_{β}(±∞) = lim_{t→±} φ_{β}(t) = ±β.

We also define

l′_{−1,±} = ±∞,

so that a direct computation shows that

l′_{0,±} = H_{eff}^{A}(β, h) ± bβ = H_{eff}^{A}(β, h) + bφ_{β}(l′_{−1,±}).

In other words, (2.7) is now satisfied for n = 0, 1, 2, ....

For t > t_{+} we have t > H_{eff}^{A}(β, h)+bφ_{β}(t) (compare the limits of both sides as t → ∞ and note that by continuity of the functions on both sides the inequality between them must be the same for all t > t_{+}). It is therefore clear that the recursion (2.7), started from l′_{−1,+} = +∞, produces a decreasing sequence (l′_{n,+})_{n≥−1}, bounded below by t_{+}. This decreasing sequence must then converge to a fixed point of the recursion, not smaller than t_{+}, and hence it must converge to t_{+}. This proves (2.9) in case ± is replaced with +. The case in which it is replaced with − is analogous.

From (2.4), (2.9) and the behavior of the solutions of (2.8) for different values of T and h, we conclude that the phase-coexistence region for the Ising model on 𝕋′_{b,A} is the set

{(h, T) : t_{−} < t_{+}} = {(h, T) : 0 < T < T_{c}, −h_{c}^{A}(T) ≤ h ≤ h_{c}(T)},  (2.10)

where T_{c} is still given by (1.9), and

h_{c}^{A}(T) = max{h ≥ 0 : H_{eff}^{A}(β, h) ≤ βh_{c}(T)},  (2.11)

with h_{c}(T) given by (1.7). In justifying the equality in (2.10), note that H_{eff}^{A}(β, h) is a strictly increasing function of h, and H_{eff}^{A}(β, 0) = 0.

Regarding the phase-coexistence region for the Ising model on the tree G = 𝕋_{b,A}, we can readily see that it is the same set (2.10) above. For this, note that the recursion (2.2) applied in case x is the root of this graph gives

l_{n,±} = βh + Aφ_{β}(βh) + (b + 1)φ_{β}(l′_{n−1,±}) = H_{eff}^{A}(β, h) + (b + 1)φ_{β}(l′_{n−1,±}).

This is so because for each site y different from the origin G_{y} is isomorphic to 𝕋′_{b,A}. Letting n → ∞ gives

l_{±} = H_{eff}^{A}(β, h) + (b + 1)φ_{β}(l′_{±}) = H_{eff}^{A}(β, h) + (b + 1)φ_{β}(t_{±}).

Since φ_{β} is strictly monotone increasing we can conclude now that

l_{−} = l_{+} ⟺ t_{−} = t_{+},

so that (2.4) implies that indeed the phase-coexistence region for the Ising model on 𝕋_{b,A} is given by (2.10). Incidentally, note that the analysis above includes a review of a proof that (1.6) is the phase coexistence region for the Ising model on the homogeneous tree 𝕋_{b} (the case A = 0). Also, from (2.11) and (2.6) it is clear that the phase-coexistence region shrinks as A grows. This justifies the claim that we made after the statement of Proposition 1.

Our problem of proving Proposition 1 is now reduced to the study of the phasecoexistence region, as described by (2.10), and in particular of the behavior of h_{c}^{A}(T), given by (2.11), as a function of T. According to the statement of that proposition we take A = 2b, and we will at some points below also have to choose b large enough.

Note that from the definition (1.9) of T_{c}, it follows that

lim_{b→∞} T_{c}/b = 1.  (2.12)

This, of course, shows that

If b is large enough, for T = 2b we have T > T_{c}.  (2.13)

We will show next that

If b is large enough, for T = b/2 we have T < T_{c} and h_{c}^{A}(T) > 1.  (2.14)

The claim about T_{c} in (2.14) is immediate from (2.12). The first step in the derivation of the claim about h_{c}^{A}(T) in (2.14) is a sort of mean-field bound. From (1.7) we have

βh_{c}(T) = max_{t≥0}(bφ_{β}(t) − t)

= max_{t≥0} (b tanh^{−1}(tanh(2/b) tanh(t)) − t) ≥ max_{t≥0} (b tanh(2/b) tanh(t) − t).

Therefore, for large enough b,

βh_{c}(T) ≥ max_{t≥0} (\frac{3}{2} tanh(t) − t) > 0,

where the last inequality is straightforward from the fact that at t = 0, d tanh(t)/dt = 1. The important fact here is that the factor 3/2 multiplying tanh(t) is larger than 1. The reader can see in this estimate the connection with the mean field model. Note that the lower bound obtained is uniform for all large b.

On the other hand, for h = 1,

H_{eff}^{A}(β, h) =H_{eff}^{A}(β, 1) = β + Aφ_{β}(β)
=2/b + 2bφ_{2/b}(2/b) = 2/b + 2b tanh^{−1}(tanh(2/b) tanh(2/b))
=2/b + b O(1/b^{2}).

From the last two displays and the definition (2.11) we conclude that for b large enough.

h_{c}^{A}(T) = max{h ≥ 0 : H_{eff}^{A}(β, h) ≤ βh_{c}(T)} > 1,

which is our claim about h_{c}^{A}(T) in (2.14).

Next we show that

For each b, for T small enough (depending on b) we have h_{c}^{A}(T) < 1. (2.15)

For this we first note that for h = 1,

H_{eff}^{A}(β, h)/β = 1 + A\frac{φ_{β}(β)}{β} → 1 + A = 1 + 2b, as T → 0.

Comparing this behavior with the behavior of h_{c}(T), as given by (1.11) (which, as explained, is derived from an explicit computation which can be found in [Geo]), we obtain the claim (2.15) from the definition (2.11) of h_{c}^{A}(T).

Proposition 1 is a consequence of (2.13), (2.14) and (2.15), and the fact that the phase-coexistence region for the the Ising model on 𝕋_{b,A} is given by (2.10). As a remark, it is worth noting that (2.14) and (2.15) are really the important estimates. In the proof of Proposition 1 we could replace the use of (2.13) with the fact that from Dobrushin’s uniqueness condition, for each b, for T large enough (depending on b), there is a unique Gibbs distribution regardless of the value of h.

3. Some open problems.

1) The graph 𝕋_{b,A} is almost-homogeneous but not homogeneous. Is there any homogeneous graph on which the Ising model presents for some value of the external field and three values of the temperature, T_{1} < T_{2} < T_{3}, phase uniqueness at temperatures T_{1} and T_{3} and phase coexistence at temperature T_{2}?

2) Is there any graph on which at some temperature T_{1} the Ising model presents phase coexistence if and only if h = 0, but at some larger temperature T_{2} > T_{1} it presents phase coexistence also for some non-null value of the external field?

Note added in revision: After this paper was completed, Problem 2 above was solved by M. Salzano, who obtained a graph with the required property. As far as we know, Problem 1 is still open.

Acknowledgements: N.I.T. thanks the warm hospitality of the UCLA Mathematics Department, during the time the research in this paper was done. The work of R.H.S. was partially supported by the N.S.F. (U.S.A.) through grants DMS-9400644 and DMS-9703814, and that of N.I.T. by FAPESP (Brazil), Proc. #96/2769-2.

References

[Geo] H.-O. Georgii, Gibbs measures and phase transitions, Walter de Gruyter, 1988.

[Hig1] Y. Higuchi, Remarks on the limiting Gibbs states on a (d + 1)-tree, Publ. RIMS, Kyoto Univ. 13 (1977), 335–348.

[Hig2] Y. Higuchi, Coexistence of infinite (∗)-clusters II:– Ising percolation in two dimension, Probability Theory and related Fields 97 (1993), 1–33.

[Leb] J. L. Lebowitz, GHS and other inequalities, Communications in Mathematical Physics 35 (1974), 87–92.

[Lig] T. M. Liggett, Interacting Particle Systems, Springer Verlag, 1985.

R. H. Schonmann Mathematics Department University of California at Los Angeles Los Angeles, CA 90095, U.S.A. rhs@math.ucla.edu

N. I. Tanaka Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal 66281-Ag. Cidade de São Paulo São Paulo, SP 05315-970, Brazil nitanaka@ime.usp.br