Ising Models on Static Inhomogeneous Random graphs

arXiv:1203.5823v1 [math-ph] 26 Mar 2012

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS By Kwabena Doku-Amponsah Statistics Department University of Ghana Legon Abstract On a finite inhomogeneous graph model for complex network, we define the Ising model, which is a paradigm model in statistical mechanics. For the ferromagnetic Ising model,we calculate the Thermodynamic limit of pressure per particle. From our results, we compute other physical quantities such as the magnetization and susceptibility, and investigate the critical behaviour of this model. Our calculations use large deviation principles (developed recently) for suitably defined empirical neighbourhood measures on inhomogeneous random graph.

1. Introduction Statistical ferromagnetic models or ferromagnetic ising models on Power-law random graphs such as the configuration model (CM),have received an interestingly increasing attention in the last few years , see e.g. Montanari et al. [2010], Dommers et al. [2010] and the reference there in. For instance in Dembo and Montanari [2010] an explicit expression for the free-energy density and other important physical quantities such the internal energy in thermodynamic limits of n → ∞, where n is number of sites on the graph, were obtained if the degree distribution of the model has finite mean. Their analysis makes use of the local neighbourhood properties of a randomly chosen edge which looks like a homogeneous tree where the offspring have size-biased degree distribution. The Ising model on a tree is simplier to analyse, and many intense scientific research work on it has been taking place. This is because the effective field of a vertex can be expressed in terms of that of its neighbours via the distribution recursion. The relation between the CM and a tree allows for the calculation of the internal energy and upon integrating over all possible values of the inverse temperature parameter we obtain the pressure. However, their technique cannot be applied to study the statistical mechanic on static inhomogeneous random graphs (SIRG) or coloured random graphs, see Penman [1998] or Cannings and Penman [2003] which have the Erdos-Renyi graphs as a special case, because it is not locally-tree like with an asymptotic distribution. Some research, see e.g Agliari et al. [2010], have use the Monte Carlo simulations to investigate the critical properties of the Ising model on Erdos-Renyi graphs. Mathematics Subject Classification : 82B30,60F10, 05C80 Keywords: Thermodynamic limits, free -energy density, relative entropy, coloured random graph, empirical cooperative measures, large deviation principles.

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KWABENA DOKU-AMPONSAH

In this work we use the LDP techniques developed in Biggins and Penman [2003], Doku-Amponsah and Moeters [2010] or Doku-Amponsah [2006] and furthered in Doku-Amponsah [2011] to carry out Thermodynamic analysis of the Ising model defined on SIRG. To be specific, we obtain a thermodynamic limiting result of free-energy density via the random partition function. Using this results, we compute other physical quantities such as the, internal energy, the magnetization and the susceptibility of the system. Our main motivation for looking at these models is the study of the two-populations Ising model. The two-populations Ising model is relevant in many phenomena, ranging from the study of anistropic magnetic materials to social economic models, to biological systems. In the case of the latter , an interesting application concerns cooperativity effects in the context of bacterial chemotaxis, where lasting puzzle, namely how a small change in extrenal concentration of attractant or repellent can cause significant amplification in receptor signal. In theory, models onthe two-population Ising model, although simplified, are able to show that a coupled system of receptors has the capacity to greatly amplify signals. These values in limit, approache the critical coupling energy of an analogous Ising models.See, e.g. Agliari et al. [2010] and the reference there in 2. Materials and Methods  2.1 The Ising model. Let V be a fixed set of n sites, say V = {1, . . . , n} and E ⊂ E := (u, v) ∈ V × V : u < v , where the formal ordering of links or bonds is introduced as a means to simply describe unordered bonds. Let X = {−1, +1} be the spin set and and denote by Gn (X ) the set of all spinned graphs with spin set X and n sites. Given a symmetric function pn : R × R → [0, 1], a continuous function s : X → R and a probability measure ℓ on s(X ) we may define the Inhomogeneous spinned random graph or simply spinned random graph X with n sites as follows:

Assign to each site v ∈ V magnetic ising spin s(η(v)) independently according to the spin law ℓ. Given the spins, we connect any two sites u, v ∈ V , independently of everything else, with a bond probability pn [s(η(u)), s(η(v))] otherwise keep them disconnected.
We always consider X = { s(η(v)) : v ∈ V , E} under the joint law of graph and spin. We shall interpret X as spinned random graph and consider s(η(v)) as the spin of the site v. On a spinned graph X, we define the ferromagnetic Ising Model by the following Boltzmann distributions over X V , n X X s(η(u)) o s(η(u)) s(η(v)) 1 √ exp + µX (η) = ZX (β,B) B(u) B(v) β (u,v)∈E

u∈V

√

where s(η(u)) = βB(η(u))η(u), β ≥ 0 is the inverse temperature, B = {b(x) : x ∈ {−1, 1}n } is the vector external of magnetic fields, η(u) ∈ {−1, 1}, and ZX the random partition function is given by n X X X s(η(u)) o s(η(u)) s(η(v)) √ ZX (β, B) := exp + B(u) B(v) β η∈{−1,1}V

(u,v)∈E

The free-energy density or the pressure per particle is defined by Φn (β, B) =

1 n

log Zn (β, B).

u∈V

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS

3

Our main concern in this paper is the study of the thermodynamic limiting behaviour of the freeenergy. ie The behaviour of φ(β, B) := lim Φn (β, B) n→∞

at different level of temperature. Throughout the rest of the paper we assume that, for large n, X is sparse or near critical case i.e. The bond probabilities satisfy npn [s(x), s(y)] → C[s(x), s(y)], for all x, y ∈ {−1, 1} and C : R × R → [0, ∞) is a continuous symmetric function, which is not identically equal to zero, and also satisfy the following energetic preference condition:   p p p p (2.1) (eβ − 1) C[− βB(−1), − βB(−1)] − C[ βB(1), βB(1)] = 2(B(−1) + B(1)). 2.2 Large deviation principles for spinned random graphs. In this subsection, we review some large deviation results of Doku et al. [2010] or Doku [2006], and extend the joint LDP for empirical measures of coloured random graph to spinned random graphs.i.e. we assume a more general spin law ℓ : R → [0, 1] with all its exponential moments finite and prove an LDP for this model in a topology generated by the total variation norm. To begin, we recall some useful definitions and notations from Doku et al. [2010] or Doku [2006]. A rate function is a non-constant, lower semicontinuous function I from a polish space M into [0, ∞], it is called good if the level sets {I(m) ≤ x} are compact for every x ∈ [0, ∞). A functional M from the set of finite spinned graphs to M is said to satisfy a large deviation principle with rate function I if, for all Borel sets B ⊂ M,  − inf I(m) ≤ lim inf n1 log Pn M (X) ∈ B n→∞ m∈int B  ≤ lim sup n1 log Pn M (X) ∈ B ≤ − inf I(m) , m∈cl B

n→∞

where X under Pn is a spinned random graph with n vertices and int B and cl B refer to the interior, resp. closure, of the set B.

And, for any finite or countable set Y we denote by M(Y) the space of probability measures, and ˜ by M(Y) the space of finite measures on Y, both endowed with the topology generated by the total ˜ ˜ variation norm. For ω ∈ M(Y) we denote by kωk its total mass. Further, if ℓ ∈ M(Y) and ω ≪ ℓ we denote by Z
ω[dy] log ω[dy] H(ω k ℓ) = ℓ[dy] R

the relative entropy of ω with respect to ℓ. We set H(ω k ℓ) = ∞ if ω 6≪ ℓ. Finally, we denote by ˜ ∗ (Y × Y) the subspace of symmetric measures in M(Y ˜ M × Y).

On each spinned graph X = ((s(η(v)) : v ∈ V ), E) with n vertices, we define a probability measure, the empirical spin measure L1 ∈ M(X ), by 1X δs(η(v)) [x], for x ∈ R, L1 [x] := n v∈V

˜ ∗ (R × R), by and a symmetric finite measure, the empirical co-operative measure L2 ∈ M 1 X (δ(s(η(u), s(η(v)) + δ(s(η(v), s(η(u)) )[x, y], for a, b ∈ X . L2 [x, y] := n (u,v)∈E

The total mass

kL2 k

of the empirical pair measure is 2|E|/n.

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KWABENA DOKU-AMPONSAH

Theorem 2.1. Suppose that X is a spinned random graph with spin law ℓ such that n−1 log ℓ(n) → −∞ and link probabilities pn : R × R → [0, 1] satisfying npn [dx, dy] → C[dx, dy], for all x, y ∈ R and some bounded symmetric measure C : R × R → [0, ∞). Then, as n → ∞, the pair (L1 , L2 ) satisfies a large ˜ ∗ (R × R) with good rate function deviation principle in M(R) × M i
1h (2.2) I(ω, ̟) = H(ω k µ) + H ̟ k Cω ⊗ ω + kCω ⊗ ωk − k̟k , 2 ˜ where the measure Cω ⊗ ω ∈ M(R × R) is defined by Cω ⊗ ω[dx, dy] = C(x, y)ω[dx]ω[dy]. 2.3 Exponential Change-of-Measure. Given a bounded function f˜: X → R and a symmetric bounded function g˜ : X × X → R, we define the constant Uf˜ by Z ˜ Uf˜ = log ef [x] ℓ[dx], R

˜ n : R × R → R by and the function h

˜ n [x, y] = log h

h

1 − pn [dx, dy] + pn [dx, dy]eg˜[x,y]


−n i

,

(2.3)

for a, b ∈ X . We use f˜ and g˜ to define (for sufficiently large n) a new coloured random graph as follows: • To the n labelled sites in V we assign spins from s(X ) independently and identically according to the spin law ℓ defined by ˜ ˜ ℓ[dx] = ef [x]−Uf˜ℓ[dx].

• Given any two sites u, v ∈ V, with u carrying spin a and v carrying spin b, we connect site u to site v with probability p˜n [dx, dy] =

pn [dx, dy]eg˜[x,y] . 1 − pn [dx, dy] + pn [dx, dy])eg˜[x,y]

˜ We observe that ℓ˜ is a probability measure and that P ˜ is We denote the transformed law by P. absolutely continuous with respect to P as, for any spinned graph X = ((s(η(v)) : v ∈ V ), E), Y ˜ dP (X) = dP u∈V Y =

˜ ℓ[ds(η(u)] ℓ[ds(η(u)]

=

Y

p˜n [ds(η(u)),ds(η(v))] pn [ds(η(u)),ds(η(v))]

e

Y

1−˜ pn [ds(η(u)),ds(η(v))] 1−pn [ds(η(u)),ds(η(v))]

(u,v)6∈E

(u,v)∈E f˜[s(η(u)]−Uf˜

u∈V

where

p˜n [ds(η(u)),ds(η(v))] pn [ds(η(u)),ds(η(v))]

(u,v)∈E ˜ ℓ[ds(η(u)] ℓ[ds(η(u)]

u∈V

Y

Y

Y

×

n−npn [ds(η(u)),ds(η(v))] n−n˜ pn [ds(η(u)),ds(η(v))]

e

1 n

˜ n [s(η(u)),s(η(v))] h

(u,v)∈E

(u,v)∈E


˜ n i − h 1 L1 , h ˜ = exp nhL , f˜ − Uf˜i + nh 21 L2 , g˜i + nh 12 L1 ⊗ L1 , h 2 ∆ ni , 1

L1∆ =

1 n

X

δ(X(u),X(u)) .

u∈V

We write hf, ωi :=

R

R f [x]ω[dx]

n−n˜ pn [ds(η(u)),ds(η(v))] n−npn [ds(η(u)),ds(η(v))]

(u,v)∈E

Y

eg˜[s(η(u)),s(η(v))]

Y

and hg, ̟i :=

Z

R×R

g[x, y]̟[dx, dy].

(2.4)

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS

5

Lemma 2.2 (Euler’s lemma). If npn [dx, dy] → C[dx, dy] for every x, y ∈ R, then  n lim 1 + αpn [dx, dy] = eαC[dx,dy] , for all α, x, y ∈ R and for α ∈ R. n→∞

Proof.

Observe that, for any ε > 0 and for large n we have n  n   ≤ 1 + αp [dx, dy] ≤ 1+ 1 + αC[dx,dy]−ε n n

αC[dx,dy]+ε n

n

(2.5)

,

by the pointwise convergence. Hence by the sandwich theorem and Euler’s formula we get (2.5). Lemma 2.3 (Exponential tightness). For every θ > 0 there exists N ∈ N such that n o lim sup n1 log P |E| > nN ≤ −θ. n→∞

Proof. Let c > supx,y∈R C[dx, dy] > 0. By a simple coupling argument we can define, for all ˜ with spin law ℓ and connection probability c , sufficiently large n, a new coloured random graph X n ˜ Let |E| ˜ be the number of edges of X. ˜ Using such that any edge present in X is also present in X. Chebyshev’s inequality, the binomial formula, and Lemma 2.2, we have that n(n−1)

    2 n o X  |E| c n(n−1)/2−k c k ˜ k n(n − 1)/2 −nl −nl ˜ 1− e P |E| ≥ nl ≤ e E e =e n n k k=0   c c n(n−1)/2 = e−nl 1 − + e ≤ e−nl enc(e−1+o(1)) . n n Now given θ > 0 choose N ∈ N such that N > θ + c(e − 1) and observe that, for sufficiently large n,   ˜ ≥ nN ≤ e−nθ , P |E| ≥ nN ≤ P |E|

which implies the statement.

Lemma 2.4 (Exponential tightness). For every θ > 0 there exists Kθ ⊂ M(R) such that n o lim sup n1 log P L1 6∈ Kθ ≤ −θ. n→∞

Proof.

Let 1 < l ∈ N and choose k(l) ∈ N, large enough, such that ℓ[el

2 1l

{s>k(l)}

] ≤ 2l

Then, using exponential Chebyschev’s inequality, we have that o o n 2P nZ
n 2 P L1 (dx) ≥ 1l ≤ e−nl E el u∈V 1l{s(η(u))≥k(l)} ≤ e−nl ℓ[el 1l{s>k(l)} ] ≤ e−n(l−log 2) . {x>k(l)}

Now we fix θ > 0, choose M > θ + log 2, define the set ΓM by n

ΓM := ω :

Z

{x≥k(l)}

ω(dx) > 1l , for all l ≥ M

o

As {x ≤ k(l)} ⊂ R is compact, the set ΓM is pre-compact in the weak topology, by Prohorov’s criterion. As o n 1 exp(−n[M − log 2]), P L1 6∈ ΓM ≤ 1 − e−1

6

KWABENA DOKU-AMPONSAH

we conclude that

n o 1 log P L1 6∈ Kθ ≤ −θ, n→∞ n as required for the proof. lim sup

for the closure Kθ of ΓM

2.4 Proof of the upper bound in Theorem 3.4. We denote by C1 the space of bounded functions on R and by C2 the space of bounded symmetric functions on R × R, and define Z Z nZ o
ˆ ̟) = sup I(ω, f [x]−Uf ω[dx]+ 12 g[x, y]̟[dx, dy]+ 12 (1−eg[x,y] )C[dx, dy]ω[dx]ω[dy] . f ∈C1 g∈C2

R

R×R

R×R

˜ ∗ (R × R), Lemma 2.5. For each closed set F ⊂ M(R) × M  lim sup n1 log P (L1 , L2 ) ∈ F ≤ − inf n→∞

Proof.

(ω,̟)∈F

ˆ ̟). I(ω,

First let f˜ ∈ C1 and g˜ ∈ C2 be arbitrary. Define β˜ : R × R → R by ˜ y] = (1 − eg˜[x,y] )C[dx, dy]. β[x,

˜ y] = limn→∞ h ˜ n [x, y] for all a, b ∈ R, recalling the definition of h ˜n Observe that, by Lemma 2.2, β[x, from (2.3). Hence, by (2.4), for sufficiently large n, Z o n 1 1 1 1 ˜ ˜ supx∈R |β(x,x)| ˜ = E enhL1 ,f˜−Uf˜i+nh 2 L2 ,˜gi+nh 2 L1 ⊗L1 ,h˜ n i , e ≥ eh 2 L∆ , hn i dP

where L1∆ =

1 n

P

u∈V

δ(X(u),X(u)) and therefore, n o 1 2 1 1 1 ˜ 1 ˜ lim sup n1 log E enhL ,f −Uf˜i+nh 2 L ,˜gi+nh 2 L ⊗L ,hni ≤ 0.

(2.6)

n→∞

ˆ ̟), ε−1 }−ε. Suppose that (ω, ̟) ∈ F and observe that I(ω, ˆ ̟) > Given ε > 0 let Iˆε (ω, ̟) = min{I(ω, ˜ ˆ Iε (ω, ̟). We now fix f ∈ C1 and g˜ ∈ C2 such that ˜ ω ⊗ ωi ≥ Iˆε (ω, ̟). g , ̟i + 12 hβ, hf˜ − Uf˜, ωi + 12 h˜

2 and B 1 of ̟, ω such that As f˜, g˜ are bounded functions, there exist open neighbourhoods B̟ ω  1 1 ˜ ˜ ˆ inf hf − Uf˜, ω ˜ i + 2 h˜ g , ̟i ˜ + 2 hβ, ω ˜ ⊗ω ˜ i ≥ Iε (ω, ̟) − ε. 1 ω∈B ˜ ω 2 ̟∈B ˜ ̟

Using Chebyshev’s inequality and (2.6) we have that  2 lim sup n1 log P (L1 , L2 ) ∈ Bω1 × B̟ n→∞ o n 1 1 1 2 1 ˜ 1 ˜ ≤ lim sup n1 log E enhL ,f −Uf˜i+nh 2 L ,˜gi+nh 2 L ⊗L ,hn i − Iˆε (ω, ̟) + ε

(2.7)

n→∞

≤ −Iˆε (ω, ̟) + ε.

Now we use Lemma 2.3 with θ = ε−1 , to choose N (ε) ∈ N and Kε such that o n o n lim sup n1 log P |E| > nN (ε) ≤ −ε−1 and lim sup n1 log P L1 6∈ Kε ≤ −ε−1 n→∞

n→∞

(2.8)

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS

7

For this N (ε) and Kε we define the set Γε by n o c ˜ Γε = (ω, ̟) ∈ M(R) × M∗ (R × R) : ω ∈ Kε , k̟k ≤ 2N (ε) , and recall that kL2 k = 2|E|/n. The set Γε ∩ F is compact and therefore may be covered by finitely 2 , r = 1, . . . , m with (ω , ̟ ) ∈ F for r = 1, . . . , m. Consequently, many sets Bω1 r × B̟ r r r m    X 2 P (L1 , L2 ) ∈ Bω1 r × B̟ + P (L1 , L2 ) 6∈ Γε . P (L1 , L2 ) ∈ F ≤ r r=1

We may now use (2.7) and (2.8) to obtain, for all sufficiently small ε > 0,     m 2 ∨ (−ε)−1 lim sup n1 log P (L1 , L2 ) ∈ F ≤ max lim sup n1 log P (L1 , L2 ) ∈ Bω1 r × B̟ r r=1 n→∞ n→∞   ≤ − inf Iˆε (ω, ̟) + ε ∨ (−ε)−1 . (ω,̟)∈F

Taking ε ↓ 0 we get the desired statement. Next, we express the rate function in term of relative entropies, see for example Dembo et al. [1998, (2.15)], and consequently show that it is a good rate function. Recall the definition of the function I from Theorem 3.4. Lemma 2.6. ˆ ̟) = I(ω, ̟), for any (ω, ̟) ∈ M(R) × M ˜ ∗ (R × R), (i) I(ω, (ii) I is a good rate function and (iii) HC (̟ k ω) ≥ 0 with equality if and only if ̟ = Cω ⊗ ω. Proof. (i) Suppose that ̟ 6≪ Cω ⊗ ω. Then, there exists a0 , b0 ∈ R with Cω ⊗ ω(a0 , b0 ) = 0 and ̟(a0 , b0 ) > 0. Define gˆ : R × R → R by   gˆ[x, y] = log K(1l(a0 ,b0 ) [x, y] + 1l(b0 ,a0 ) [x, y]) + 1 , for a, b ∈ R and K > 0.

For this choice of gˆ and f = 0 we have Z Z Z
1 f [x] − Uf ω(dx) + g ˆ [x, y]̟[dx, dy] + 2 R×R

R

≥

1 2

log(K + 1)̟(a0 , b0 ) → ∞,

− egˆ[x,y] )C[dx, dy]ω[dx]ω[dy]

for K ↑ ∞.

Now suppose that ̟ ≪ Cω ⊗ ω. We have Z  o nZ  ˆ f [x] − log ef [x] ℓ[dx] ω[dx] I(ω, ̟) = sup f ∈C1 R R Z nZ + 21 C[dx, dy]ω[dx]ω[dy] + 21 sup R×R

R×R

1 2 (1

g∈C2

R×R

g[x, y]̟[dx, dy] −

Z

R×R

o eg[x,y] C[dx, dy]ω[dx]ω[dy] .

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KWABENA DOKU-AMPONSAH

By the variational characterization of relative entropy, the first term equals H(ω k ℓ). By the substithe last term equals tution h = eg Cω⊗ω ̟ Z h   i ̟[dx, dy] log h[x, y] sup − h[x, y] ̟[dx, dy] C[dx, dy]ω[dx]ω[dy] h∈C2 R×R h≥0

= sup h∈C2 h≥0

X

a,b∈R

 X
log h[x, y] − h[x, y] ̟[dx, dy] + log a,b∈R

 ̟[dx, dy] ̟[dx, dy] C[dx, dy]ω[dx]ω[dy]

= −k̟k + H(̟ k Cω ⊗ ω),

ˆ ̟) = I(ω, ̟). where we have used supx>0 log x − x = −1 in the last step. This yields that I(ω,

(ii) Recall from (2.2) that


I(ω, ̟) = H(ω k ℓ) + 12 H ̟ k Cω ⊗ ω + 21 kCω ⊗ ωk −

1 2

k̟k.

All summands are continuous in ω, ̟ and thus I is a rate function. Moreover, for all α < ∞, the level ˜ ∗ (R × R) : HC (̟ k ω) ≤ α} and sets {I ≤ α} are contained in the bounded set {(ω, ̟) ∈ M(R) × M are therefore compact. Consequently, I is a good rate function. (iii) Consider the nonnegative function ξ(x) = x log x − x + 1, for x > 0, ξ(0) = 1, which has its only root in x = 1. Note that  R d̟ ≥ 0 exists, ξ ◦ g dCω ⊗ ω if g := dCω⊗ω (2.9) HC (̟ k ω) = ∞ otherwise.

d̟ Hence HC (̟ k ω) ≥ 0, and, if ̟ = Cω ⊗ ω, then ξ( dCω⊗ω ) = ξ(1) = 0 and so HC (Cω ⊗ ω k ω) = 0. Conversely, if HC (̟ k ω) = 0, then ̟[dx, dy] > 0 implies Cω ⊗ ω[x, y] > 0, which then implies ξ ◦ g[x, y] = 0 and further g[x, y] = 1. Hence ̟ = Cω ⊗ ω, which completes the proof of (iii).

2.5 Proof of the lower bound in Theorem 3.4. ˜ ∗ (R × R), Lemma 2.7. For every open set O ⊂ M(R) × M n o lim inf n1 log P (L1 , L2 ) ∈ O ≥ − inf n→∞

Proof.

(ω,̟)∈O

I(ω, ̟).

Suppose (ω, ̟) ∈ O, with ̟ ≪ Cω ⊗ ω. Define f˜ω : R → R by ( log ω[dx] ℓ[dx] , if ω[dx] > 0, f˜ω (a) = 0, otherwise.

and g˜̟ : R × R → R by g˜̟ [x, y] =

(

log 0,

̟[dx,dy] C[dx,dy]ω[dx]ω[dy] ,

if ̟[dx, dy] > 0, otherwise.

˜ ̟,n [x, y], for In addition, we let β˜̟ [x, y] = C[dx, dy](1 − eg˜̟ [x,y] ) and note that β˜̟ [x, y] = limn→∞ h all a, b ∈ R where h
i ˜ ̟,n [x, y] = log 1 − pn [dx, dy] + pn [dx, dy]eg˜̟ [x,y] −n . h 2 open neighbourhoods of ω, ̟, such that B 1 × B 2 ⊂ O and for all (˜ 2 Choose Bω1 , B̟ ω , ̟) ˜ ∈ Bω1 × B̟ ω ̟

g̟ , ̟i + hf˜ω , ωi + 12 h˜

1 2

hβ˜̟ , ω ⊗ ωi − ε ≤ hf˜ω , ω ˜ i + 12 h˜ g̟ , ̟i ˜ + 21 hβ˜̟ , ω ˜ ⊗ω ˜ i.

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS

9

˜ the probability measure obtained by transforming P using the functions f˜ω , g˜̟ . Note We now use P, that the spin law in the transformed measure is now ω, and the connection probabilities p˜n [dx, dy] ˜ satisfy n p˜n [dx, dy] → ̟[dx, dy]/(ω[dx]ω[dy]) =: C[dx, dy], as n → ∞. Using (2.4), we obtain o n o n ˜ dP (X)1l{(L1 ,L2 )∈B 1 ×B 2 } P (L1 , L2 ) ∈ O ≥ E ˜ ω ̟ dP o nY Y − 1 h˜ [s(η(u)),s(η(v))] Y ˜ ˜ 1l{(L1 ,L2 )∈Bω1 ×B̟ e n ̟,n e−˜g̟ [s(η(u)),s(η(v))] =E e−fω [s(η(u))] 2 } u∈V

(u,v)∈E

(u,v)∈E

n o ˜ e−nhL1 ,f˜ω i−n 21 hL2 ,˜g̟ i−n 12 hL1 ⊗L1 ,˜g̟ i+ 12 hL1∆ ,h˜ ̟,ni × 1l{(L1 ,L2 )∈B 1 ×B 2 } =E ω ̟ o n
˜ (L1 , L2 ) ∈ B 1 × B 2 , g̟ , ̟i − n 21 hβ˜̟ , ω ⊗ ωi + m − nε × P ≥ exp − nhf˜ω , ωi − n 21 h˜ ω ̟

˜ x]. Therefore, by (2.5), we have where m := 0 ∧ inf x∈R β[x, n o lim inf n1 log P (L1 , L2 ) ∈ O n→∞

≥ −hf˜ω , ωi −

1 2

h˜ g̟ , ̟i − 21 hβ˜̟ , ω ⊗ ωi − ε + lim inf

1 n→∞ n

The result follows once we prove that 1 n→∞ n

lim inf

o n ˜ (L1 , L2 ) ∈ B 1 × B 2 . log P ̟ ω

o n ˜ (L1 , L2 ) ∈ B 1 × B 2 = 0. log P ω ̟

(2.10)

˜ to prove (2.10). Then we obtain We use the upper bound (but now with the law P replaced by P)  ˜ (L1 , L2 ) ∈ (B 1 × B 2 )c ≤ − inf I(˜ ˜ ω , ̟), ˜ lim sup n1 log P ω ̟ (˜ ω ,̟)∈ ˜ F˜

n→∞

2 )c and where F˜ = (Bω1 × B̟
˜ ω , ̟) I(˜ ˜ := H(˜ ω k ω) + 12 H ̟ ˜ k C˜ ω ˜ ⊗ω ˜ +

1 2

kC˜ ω ˜ ⊗ω ˜ k − 21 k̟k. ˜

It therefore suffices to show that the infimum is positive. Suppose for contradiction that there exists ˜ ωn , ̟ a sequence (˜ ωn , ̟ ˜ n ) ∈ F˜ with I(˜ ˜ n ) ↓ 0. Then, because I˜ is a good rate function and its level sets ˜ ω , ̟), are compact, and by lower semicontinuity of the mapping (˜ ω , ̟) ˜ 7→ I(˜ ˜ we can construct a limit ˜ ˜ point (˜ ω , ̟) ˜ ∈ F with I(˜ ω , ̟) ˜ = 0 . By Lemma 2.6 this implies H(˜ ω k ω) = 0 and HC (̟ ˜ kω ˜ ) = 0, hence ω ˜ = ω, and ̟ ˜ = C˜ ω ˜ ⊗ω ˜ = ̟ contradicting (˜ ω , ̟) ˜ ∈ F˜ . 3. Results and Discussion.

3.1 Thermodynamic Limits Theorem 3.1. Suppose that X is a spinned random graph with bond probabilities pn : X × X → [0, 1] satisfying npn [s(x), s(y)] → C[s(x), s(y)], for all x, y ∈ {−1, 1} and some continuous symmetric function C : R × R → [0, ∞) that satisfies (2.1) above. Then, the asymptotic pressure per particle is o n p p p p φ(β, B) = 41 (eβ − 1)C[ βB(1), βB(1)] + (e−β − 1)C[− βB(−1), βB(1)] + (3B(1)−B(−1)) 4

Corollary 3.2. Suppose that X is a spinned random graph with bond probabilities pn : X × X → [0, 1] satisfying npn [s(x), s(y)] → C[s(x), s(y)], for all x, y ∈ {−1, 1} and some twice differential symmetric function C : R × R → [0, ∞) that satisfies (2.1) above. Then, for all β ≥ 0, each of the following statements holds a.s.:

10

KWABENA DOKU-AMPONSAH

(i) Asymptotic total magnetization per site is given by dC

dC

−1,1 1,1 −β β −β β 1 (e C1,1 − e C−1,1 ) + (e − 1) dβ + (e − 1) dβ M (β, B) = − dC dC 2 eβ (C1,1 − C−1,−1 ) + (eβ − 1)( 1,1 − −1,−1 )

dβ

dβ

(ii) Asymptotic internal energy is given by i 1h dC dC−1,1 + (eβ − 1) −1,−1 U (β, B) = − C−1,−1 eβ − C−1,1 e−β + (e−β − 1) dβ dβ 4 (iii) Asymptotic specific heat is given by dC−1,1 dC−1,1 dC−1,−1 β dC−1,−1  dC−1,−1 β2  H(β, B) = −2e−β +(e−β −1) +(eβ −1) +e C−1,−1 eβ +C−1,1 eβ +eβ 4 dβ dβ dβ dβ dβ Remark 1 :Asymptotic susceptibility of the system is given by S(β, B) :=

∂2φ ∂2φ + ∂B(−1)2 ∂B(1)2

Theorem 3.3. Let βc be the solution of equation C−1,1 = 21 C1,1 + 21 C−1,−1 . Then, β = βc , separates the non magnetized phase ( diamagnetic state) from the magnetized one ( paramagnetic state).Further, for B(1) + B(−1) = 0, the continuous function B → φ(β, B) exhibits a discontinuous derivative.i.e. the susceptibility becomes infinite at β = βc in B(1) + B(−1) = 0. Corollary 3.4. Let λ be the asymptotic average number of connectivity of erdos-renyi graphs. Then, there exists βc (λ) such the ferromagnetic Ising model on the erdos-renyi graphs exhibit (i) diamagnetic properties if 0 < β < βc (λ) (ii) paramagnetic properties if βc (λ) < β ≤ ∞ (iii) zero-magnetization properties if β = 0 or β = βc (λ). 3.2 Proof of Thermodynamics Limits. We kick start the proof of our Thermodynamics limit results by stating an important Lemma (Varadhan’s Lemma, see Dembo et al. [1998, Theorem 4.3.1] ), which is a key step in establishing our first result Theorem 3.1, without proof. Lemma 3.5 (Varadhan). Suppose the functional Mn from the space of finite spinned graphs to M satisfies the LDP withh good rate function I : M → [0, ∞] and let Ψ : M → R be any continuous function. Assume further the following moment condition for some λ > 1,   lim sup n1 log E enλΨ(Mn [X]) < ∞. n→∞

Then,

1 n→∞ n

lim

   log E enλΨ(Mn [X]) = sup Ψ(m) − I(m) . m∈M

Next we provide annealed asymptotics of the random partition fuction for the ferromagnetic Ising model on spinned random graphs, as the graph size goes to infinity. Lemma 3.6. Suppose that X is a spinned random graph with spin law ℓ and link probabilities pn : X × X → [0, 1] satisfying npn [s(a), s(b)] → C[s(a), s(b)], for all a, b ∈ {−1, 1} and some continuous symmetric function C : R × R → [0, ∞) with supx,y∈R C[x, y] < ∞ and satisfies (2.1) above. Then, n o p p p p lim n1 log E[ZX (β, B)] = log 2 + 14 (eβ − 1)C[ βB(1), βB(1)] + (e−β − 1)C[− βB(−1), βB(1)] n→∞

+

(3B(1)−B(−1)) 4

ISING MODELS ON STATIC INHOMOGENEOUS RANDOM GRAPHS

11

Proof. Recall the random partition function of the Ising model on the spinned graph from section 2 and write it as integral of some function with respect to our empirical measures: Z h n Z oi s(x) s(x) 2 s(x) 1 n n √ L [ds(x)] L [ds(x), ds(y)] + n E[ZX (β, B)] := 2 E exp 2 (3.1) B(x) B(y) β

Now using the (Varadhan) Lemma 3.5 and Theorem 3.4 we obtain, 1 n→∞ n

lim

log E[ZX (β, B)] = log 2 + sup

n Z 1 2

s(x) s(x) B(x) B(y) ̟[ds(x),

ds(y)] +

Z

s(x) √ ω[ds(x)] β

− I(ω, ̟)

o : ω ∈ X ({s(−1), s(1)}), ̟ ∈ M∗ ({s(−1), s(1)} × {s(−1), s(1)}) n
= sup β2 ̟(∆) − ̟(∆c ) + B(1)x − B(−1)(1 − x) − x log(x) − (1 − x) log(1 − x)
o − 12 H(̟ k ωx ) + C1,1 x + C−1,−1 (1 − x) + 2C−1,1 x(1 − x) − k̟k , (3.2) where ∆ is the diagonal in {s(−1), s(1)} × {s(−1), s(1)}, and the supremum is over all x ∈ [0, 1] and ˜ ∗ ({s(−1), s(1)} × {s(−1), s(1)}) is ̟ ∈ M∗ ({s(−1), s(1)} × {s(−1), s(1)}), and the measure ωx ∈ M defined by ωx [s(i), s(j)] = Ci,j x(2+i+j)/2 (1 − x)(2−i−j)/2 for i, j ∈ {−1, 1} . We take the partial derivatives of (3.2) with respect to ̟, and set our results to zero to get, ̟[s(1), s(1)] = eβ ωx [s(1), s(1)], ̟[s(−1), s(−1)] = eβ ωx [s(−1), s(−1)] ̟[s(1), s(−1)] = e−β ωx [s(1), s(−1)] ̟[s(−1), s(1)] = e−β ωx [s(−1), s(1)]. lim 1 n→∞ n

log E[ZX (β, B)] n β β β β −β ωx [s(−1), s(1)] + B(1)x − B(−1)(1 − x) = sup 2 e ωx [s(1), s(1)] + 2 e ωx [s(−1), s(−1) − βe x∈[0,1]

− x log(x) − (1 − x) log(1 − x) − β2 eβ ωx [s(1), s(1)] − β2 eβ ωx [s(−1), s(−1)] + βe−β ωx [s(−1), s(1)] − 12 C1,1 x o − 21 C−1,−1 (1 − x) − C−1,1 x(1 − x) + 12 eβ ωx [s(1), s(1)] + 21 eβ ωx [s(−1), s(−1)] + e−β ωx [s(−1), s(1)] n − x log(x) − (1 − x) log(1 − x) − 12 C1,1 x2 − 21 C−1,−1 (1 − x)2 + B(1)x − B(−1)(1 − x) = sup x∈[0,1]

o − C−1,1 x(1 − x) + 12 eβ C1,1 x2 + 12 eβ C−1,−1 (1 − x)2 + e−β C−1,1 x(1 − x) n o = sup λ1 (x) + λ2 (x) x∈[0,1]

where λ1 (x) = −x log(x) − (1 − x) log(1 − x) λ2 (x) = a1 x + a2 (1 − x)2 + a3 x(1 − x) + B(1)x − B(−1)(1 − x) a1 = 12 C1,1 (eβ − 1), a2 = 12 C−1,−1 (eβ − 1) and a3 = C−1,1 (e−β − 1). 2

(3.3)

12

KWABENA DOKU-AMPONSAH

Elementary calculus shows that the grobal maxima of the functions λ1 and λ2 are attained at the value x = 12 , when a2 − a1 = B(1) + B(−1). Therefore we have log E[ZX (β, B)] = log 2 + 14 (2a1 + a3 ) + 41 (3B(1) − B(−1)) o n p p p p = log 2 + 14 (eβ − 1)C[ βB(1), βB(1)] + (e−β − 1)C[− βB(−1), βB(1)] +

lim 1 n→∞ n

Lemma 3.7. Φn (β, B) − log 2 = lim n→∞ n→∞ n4 lim

1 n

3B(1)−B(−1)) 4

log E[ZX (β, B)/2n ] with prob. 1 n4

Proof. Φn (β, B) − log 2 E − n4

for some A = A(β, B, ε). Now

1 n

log E[ZX (β, B)/2n ] ≤ n4

∞ X Φn (β, B) − log 2 E − n4

1 n

n=1

β 4n2

+

B n3

+

φ(β,B)−log 2+ε n4

≤

A n2

∞ log E[ZX (β, B)/2n ] X A ≤