Approximating partition functions of the two-state spin system
Information Processing Letters 111 (2011) 702–710
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Information Processing Letters www.elsevier.com/locate/ipl
Approximating partition functions of the two-state spin system ✩ Jinshan Zhang a,∗ , Heng Liang b , Fengshan Bai b a b
Department of Computer Science, University of Liverpool, UK Department of Mathematical Sciences, Tsinghua University, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 10 April 2010 Received in revised form 26 April 2011 Accepted 26 April 2011 Available online 29 April 2011 Communicated by A. Tarlecki Keywords: Approximation algorithms Two-state spin system Ising model Gibbs measure Uniqueness of Gibbs measure FPTAS Strong correlation decay
Two-state spin system is a classical topic in statistical physics. We consider the problem of computing the partition function of the system on a bounded degree graph. Based on the self-avoiding tree, we prove the system exhibits strong correlation decay under the condition that the absolute value of inverse temperature is small. Due to strong correlation decay property, an FPTAS for the partition function is presented and uniqueness of Gibbs measure of the two-state spin system on a bounded degree infinite graph is proved, under the same condition. This condition is sharp for Ising model. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Spin model with p states is a classical mathematical model in statistical physics. Such models describe and explain the behavior of ferromagnets, lattice gas and certain other phenomena of statistical physics. In this paper, we focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field). In statistical mechanics, the partition function is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. However, partition functions are normally hard to compute, even for the two-state spin system [5]. Hence approximate algorithms, which can give a reasonable estimation within acceptable computing time, attract more attention. Markov Chain Monte Carlo methods [6,10] are the existing powerful approach for approximating the partition functions. Exploiting the structure property of Gibbs measure, Weitz [13] and Bandyopadhyay and Gamarnik [1] introduce new deterministic algorithms for counting the number of independent sets and colorings. The key point of this method is to establish the strong spatial mixing property, which is also known as strong correlation decay, on a self -avoiding tree or a computation tree in a dynamic programming type recursion. In [13], Weitz proves the strong correlation decay for hard-core model on bounded degree trees and establishes the result for general graph using the self -avoiding tree technique. Weitz’s approach is applicable to some kinds of statistical systems, such as Ising model [11,8], hard-core model [13] and coloring model [7].
✩
*
Supported by National Science Foundation of China, no. 10871115. Corresponding author. E-mail address: [email protected] (J. Zhang).
0020-0190/$ – see front matter doi:10.1016/j.ipl.2011.04.012
© 2011
Elsevier B.V. All rights reserved.
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
703
It is natural to ask whether the more general two-state spin system exhibits strong correlation decay. In this paper, our results give a positive answer to this question. We show that for arbitrary external field, the Gibbs measure exhibits strong correlation decay on a bounded degree tree under the condition (d − 1) tanh J < 1. Here d is the maximum degree of the graph and J is a system parameter, which is also known as inverse temperature. Our result extends Dobrushion’s condition for the uniqueness of Gibbs measure of antiferromagnetic and ferromagnetic Ising models [3,4,12]. This condition is sharp for Ising model on a bounded degree infinite graph. A fully polynomial time approximation scheme (FPTAS) for partition functions of two-state spin system on a bounded degree graph is also constructed in this paper. This is natural and reasonable when the strong correlation decay holds. Jerrum and Sinclair [6] provide an FPRAS for ferromagnetic Ising model on graphs with uniform positive inverse temperature and identical external field for all the vertices. Their results neither include the case where different vertices have different external field, nor are applicable to antiferromagnetic Ising model. Very recently Dembo and Montanari propose an explicit formula for partition function of ferromagnetic Ising model with external field on locally tree-like graphs, which still does not include the antiferromagnetic case [2]. The remainder of the paper has the following structure. In Section 2, we present some preliminary definitions. We go on to prove the main theorem in Section 3. Section 4 is devoted to propose an FPTAS for the partition functions. Discussion and conclusions are given in Section 5. 2. Notations and definitions Let G = ( V , E ) be a finite graph with vertex set V = {1, 2, . . . , n} and edge set E. If there is no specific statement, all the graphs considered in this paper are connected. Let d G (u , v ) denote the distance between u and v, for any u, v ∈ V . A path v 1 → v 2 → · · · is called a self-avoiding path if v i = v j for all i = j. The distance between a vertex v ∈ V and a subset Λ ⊂ V is defined by
d G ( v , Λ) = min d G ( v , u ): u ∈ Λ . The set of vertices with distance l to the vertex v is denoted by
S (G , v , l) = u: d G ( v , u ) = l . The set of vertices which are no more than l away from v is denoted by
V (G , v , l) = u: d G ( v , u ) l . Let δ v denote the degree of v in G and (G ) = max{δ v : v ∈ V }. A \ B denotes the set obtained by removing elements of B from the set A. Let all the vertices in graph G = ( V , E ) be numbered, where V and E are respectively vertex set and edge set of G. We define the partial order on E, where (i , j ) > (k, l) if and only if (i , j ) and (k, l) share a common vertex and i + j > k + l. In two-state spin system on G, each vertex i ∈ V is associated with a random variable X i on Ω = {±1} (briefly we use + and − to denote +1 and −1 respectively). Definition 1. The Gibbs measure of two-state spin system on G is defined by the joint distribution of X = { X 1 , X 2 , . . . , X n }
PG(X = σ ) =
1 Z (G )
e
(i , j )∈ E
βi j (σi ,σ j )+
i∈V
h i (σ i )
,
(1)
where h i is a map Ω → R and βi j is a map Ω 2 → R. Z (G ) is a normalizing constant, which is called the partition function of the system. For any vertex i (i = 1, 2, . . . , n), the probability P G ( X i = σi ) is called the marginal probability that the vertex i is assigned σi . Note that the Gibbs measure satisfies σ ∈Ω n P G ( X = σ ) = 1. We define βi j (a, b) = β ji (b, a). For any Λ ⊆ V , σΛ denotes the set {σi , i ∈ Λ}. With a little abuse of notation, σΛ also denotes the condition or configuration with fixed σi for any i ∈ Λ. Let Z (G , Φ) denote the partition function under the condition Φ . For example, Z (G , X 1 = +) represents the partition function under the condition that the vertex 1 is fixed to be +. The partition function can be represented in terms of the marginal probabilities of Gibbs measure, which provides a powerful tool for counting [1,13]. Local Markov property is in the nature of Gibbs measure, which is useful in our analysis. Proposition 1 (Local Markov Property). Suppose there is a two-state spin system on G = ( V , E ) and P is the Gibbs measure of the system. Let Λ ⊂ V and ∂Λ = {i ∈ V : (i , j ) ∈ E , i ∈ / Λ, j ∈ Λ}. Denote G to be the induced graph by Λ ∪ ∂Λ in G. It is well known that Gibbs measure P satisfies the local Markov property which is P G (·|σ V \Λ ) = P G (·|σ∂Λ ), for any Λ ⊂ V . A self-avoiding walk (SAW) is a sequence of moves (on a graph) which does not visit the same point more than once. The following gives an important tool in proving our results. It is introduced in [13].
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Fig. 1. The graph with one vertex assigned + (right) and its corresponding self-avoiding tree T saw(1) (left).
Definition 2 (Self-Avoiding Tree). The self-avoiding tree T saw( v ) (G ) corresponding to the vertex v of G is a tree with root v. The tree is generated through the self-avoiding walks originating at v, and can be denoted by T saw( v ) for simplicity. A vertex closing a cycle is included as a leaf of the tree and is assigned to be +, if the edge ending the cycle is larger than the edge starting the cycle, and − otherwise. Remark. Given any configuration σΛ of G, Λ ⊂ V , the self-avoiding tree is constructed in the same way as the above procedure except that the vertex, which is a copy of the vertex i in Λ, is fixed to the same spin σi as i and the subtree below it is not constructed due to the local Markov property, see Fig. 1 for example, where vertex 5 is fixed + in G. The strong correlation decay property implies that the marginal probability that the root is assigned + is asymptotically independent of the configuration on the leaves far below. Definition 3 (Strong Correlation Decay). The Gibbs distribution of two-state spin system on G exhibits strong correlation decay if and only if for any vertex v ∈ V , subset Λ ⊂ V , any two configurations σΛ and ηΛ on Λ, there exist positive numbers a, b independent of n such that
log P G ( X v = +|σΛ ) − log P G ( X v = +|ηΛ ) f (t ),
where t = d G ( v , { v ∈ Λ:
σ v = η v }), and decay function is given by f (t ) = ae −bt .
Definition 4 (FPTAS). For a problem to compute a (numerical) value M ( I ) for any input I , an approximation algorithm is called a fully polynomial time approximation scheme (FPTAS) for this problem if and only if for any > 0 and input I , the ¯ ( I , ) satisfying algorithm takes polynomial time in size of I and −1 to output a (numerical) value M
e −
¯ (I , ) M M(I )
e .
3. Strong correlation decay Consider a special two-state spin system with βi j (σi , σ j ) = J i j σi σ j , h i = B i σi for all the edges (i , j ) ∈ E and all the vertices i ∈ V . If J i j is negative (positive) for all (i , j ) ∈ E, the system is called antiferromagnetic (ferromagnetic) Ising model. Ising model is a well-known mathematical model of ferromagnetism, where J i j is precisely the inverse of the temperature, and B i is the external field of the system. Now for a general two-state spin system, define
Jij =
βi j (+, +) + βi j (−, −) − βi j (−, +) − βi j (+, −) 4
,
h (+)−h (−)
and B i = i 2 i for all edges and vertices. The two-state spin system is Ising model if βi j (σi , σ j ) = J i j σi σ j . Hence we call J i j and B i ‘inverse temperature’ and ‘external field’ of the two-state spin system. Let J = max(i , j )∈ E | J i j |. The main theorem in this section is given as follows: Theorem 1. Let G = ( V , E ) be a graph with vertex set V = {1, 2, . . . , n} and edge set E. There exists a number d > 0 such that (G ) d. Suppose
(d − 1) tanh J < 1, d which is equivalent to J < J d = 12 log( d− ). Then the Gibbs distribution of the two-state spin system on G exhibits strong correlation 2 decay for arbitrary external field. Specifically, the decay function is given by
f (t ) = 4 J d (d − 1) tanh J
t −1
.
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
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In order to prove Theorem 1, four technical lemmas are given first. The inequality in Lemma 1 is inspired by a similar result in [9]. ax+b cx+d
Lemma 1. Let a, b, c, d, x, y be positive numbers, g (x) =
max
g (x) g ( y )
max
,
g ( y ) g (x)
x y
t
,
√
√
and t = | √ad−√bc |. Then ad+ bc
.
y x
Proof. We separate the proof into two cases: Case 1. ad bc. Consider the function
g (x) =
ax + b cx + d
=
a c
ad − bc
−
c (cx + d)
.
It is clearly an increasing function. Without loss of generality, suppose x y and let x = zy, where z 1. Then one obtains
log
g (x)
=
g ( y)
z d(log( g (α y ) )) g ( y) dα
z dα =
1
1
z 1
1
where t =
cα y + d
dα
√ √ (ad − bc ) y ad − bc dα = √ √ √ √ log z. ( bc + ad)2 α y ad + bc
z
max
cy
(ad − bc ) y dα √ √ √ √ ( ac α y − bd)2 + ( bc + ad)2 α y
=
−
(ad − bc ) y dα (aα y + b)(c α y + d)
=
Hence
aα y + b
1
z
ay
g (x) g ( y )
=
,
g ( y ) g (x)
√ √ √ad−√bc . ad+ bc
g (x) g ( y)
t
x
y
= max
x y
t
,
y x
,
Case 2. ad bc. Now g (x) is a decreasing function. Let h(x) = 1/ g (x). Now h(x) is an increasing function. Suppose x y. It is easy to see √
h(x) h( y ) Hence
max
where t =
√
√bc−√ad x
ad+ bc
y
g (x) g ( y )
,
g ( y ) g (x)
√ √ √bc −√ad . ad+ bc
.
=
g ( y) g (x)
=
h(x) h( y )
t
x
y
t x y = max , , y x
2
Let T = T ( V , E ) be a rooted tree with vertex set V = {0, 1, 2, . . . , n} and edge set E. The root of the tree is assumed to be the vertex 0. Suppose some vertices of T have fixed spins. Now remove an edge (k, l) from T with d T (k, 0) < d T (l, 0). Then the tree T is separated into two trees. Let T k and T l denote the induced subgraphs of T , which include vertex k and l respectively. Lemma 2. Let the vertices with fixed spins remain fixed on T k and T l . Then the probability P ( X 0 = + on T ) equals the probability P ( X 0 = + on the subtree T k ), if the ‘external field’ hk is modified to hk on T k as
hk (σk ) = hk (σk ) + log
e
βkl (σk ,τl )+
(i , j )∈ E l
βi j (τi ,τ j )+
i∈V l
h i (τ i )
,
τ ∈ΩT l
where E l and V l denotes the edge set and vertex set of T l , and Ω T l denotes the configuration space of T l .
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J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
Proof. The result can be obtained through simple calculations. So we omit the technical details here.
2
With Lemma 1 and Lemma 2, strong correlation decay property on trees can be proved. Lemma 3. Let T = T ( V , E ) be a rooted tree with vertex set V = {0, 1, 2, . . . , n} and edge set E. The root is vertex 0. Consider a twostate spin system on T . Let ζΛ and ηΛ be any two configurations on Λ, where Λ ⊂ V . Suppose Θ = {i: ζi = ηi , i ∈ Λ}, t = d T (0, Θ) and s = | S ( T , 0, t )| = |{i: d T (0, i ) = t , i ∈ T }|. Then
max
P T ( X 0 = +|ζΛ ) P T ( X 0 = +|ηΛ )
t −1
e 4 J s(tanh J )
,
P T ( X 0 = +|ηΛ ) P T ( X 0 = +|ζΛ )
.
Proof. For any i ∈ V , let T i denote the subtree below i with i as its root and Y (i ) be the two-state spin system induced on P ( X =+|ζ ) T i by that on T . Note that T 0 is equal to T . To prove the result, it is convenient to deal with the ratio P T ( X 0 =−|ζΛ ) rather When x1 , x2 ∈ (0, 1),
max
x1 x2
x1 x2
1 if and only if
max
,
x2 x1
T
P T i ( X i =+|ζΛi ) , P T i ( X i =−|ζΛi ) x1 x2 1−x2 . 1−x1
ζ
than P T ( X 0 = +|ζΛ ) itself. Denote R i Λ ≡
0
Λ
where ζΛi is the condition by imposing the configuration ζΛ on T i . Then
x1 /(1 − x1 ) x2 /(1 − x2 )
,
x2 /(1 − x2 ) x1 /(1 − x1 )
.
Replace x1 and x2 by P T ( X 0 = +|ζΛ ) and P T ( X 0 = +|ηΛ ). Then Lemma 3 follows by
max
P T ( X 0 = +|ζΛ ) P T ( X 0 = +|ηΛ )
,
P T ( X 0 = +|ηΛ ) P T ( X 0 = +|ζΛ )
max
η
ζ
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
.
Hence what is needed to be proved becomes
max
η
ζ
R 0Λ
R 0Λ
R 0Λ
ζΛ
η ,
t −1
e 4 J s(tanh J )
R0
(2)
.
The inequality (2) is proved by induction on t. Before doing so, some trivial cases need to be clarified. We are interested in the case t 1 and 0 is unfixed. Let Γkl denote the unique self-avoiding path from k to l on T . If i is a leaf on T and d T (0, i ) < t, where t = d T (0, Θ), define U i = { j ∈ V : j ∈ Γ0i , ∃k ∈ S ( T , 0, t ), s.t. j ∈ Γ0k }. U i = ∅ because of 0 ∈ U i . Let j i ∈ U i such that d T (i , j i ) = d T (i , U i ). By Lemma 2, we can remove the subtree below j i and change external field from h j i to hj at i j i without changing the probability P ( X 0 = +). It is noted that this procedure removes at least one leaf with the hight < t, and does not remove any vertex with the hight t. We can suppose that T is a tree rooted at 0 and the height of every leaf on the tree is no less than t. Let 01 , 02 , . . . , 0q be the neighbors connecting with 0. The recursive formula can be presented. Let Ω T i denote the configuration space in T i under the condition ζΛ , i = 1, 2, . . . , q and Ω0 denote the configuration space of T 0 under the condition ζΛ ∪ {σ0 }. We have ζ
R 0Λ =
=
Z ( T 0 , X 0 = +, ζΛ ) Z ( T 0 , X 0 = −, ζΛ ) eh0 (+) eh0 (−)
=e
2B 0
q
q
q i =1
= e 2B 0
σ ∈ΩT i e σ ∈ΩT i e
(β00i (+,σ0i )+ (β00i (−,σ0i )+
β00i (+,σ0i )+ β00i (−,σ0i )+
(k,l)∈ T i
(k,l)∈ T i
(k,l)∈ T i
(k,l)∈ T i
i
βkl (σk ,σl )+
βkl (σk ,σl )+ βkl (σk ,σl )+
i
ζ q ai R 0Λ + b i i ζ
c i R 0Λ + di
k∈ T i
k∈ T i
k∈ T i
k∈ T i
hk (σk )) hk (σk ))
hk (σk ) hk (σk )
i
i
(3)
,
i
ai = e β00i (+,+) , b i = e β00i (+,−) , c i = e β00i (−,+) , di = e β00i (−,−) . Now checking the base case t = 1
where R 0Λ , R 0 Λ ∈ [0, +∞], by the monotonicity of i
βkl (σk ,σl )+
c i Z ( T 0i , X i = +, ζΛi ) + di Z ( T 0i , X i = −, ζΛi )
h0 (+)−h0 (−) , 2
η
ζ
i =1
i
i =1
where B 0 =
σ ∈Ω0 e
i =1
q ai Z ( T 0 , X i = +, ζΛ ) + b i Z ( T 0 , X i = −, ζΛ ) i =1
= e 2B 0
σ ∈Ω0 e
ζ
ai R 0Λ +b i i
ζΛ
c i R 0 +di i
and
η ai R 0 Λ +b i i ηΛ c i R 0 +di i
,
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
max
ζ
η
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
q
max
ai di b i c i
,
b i c i ai di
i =1
707
e 4q J .
Hence, (2) holds when t = 1. By induction, assume that (2) holds for t − 1, we will show that it holds for t. Let si = | S ( T 0i , 0i , t − 1)|, i = 1, 2, . . . , q, repeating above recursive procedure, then
⎛
max
ζ
η
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
q
ζ
ai R 0Λ +b i i
⎜ ci R ζΛ +di ⎜ 0i , ⎝ ai R η0iΛ +bi
max ⎜
i =1
ηΛ
c i R 0 +di i
q
max
i =1
q
max
i =1
⎞
η ai R 0 Λ +b i i ηΛ c i R 0 +di
⎟ ⎟ ⎟ ⎠
i
ζ
ai R 0Λ +b i i
ζ c i R 0Λ +di
√
i
√
η √ai di −√bi ci
ζ
R 0Λ
R 0Λ
R 0Λ
R 0Λ
R 0Λ
ζ
R 0Λ
R 0Λ
ζΛ
i η ,
i
ai di +
bi ci
ζ
i
i
η tanh J
i η ,
i
R0
i
,
i
where the second inequality comes from Lemma 1. According to the hypothesis of induction max t −2
e 4 J si (tanh J )
max
R ζΛ 0i
ηΛ
R0
i
,
η R 0Λ i
ζΛ
R0
i
, it’s sufficient to show ζ
R 0Λ
ηΛ ,
R0
η
R 0Λ ζΛ
R0
q
t −1
e 4 J si (tanh J )
i =1
where the last equation follows by
q
s i =1 i
t −1
= e 4 J s(tanh J )
,
= s. This completes the proof. 2
To generalize the strong correlation decay property on trees to the general graphs, we need to utilize the remarkable property of the self-avoiding tree, which is implicitly stated in [13] and explicitly stated in [8]. Lemma 4. (See [8].) Consider a two-state spin system on a graph G = ( V , E ). For any configuration σΛ , Λ ⊂ V and any vertex v ∈ V , one has
P G ( X v = +|σΛ ) = P T saw( v ) ( X v = +|σΛ ), where T saw( v ) is the self-avoiding tree corresponding to the vertex v. Lemma 3 and Lemma 4 are sufficient to prove Theorem 1. Proof of Theorem 1. Since the maximum degree of T saw(i ) is also bounded by d, one easily has
s = S ( T saw(i ) , i , t ) d(d − 1)t −1 . By Lemma 3 and Lemma 4, Theorem 1 is proved.
2
Remark. From the proof of Theorem 1, by a similar argument, we can get
log P G ( X v = −|σΛ ) − log P G ( X v = −|ηΛ ) f (t ), where f (t ) = 4 J d((d − 1) tanh J )t −1 . As one of the corollaries of strong correlation decay property, we prove there is unique Gibbs measure on a bounded degree infinite graph (an infinite graph with maximum degree d). This generalizes original Dobrushion’s condition
d tanh J < 1 to
(d − 1) tanh J < 1
for uniqueness of Gibbs measure of Ising models [4]. Let G = (V, E) be an infinite graph. For any finite subset V of V, the graph G = ( V , E ), where E is the edge set induced / V , j ∈ V } and recall that A \ B denotes the set obtained by V , is called a finite subgraph of G. Let ∂ V = {i: (i , j ) ∈ G, i ∈ by removing elements of B from the set A. Suppose there is a two-state spin system or a Gibbs measure P on each finite subgraph G of G which is defined by Definition 1, a probability measure P on G is called a Gibbs measure on G corresponding to such a system if P satisfies that, for any finite subgraph G = ( V , E ), G denoting the induced graph by V ∪ ∂ V in G, P(·|σV\ V ) = P G (·|σ∂ V ).
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J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
Theorem 2. Let G = (V, E) be a countable infinite connected graph and (G) = supi ∈G {δi }. Assume there exists a constant d such that (G) d. There is a two-state spin system on each finite subgraph G of G which is given by Definition 1. If (d − 1) tanh J < 1, then the Gibbs measure on G corresponding to the two-state spin system on finite subgraphs is unique. Proof. For any given finite subgraph G = ( V , E ) of G, let G 0 = ( V 0 , E 0 ) = G. We construct a sequence of finite subgraphs G n = ( V n , E n ) of G, n = 1, 2, . . . , ∞, recursively as follows: let G n+1 = ( V n+1 , E n+1 ) be the induced graph by V n ∪ ∂ V n , n = 0, 1, 2, . . . , ∞. A simple mathematical induction would show that
∂ V n = v: dG ( v , G ) = n + 1 ,
n = 0, 1, 2, . . . , ∞.
For any v ∈ V, suppose dG ( v , G ) = d v , then there is a subgraph G d v ⊂ G such that v ∈ G d v . Besides, obviously, G n ⊂ G n+1 , n = 0, 1, 2, . . . , ∞ and the distance dn = dG (G , ∂ V n−1 ) between G and ∂ V n−1 goes to infinity as n → ∞. Denote ζn , ηn any two configurations on ∂ V n−1 , for n = 1, 2, . . . , ∞. Let σΛ be any configuration on Λ, Λ ⊂ V . By Proposition 2.2 in [12], the Gibbs measure is unique on G if
lim sup max P G n ( X Λ = σΛ |ζn ) − P G n ( X Λ = σΛ |ηn ) = 0
n→∞ ζ ,η Λ⊂ V ,σΛ n n
holds. Let Λ = {1, 2, . . . , m}, where m = |Λ|. Set αn (1) = P G n ( X 1 = σ1 |ζn ) and βn (1) = P G n ( X 1 = σ1 |ηn ). Let αn (i ) = P G n ( X i = σi |ζn , σ j , 1 j i − 1) and βn (i ) = P G n ( X i = σi |ηn , σ j , 1 j i − 1), i = 2, 3, . . . , n. The telescoping trick gives
P G n ( X Λ = σΛ |ζn ) =
m
αn (i )
i =1
and P G n ( X Λ = σΛ |ηn ) =
m
i =1 βn (i ).
log α (i ) f (dn ), β(i )
By Theorem 1 and above remark, we know, for each i ∈ Λ,
where f (t ) is the decay function f (t ) = 4 J d((d − 1) tanh J )t −1 . Then
m α (i ) mf (dn ), log β(i ) i =1
where m = |Λ|. Hence
P G n ( X Λ = σΛ |ζn ) | V | f (dn ). sup max log P G n ( X Λ = σΛ |ηn ) ζn ,ηn Λ⊂ V ,σΛ Therefore, if (d − 1) tanh J < 1, then
P G n ( X Λ = σΛ |ζn ) = 0, lim sup max log n→∞ Λ⊂ V ,σΛ P ( X = σ |η ) ζn ,ηn
which implies
Gn
Λ
Λ
n
lim sup max P G n ( X Λ = σΛ |ζn ) − P G n ( X Λ = σΛ |ηn ) = 0.
n→∞ ζ ,η Λ⊂ V ,σΛ n n
This completes the proof.
2
Remark. Our condition (d − 1) tanh J < 1 is sharp for the uniqueness of Gibbs measure of Ising model on bounded degree infinite graph. First, by Theorem 2, the condition (d − 1) tanh J < 1 implies uniqueness of Gibbs measure of Ising model on bounded degree infinite graph. Second, if Gibbs measure of Ising model on bounded degree infinite graph is unique, then the Gibbs measure of Ising model on d regular infinite tree is unique since d regular infinite tree is a special bounded degree infinite graph. By [4], we know the condition (d − 1) tanh J < 1 is sharp for uniqueness of Gibbs measure of Ising model on d regular infinite tree, hence uniqueness of Gibbs measure of Ising model on bounded degree infinite graph implies the condition (d − 1) tanh J < 1. 4. Approximating the partition function In the proof of Lemma 3, the calculation of the marginal probability that the root vertex is fixed + yields a local recursive procedure. We will use the recursive formula (3) to compute the marginal probability that the root is set to +. If the tree is truncated at a height t, it is easy to see that the computational complexity of this procedure is linear in the number of vertices of the truncated tree. Let Φ1 denote the whole state space, which means P G ( X 1 = +|Φ1 ) = P G ( X 1 = +), and Φ j = { X i = +, 1 i j − 1}, 2 j n + 1. Let p j be an estimator of conditional marginal probability p j = P G ( X j = +|Φ j ), j = 1, 2, . . . , n. The algorithm for approximating the partition function is proposed as follows:
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710 Algorithm for Partition Function Z (G) Input: G, a graph with vertices {1, . . . , n}, the two-state spin system on G, Z (G ), the estimator of partition function Z (G ). Output: Begin For j from 1 to n do step 1. Set t j =
log(4n J d −1 ) log((d−1)tanh J )−1
709
> 0, precision;
+ 1,
step 2. Take the vertex j as root and generate the truncated subtree T saw ( j ) with height t j under the condition Φ j , step 3. Set initial values be 0 s for all the vertices of T saw( j ) at height t j ,
p j through T saw step 4. Computing ( j ) by recursive formula (3). End For n 1 Z (G ) = Z (G , Φn+1 ) i =1 p− Compute i . End
Theorem 3. Let G = ( V , E ) be a graph with vertex set V = {1, 2, . . . , n} and edge set E. There exists a positive number d > 0 such that (G ) d. If J < J d , then the above algorithm provides an FPTAS for the partition function of the two-state spin system on G. Proof. According to the results of Theorem 1, for any j = 1, 2, . . . , n, there is a function f (t j ) n such that
e− n
pj
pj
en,
under the condition that
tj = Since p j =
log(4n J d −1 ) log((d − 1) tanh J )−1 Z (G ,Φ j +1 ) , Z (G ,Φ j )
+ 1 = O log n + log −1 .
Z (G ) can be expressed as the product Z (G ) = Z (G , Φn+1 )
e − e (− n )n
n p −1 i
−1
p i =1 i
=
Z (G ) Z (G )
n
i =1
1 p− i . Hence,
e ( n )n e .
The complexity of the algorithm for each j in the loop is
O V ( T saw( j ) , j , t j ) = O (d − 1)t j . Thus, the total complexity of the algorithm is
O (1 ) −1 −1 , O (d − 1) O (log n+log( )) = nO (d − 1) O (log n+log( )) = O n O (1) + n −1 j
which completes the proof.
2
5. Conclusions and discussion In this paper, we show that for arbitrary external field, the Gibbs measure exhibits strong correlation decay on a bounded degree tree when J < J d , where J d is the critical point for the uniqueness of Gibbs measures of (anti)ferromagnetic Ising model on an infinite d regular tree [4,12]. This generalizes the recent result by Mossel and Sly [10] to the antiferromagnetic cases. By the strong correlation decay, it is proved that there exists a unique Gibbs measure of the two-state spin system on a bounded degree infinite graph. We tighten Dobrushion’s condition d tanh( J ) < 1 to (d − 1) tanh( J ) < 1 for the uniqueness of Gibbs measure of antiferromagnetic and ferromagnetic Ising models [3,4,12]. This condition turns out to be sharp for the uniqueness of Gibbs measure of Ising model on an bounded degree infinite graph. An interesting investigation could be whether the condition (d − 1) tanh( J ) < 1 is sharp for the general two-state spin system. An FPTAS for the partition functions of two-state spin systems on the bounded degree graphs is also presented. Acknowledgement The authors would like to thank the anonymous referees for their careful reading and helpful comments and suggestions. References [1] A. Bandyopadhyay, D. Gamarnik, Counting without sampling: New algorithms for enumeration problems using statistical physics, Random Structures and Algorithms 33 (2008) 452–479. [2] A. Dembo, A. Montanari, Ising models on locally tree-like graphs, The Annals of Applied Probability 20 (2) (2010) 565–592.
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[3] R.L. Dobrushin, Prescribing a system of random variables by the help of conditional distributions, Theory Probability and Its Application 15 (1970) 469–497. [4] H.O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. [5] L.A. Goldberg, M. Jerrum, M. Paterson, The computational complexity of two-state spin systems, Random Structures and Algorithms 23 (2003) 133–154. [6] M. Jerrum, A. Sinclair, Polynomial-time approximation algorithms for Ising model, SIAM Journal on Computing 22 (5) (1993) 1087–1116. [7] J. Jonasson, Uniqueness of uniform random colorings of regular trees, Statistics and Probability Letters 57 (2002) 243–248. [8] K. Jung, D. Shah, Inference in binary pair-wise Markov random field through self-avoiding walk, http://arxiv.org/abs/cs.AI/0610111v2. [9] R. Lyons, The Ising model and percolation on trees and treelike graphs, Communications in Mathematical Physics 125 (2) (1989) 337–353. [10] E. Mossel, A. Sly, Rapid mixing of Gibbs sampling on graphs that are sparse on average, Random Structures and Algorithms 35 (2) (2009) 250–270. [11] R. Pemantle, Y. Peres, The critical Ising model on trees, concave recursions and nonlinear capacity, The Annals of Probability 38 (1) (2010) 184–206. [12] D. Weitz, Combinatorial criteria for uniqueness of Gibbs measures, Random Structures and Algorithms 27 (2005) 445–475. [13] D. Weitz, Counting independent sets up to the tree threshold, in: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149.
Further reading [14] M. Jerrum, L. Valiant, V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science 43 (1986) 169–188. [15] L.G. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8 (1979) 189–201.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Approximating partition functions of the two-state spin system ✩ Jinshan Zhang a,∗ , Heng Liang b , Fengshan Bai b a b
Department of Computer Science, University of Liverpool, UK Department of Mathematical Sciences, Tsinghua University, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 10 April 2010 Received in revised form 26 April 2011 Accepted 26 April 2011 Available online 29 April 2011 Communicated by A. Tarlecki Keywords: Approximation algorithms Two-state spin system Ising model Gibbs measure Uniqueness of Gibbs measure FPTAS Strong correlation decay
Two-state spin system is a classical topic in statistical physics. We consider the problem of computing the partition function of the system on a bounded degree graph. Based on the self-avoiding tree, we prove the system exhibits strong correlation decay under the condition that the absolute value of inverse temperature is small. Due to strong correlation decay property, an FPTAS for the partition function is presented and uniqueness of Gibbs measure of the two-state spin system on a bounded degree infinite graph is proved, under the same condition. This condition is sharp for Ising model. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Spin model with p states is a classical mathematical model in statistical physics. Such models describe and explain the behavior of ferromagnets, lattice gas and certain other phenomena of statistical physics. In this paper, we focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field). In statistical mechanics, the partition function is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. However, partition functions are normally hard to compute, even for the two-state spin system [5]. Hence approximate algorithms, which can give a reasonable estimation within acceptable computing time, attract more attention. Markov Chain Monte Carlo methods [6,10] are the existing powerful approach for approximating the partition functions. Exploiting the structure property of Gibbs measure, Weitz [13] and Bandyopadhyay and Gamarnik [1] introduce new deterministic algorithms for counting the number of independent sets and colorings. The key point of this method is to establish the strong spatial mixing property, which is also known as strong correlation decay, on a self -avoiding tree or a computation tree in a dynamic programming type recursion. In [13], Weitz proves the strong correlation decay for hard-core model on bounded degree trees and establishes the result for general graph using the self -avoiding tree technique. Weitz’s approach is applicable to some kinds of statistical systems, such as Ising model [11,8], hard-core model [13] and coloring model [7].
✩
*
Supported by National Science Foundation of China, no. 10871115. Corresponding author. E-mail address: [email protected] (J. Zhang).
0020-0190/$ – see front matter doi:10.1016/j.ipl.2011.04.012
© 2011
Elsevier B.V. All rights reserved.
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
703
It is natural to ask whether the more general two-state spin system exhibits strong correlation decay. In this paper, our results give a positive answer to this question. We show that for arbitrary external field, the Gibbs measure exhibits strong correlation decay on a bounded degree tree under the condition (d − 1) tanh J < 1. Here d is the maximum degree of the graph and J is a system parameter, which is also known as inverse temperature. Our result extends Dobrushion’s condition for the uniqueness of Gibbs measure of antiferromagnetic and ferromagnetic Ising models [3,4,12]. This condition is sharp for Ising model on a bounded degree infinite graph. A fully polynomial time approximation scheme (FPTAS) for partition functions of two-state spin system on a bounded degree graph is also constructed in this paper. This is natural and reasonable when the strong correlation decay holds. Jerrum and Sinclair [6] provide an FPRAS for ferromagnetic Ising model on graphs with uniform positive inverse temperature and identical external field for all the vertices. Their results neither include the case where different vertices have different external field, nor are applicable to antiferromagnetic Ising model. Very recently Dembo and Montanari propose an explicit formula for partition function of ferromagnetic Ising model with external field on locally tree-like graphs, which still does not include the antiferromagnetic case [2]. The remainder of the paper has the following structure. In Section 2, we present some preliminary definitions. We go on to prove the main theorem in Section 3. Section 4 is devoted to propose an FPTAS for the partition functions. Discussion and conclusions are given in Section 5. 2. Notations and definitions Let G = ( V , E ) be a finite graph with vertex set V = {1, 2, . . . , n} and edge set E. If there is no specific statement, all the graphs considered in this paper are connected. Let d G (u , v ) denote the distance between u and v, for any u, v ∈ V . A path v 1 → v 2 → · · · is called a self-avoiding path if v i = v j for all i = j. The distance between a vertex v ∈ V and a subset Λ ⊂ V is defined by
d G ( v , Λ) = min d G ( v , u ): u ∈ Λ . The set of vertices with distance l to the vertex v is denoted by
S (G , v , l) = u: d G ( v , u ) = l . The set of vertices which are no more than l away from v is denoted by
V (G , v , l) = u: d G ( v , u ) l . Let δ v denote the degree of v in G and (G ) = max{δ v : v ∈ V }. A \ B denotes the set obtained by removing elements of B from the set A. Let all the vertices in graph G = ( V , E ) be numbered, where V and E are respectively vertex set and edge set of G. We define the partial order on E, where (i , j ) > (k, l) if and only if (i , j ) and (k, l) share a common vertex and i + j > k + l. In two-state spin system on G, each vertex i ∈ V is associated with a random variable X i on Ω = {±1} (briefly we use + and − to denote +1 and −1 respectively). Definition 1. The Gibbs measure of two-state spin system on G is defined by the joint distribution of X = { X 1 , X 2 , . . . , X n }
PG(X = σ ) =
1 Z (G )
e
(i , j )∈ E
βi j (σi ,σ j )+
i∈V
h i (σ i )
,
(1)
where h i is a map Ω → R and βi j is a map Ω 2 → R. Z (G ) is a normalizing constant, which is called the partition function of the system. For any vertex i (i = 1, 2, . . . , n), the probability P G ( X i = σi ) is called the marginal probability that the vertex i is assigned σi . Note that the Gibbs measure satisfies σ ∈Ω n P G ( X = σ ) = 1. We define βi j (a, b) = β ji (b, a). For any Λ ⊆ V , σΛ denotes the set {σi , i ∈ Λ}. With a little abuse of notation, σΛ also denotes the condition or configuration with fixed σi for any i ∈ Λ. Let Z (G , Φ) denote the partition function under the condition Φ . For example, Z (G , X 1 = +) represents the partition function under the condition that the vertex 1 is fixed to be +. The partition function can be represented in terms of the marginal probabilities of Gibbs measure, which provides a powerful tool for counting [1,13]. Local Markov property is in the nature of Gibbs measure, which is useful in our analysis. Proposition 1 (Local Markov Property). Suppose there is a two-state spin system on G = ( V , E ) and P is the Gibbs measure of the system. Let Λ ⊂ V and ∂Λ = {i ∈ V : (i , j ) ∈ E , i ∈ / Λ, j ∈ Λ}. Denote G to be the induced graph by Λ ∪ ∂Λ in G. It is well known that Gibbs measure P satisfies the local Markov property which is P G (·|σ V \Λ ) = P G (·|σ∂Λ ), for any Λ ⊂ V . A self-avoiding walk (SAW) is a sequence of moves (on a graph) which does not visit the same point more than once. The following gives an important tool in proving our results. It is introduced in [13].
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J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
Fig. 1. The graph with one vertex assigned + (right) and its corresponding self-avoiding tree T saw(1) (left).
Definition 2 (Self-Avoiding Tree). The self-avoiding tree T saw( v ) (G ) corresponding to the vertex v of G is a tree with root v. The tree is generated through the self-avoiding walks originating at v, and can be denoted by T saw( v ) for simplicity. A vertex closing a cycle is included as a leaf of the tree and is assigned to be +, if the edge ending the cycle is larger than the edge starting the cycle, and − otherwise. Remark. Given any configuration σΛ of G, Λ ⊂ V , the self-avoiding tree is constructed in the same way as the above procedure except that the vertex, which is a copy of the vertex i in Λ, is fixed to the same spin σi as i and the subtree below it is not constructed due to the local Markov property, see Fig. 1 for example, where vertex 5 is fixed + in G. The strong correlation decay property implies that the marginal probability that the root is assigned + is asymptotically independent of the configuration on the leaves far below. Definition 3 (Strong Correlation Decay). The Gibbs distribution of two-state spin system on G exhibits strong correlation decay if and only if for any vertex v ∈ V , subset Λ ⊂ V , any two configurations σΛ and ηΛ on Λ, there exist positive numbers a, b independent of n such that
log P G ( X v = +|σΛ ) − log P G ( X v = +|ηΛ ) f (t ),
where t = d G ( v , { v ∈ Λ:
σ v = η v }), and decay function is given by f (t ) = ae −bt .
Definition 4 (FPTAS). For a problem to compute a (numerical) value M ( I ) for any input I , an approximation algorithm is called a fully polynomial time approximation scheme (FPTAS) for this problem if and only if for any > 0 and input I , the ¯ ( I , ) satisfying algorithm takes polynomial time in size of I and −1 to output a (numerical) value M
e −
¯ (I , ) M M(I )
e .
3. Strong correlation decay Consider a special two-state spin system with βi j (σi , σ j ) = J i j σi σ j , h i = B i σi for all the edges (i , j ) ∈ E and all the vertices i ∈ V . If J i j is negative (positive) for all (i , j ) ∈ E, the system is called antiferromagnetic (ferromagnetic) Ising model. Ising model is a well-known mathematical model of ferromagnetism, where J i j is precisely the inverse of the temperature, and B i is the external field of the system. Now for a general two-state spin system, define
Jij =
βi j (+, +) + βi j (−, −) − βi j (−, +) − βi j (+, −) 4
,
h (+)−h (−)
and B i = i 2 i for all edges and vertices. The two-state spin system is Ising model if βi j (σi , σ j ) = J i j σi σ j . Hence we call J i j and B i ‘inverse temperature’ and ‘external field’ of the two-state spin system. Let J = max(i , j )∈ E | J i j |. The main theorem in this section is given as follows: Theorem 1. Let G = ( V , E ) be a graph with vertex set V = {1, 2, . . . , n} and edge set E. There exists a number d > 0 such that (G ) d. Suppose
(d − 1) tanh J < 1, d which is equivalent to J < J d = 12 log( d− ). Then the Gibbs distribution of the two-state spin system on G exhibits strong correlation 2 decay for arbitrary external field. Specifically, the decay function is given by
f (t ) = 4 J d (d − 1) tanh J
t −1
.
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
705
In order to prove Theorem 1, four technical lemmas are given first. The inequality in Lemma 1 is inspired by a similar result in [9]. ax+b cx+d
Lemma 1. Let a, b, c, d, x, y be positive numbers, g (x) =
max
g (x) g ( y )
max
,
g ( y ) g (x)
x y
t
,
√
√
and t = | √ad−√bc |. Then ad+ bc
.
y x
Proof. We separate the proof into two cases: Case 1. ad bc. Consider the function
g (x) =
ax + b cx + d
=
a c
ad − bc
−
c (cx + d)
.
It is clearly an increasing function. Without loss of generality, suppose x y and let x = zy, where z 1. Then one obtains
log
g (x)
=
g ( y)
z d(log( g (α y ) )) g ( y) dα
z dα =
1
1
z 1
1
where t =
cα y + d
dα
√ √ (ad − bc ) y ad − bc dα = √ √ √ √ log z. ( bc + ad)2 α y ad + bc
z
max
cy
(ad − bc ) y dα √ √ √ √ ( ac α y − bd)2 + ( bc + ad)2 α y
=
−
(ad − bc ) y dα (aα y + b)(c α y + d)
=
Hence
aα y + b
1
z
ay
g (x) g ( y )
=
,
g ( y ) g (x)
√ √ √ad−√bc . ad+ bc
g (x) g ( y)
t
x
y
= max
x y
t
,
y x
,
Case 2. ad bc. Now g (x) is a decreasing function. Let h(x) = 1/ g (x). Now h(x) is an increasing function. Suppose x y. It is easy to see √
h(x) h( y ) Hence
max
where t =
√
√bc−√ad x
ad+ bc
y
g (x) g ( y )
,
g ( y ) g (x)
√ √ √bc −√ad . ad+ bc
.
=
g ( y) g (x)
=
h(x) h( y )
t
x
y
t x y = max , , y x
2
Let T = T ( V , E ) be a rooted tree with vertex set V = {0, 1, 2, . . . , n} and edge set E. The root of the tree is assumed to be the vertex 0. Suppose some vertices of T have fixed spins. Now remove an edge (k, l) from T with d T (k, 0) < d T (l, 0). Then the tree T is separated into two trees. Let T k and T l denote the induced subgraphs of T , which include vertex k and l respectively. Lemma 2. Let the vertices with fixed spins remain fixed on T k and T l . Then the probability P ( X 0 = + on T ) equals the probability P ( X 0 = + on the subtree T k ), if the ‘external field’ hk is modified to hk on T k as
hk (σk ) = hk (σk ) + log
e
βkl (σk ,τl )+
(i , j )∈ E l
βi j (τi ,τ j )+
i∈V l
h i (τ i )
,
τ ∈ΩT l
where E l and V l denotes the edge set and vertex set of T l , and Ω T l denotes the configuration space of T l .
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J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
Proof. The result can be obtained through simple calculations. So we omit the technical details here.
2
With Lemma 1 and Lemma 2, strong correlation decay property on trees can be proved. Lemma 3. Let T = T ( V , E ) be a rooted tree with vertex set V = {0, 1, 2, . . . , n} and edge set E. The root is vertex 0. Consider a twostate spin system on T . Let ζΛ and ηΛ be any two configurations on Λ, where Λ ⊂ V . Suppose Θ = {i: ζi = ηi , i ∈ Λ}, t = d T (0, Θ) and s = | S ( T , 0, t )| = |{i: d T (0, i ) = t , i ∈ T }|. Then
max
P T ( X 0 = +|ζΛ ) P T ( X 0 = +|ηΛ )
t −1
e 4 J s(tanh J )
,
P T ( X 0 = +|ηΛ ) P T ( X 0 = +|ζΛ )
.
Proof. For any i ∈ V , let T i denote the subtree below i with i as its root and Y (i ) be the two-state spin system induced on P ( X =+|ζ ) T i by that on T . Note that T 0 is equal to T . To prove the result, it is convenient to deal with the ratio P T ( X 0 =−|ζΛ ) rather When x1 , x2 ∈ (0, 1),
max
x1 x2
x1 x2
1 if and only if
max
,
x2 x1
T
P T i ( X i =+|ζΛi ) , P T i ( X i =−|ζΛi ) x1 x2 1−x2 . 1−x1
ζ
than P T ( X 0 = +|ζΛ ) itself. Denote R i Λ ≡
0
Λ
where ζΛi is the condition by imposing the configuration ζΛ on T i . Then
x1 /(1 − x1 ) x2 /(1 − x2 )
,
x2 /(1 − x2 ) x1 /(1 − x1 )
.
Replace x1 and x2 by P T ( X 0 = +|ζΛ ) and P T ( X 0 = +|ηΛ ). Then Lemma 3 follows by
max
P T ( X 0 = +|ζΛ ) P T ( X 0 = +|ηΛ )
,
P T ( X 0 = +|ηΛ ) P T ( X 0 = +|ζΛ )
max
η
ζ
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
.
Hence what is needed to be proved becomes
max
η
ζ
R 0Λ
R 0Λ
R 0Λ
ζΛ
η ,
t −1
e 4 J s(tanh J )
R0
(2)
.
The inequality (2) is proved by induction on t. Before doing so, some trivial cases need to be clarified. We are interested in the case t 1 and 0 is unfixed. Let Γkl denote the unique self-avoiding path from k to l on T . If i is a leaf on T and d T (0, i ) < t, where t = d T (0, Θ), define U i = { j ∈ V : j ∈ Γ0i , ∃k ∈ S ( T , 0, t ), s.t. j ∈ Γ0k }. U i = ∅ because of 0 ∈ U i . Let j i ∈ U i such that d T (i , j i ) = d T (i , U i ). By Lemma 2, we can remove the subtree below j i and change external field from h j i to hj at i j i without changing the probability P ( X 0 = +). It is noted that this procedure removes at least one leaf with the hight < t, and does not remove any vertex with the hight t. We can suppose that T is a tree rooted at 0 and the height of every leaf on the tree is no less than t. Let 01 , 02 , . . . , 0q be the neighbors connecting with 0. The recursive formula can be presented. Let Ω T i denote the configuration space in T i under the condition ζΛ , i = 1, 2, . . . , q and Ω0 denote the configuration space of T 0 under the condition ζΛ ∪ {σ0 }. We have ζ
R 0Λ =
=
Z ( T 0 , X 0 = +, ζΛ ) Z ( T 0 , X 0 = −, ζΛ ) eh0 (+) eh0 (−)
=e
2B 0
q
q
q i =1
= e 2B 0
σ ∈ΩT i e σ ∈ΩT i e
(β00i (+,σ0i )+ (β00i (−,σ0i )+
β00i (+,σ0i )+ β00i (−,σ0i )+
(k,l)∈ T i
(k,l)∈ T i
(k,l)∈ T i
(k,l)∈ T i
i
βkl (σk ,σl )+
βkl (σk ,σl )+ βkl (σk ,σl )+
i
ζ q ai R 0Λ + b i i ζ
c i R 0Λ + di
k∈ T i
k∈ T i
k∈ T i
k∈ T i
hk (σk )) hk (σk ))
hk (σk ) hk (σk )
i
i
(3)
,
i
ai = e β00i (+,+) , b i = e β00i (+,−) , c i = e β00i (−,+) , di = e β00i (−,−) . Now checking the base case t = 1
where R 0Λ , R 0 Λ ∈ [0, +∞], by the monotonicity of i
βkl (σk ,σl )+
c i Z ( T 0i , X i = +, ζΛi ) + di Z ( T 0i , X i = −, ζΛi )
h0 (+)−h0 (−) , 2
η
ζ
i =1
i
i =1
where B 0 =
σ ∈Ω0 e
i =1
q ai Z ( T 0 , X i = +, ζΛ ) + b i Z ( T 0 , X i = −, ζΛ ) i =1
= e 2B 0
σ ∈Ω0 e
ζ
ai R 0Λ +b i i
ζΛ
c i R 0 +di i
and
η ai R 0 Λ +b i i ηΛ c i R 0 +di i
,
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
max
ζ
η
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
q
max
ai di b i c i
,
b i c i ai di
i =1
707
e 4q J .
Hence, (2) holds when t = 1. By induction, assume that (2) holds for t − 1, we will show that it holds for t. Let si = | S ( T 0i , 0i , t − 1)|, i = 1, 2, . . . , q, repeating above recursive procedure, then
⎛
max
ζ
η
R 0Λ
R 0Λ
R0
R 0Λ
ηΛ ,
ζ
q
ζ
ai R 0Λ +b i i
⎜ ci R ζΛ +di ⎜ 0i , ⎝ ai R η0iΛ +bi
max ⎜
i =1
ηΛ
c i R 0 +di i
q
max
i =1
q
max
i =1
⎞
η ai R 0 Λ +b i i ηΛ c i R 0 +di
⎟ ⎟ ⎟ ⎠
i
ζ
ai R 0Λ +b i i
ζ c i R 0Λ +di
√
i
√
η √ai di −√bi ci
ζ
R 0Λ
R 0Λ
R 0Λ
R 0Λ
R 0Λ
ζ
R 0Λ
R 0Λ
ζΛ
i η ,
i
ai di +
bi ci
ζ
i
i
η tanh J
i η ,
i
R0
i
,
i
where the second inequality comes from Lemma 1. According to the hypothesis of induction max t −2
e 4 J si (tanh J )
max
R ζΛ 0i
ηΛ
R0
i
,
η R 0Λ i
ζΛ
R0
i
, it’s sufficient to show ζ
R 0Λ
ηΛ ,
R0
η
R 0Λ ζΛ
R0
q
t −1
e 4 J si (tanh J )
i =1
where the last equation follows by
q
s i =1 i
t −1
= e 4 J s(tanh J )
,
= s. This completes the proof. 2
To generalize the strong correlation decay property on trees to the general graphs, we need to utilize the remarkable property of the self-avoiding tree, which is implicitly stated in [13] and explicitly stated in [8]. Lemma 4. (See [8].) Consider a two-state spin system on a graph G = ( V , E ). For any configuration σΛ , Λ ⊂ V and any vertex v ∈ V , one has
P G ( X v = +|σΛ ) = P T saw( v ) ( X v = +|σΛ ), where T saw( v ) is the self-avoiding tree corresponding to the vertex v. Lemma 3 and Lemma 4 are sufficient to prove Theorem 1. Proof of Theorem 1. Since the maximum degree of T saw(i ) is also bounded by d, one easily has
s = S ( T saw(i ) , i , t ) d(d − 1)t −1 . By Lemma 3 and Lemma 4, Theorem 1 is proved.
2
Remark. From the proof of Theorem 1, by a similar argument, we can get
log P G ( X v = −|σΛ ) − log P G ( X v = −|ηΛ ) f (t ), where f (t ) = 4 J d((d − 1) tanh J )t −1 . As one of the corollaries of strong correlation decay property, we prove there is unique Gibbs measure on a bounded degree infinite graph (an infinite graph with maximum degree d). This generalizes original Dobrushion’s condition
d tanh J < 1 to
(d − 1) tanh J < 1
for uniqueness of Gibbs measure of Ising models [4]. Let G = (V, E) be an infinite graph. For any finite subset V of V, the graph G = ( V , E ), where E is the edge set induced / V , j ∈ V } and recall that A \ B denotes the set obtained by V , is called a finite subgraph of G. Let ∂ V = {i: (i , j ) ∈ G, i ∈ by removing elements of B from the set A. Suppose there is a two-state spin system or a Gibbs measure P on each finite subgraph G of G which is defined by Definition 1, a probability measure P on G is called a Gibbs measure on G corresponding to such a system if P satisfies that, for any finite subgraph G = ( V , E ), G denoting the induced graph by V ∪ ∂ V in G, P(·|σV\ V ) = P G (·|σ∂ V ).
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J. Zhang et al. / Information Processing Letters 111 (2011) 702–710
Theorem 2. Let G = (V, E) be a countable infinite connected graph and (G) = supi ∈G {δi }. Assume there exists a constant d such that (G) d. There is a two-state spin system on each finite subgraph G of G which is given by Definition 1. If (d − 1) tanh J < 1, then the Gibbs measure on G corresponding to the two-state spin system on finite subgraphs is unique. Proof. For any given finite subgraph G = ( V , E ) of G, let G 0 = ( V 0 , E 0 ) = G. We construct a sequence of finite subgraphs G n = ( V n , E n ) of G, n = 1, 2, . . . , ∞, recursively as follows: let G n+1 = ( V n+1 , E n+1 ) be the induced graph by V n ∪ ∂ V n , n = 0, 1, 2, . . . , ∞. A simple mathematical induction would show that
∂ V n = v: dG ( v , G ) = n + 1 ,
n = 0, 1, 2, . . . , ∞.
For any v ∈ V, suppose dG ( v , G ) = d v , then there is a subgraph G d v ⊂ G such that v ∈ G d v . Besides, obviously, G n ⊂ G n+1 , n = 0, 1, 2, . . . , ∞ and the distance dn = dG (G , ∂ V n−1 ) between G and ∂ V n−1 goes to infinity as n → ∞. Denote ζn , ηn any two configurations on ∂ V n−1 , for n = 1, 2, . . . , ∞. Let σΛ be any configuration on Λ, Λ ⊂ V . By Proposition 2.2 in [12], the Gibbs measure is unique on G if
lim sup max P G n ( X Λ = σΛ |ζn ) − P G n ( X Λ = σΛ |ηn ) = 0
n→∞ ζ ,η Λ⊂ V ,σΛ n n
holds. Let Λ = {1, 2, . . . , m}, where m = |Λ|. Set αn (1) = P G n ( X 1 = σ1 |ζn ) and βn (1) = P G n ( X 1 = σ1 |ηn ). Let αn (i ) = P G n ( X i = σi |ζn , σ j , 1 j i − 1) and βn (i ) = P G n ( X i = σi |ηn , σ j , 1 j i − 1), i = 2, 3, . . . , n. The telescoping trick gives
P G n ( X Λ = σΛ |ζn ) =
m
αn (i )
i =1
and P G n ( X Λ = σΛ |ηn ) =
m
i =1 βn (i ).
log α (i ) f (dn ), β(i )
By Theorem 1 and above remark, we know, for each i ∈ Λ,
where f (t ) is the decay function f (t ) = 4 J d((d − 1) tanh J )t −1 . Then
m α (i ) mf (dn ), log β(i ) i =1
where m = |Λ|. Hence
P G n ( X Λ = σΛ |ζn ) | V | f (dn ). sup max log P G n ( X Λ = σΛ |ηn ) ζn ,ηn Λ⊂ V ,σΛ Therefore, if (d − 1) tanh J < 1, then
P G n ( X Λ = σΛ |ζn ) = 0, lim sup max log n→∞ Λ⊂ V ,σΛ P ( X = σ |η ) ζn ,ηn
which implies
Gn
Λ
Λ
n
lim sup max P G n ( X Λ = σΛ |ζn ) − P G n ( X Λ = σΛ |ηn ) = 0.
n→∞ ζ ,η Λ⊂ V ,σΛ n n
This completes the proof.
2
Remark. Our condition (d − 1) tanh J < 1 is sharp for the uniqueness of Gibbs measure of Ising model on bounded degree infinite graph. First, by Theorem 2, the condition (d − 1) tanh J < 1 implies uniqueness of Gibbs measure of Ising model on bounded degree infinite graph. Second, if Gibbs measure of Ising model on bounded degree infinite graph is unique, then the Gibbs measure of Ising model on d regular infinite tree is unique since d regular infinite tree is a special bounded degree infinite graph. By [4], we know the condition (d − 1) tanh J < 1 is sharp for uniqueness of Gibbs measure of Ising model on d regular infinite tree, hence uniqueness of Gibbs measure of Ising model on bounded degree infinite graph implies the condition (d − 1) tanh J < 1. 4. Approximating the partition function In the proof of Lemma 3, the calculation of the marginal probability that the root vertex is fixed + yields a local recursive procedure. We will use the recursive formula (3) to compute the marginal probability that the root is set to +. If the tree is truncated at a height t, it is easy to see that the computational complexity of this procedure is linear in the number of vertices of the truncated tree. Let Φ1 denote the whole state space, which means P G ( X 1 = +|Φ1 ) = P G ( X 1 = +), and Φ j = { X i = +, 1 i j − 1}, 2 j n + 1. Let p j be an estimator of conditional marginal probability p j = P G ( X j = +|Φ j ), j = 1, 2, . . . , n. The algorithm for approximating the partition function is proposed as follows:
J. Zhang et al. / Information Processing Letters 111 (2011) 702–710 Algorithm for Partition Function Z (G) Input: G, a graph with vertices {1, . . . , n}, the two-state spin system on G, Z (G ), the estimator of partition function Z (G ). Output: Begin For j from 1 to n do step 1. Set t j =
log(4n J d −1 ) log((d−1)tanh J )−1
709
> 0, precision;
+ 1,
step 2. Take the vertex j as root and generate the truncated subtree T saw ( j ) with height t j under the condition Φ j , step 3. Set initial values be 0 s for all the vertices of T saw( j ) at height t j ,
p j through T saw step 4. Computing ( j ) by recursive formula (3). End For n 1 Z (G ) = Z (G , Φn+1 ) i =1 p− Compute i . End
Theorem 3. Let G = ( V , E ) be a graph with vertex set V = {1, 2, . . . , n} and edge set E. There exists a positive number d > 0 such that (G ) d. If J < J d , then the above algorithm provides an FPTAS for the partition function of the two-state spin system on G. Proof. According to the results of Theorem 1, for any j = 1, 2, . . . , n, there is a function f (t j ) n such that
e− n
pj
pj
en,
under the condition that
tj = Since p j =
log(4n J d −1 ) log((d − 1) tanh J )−1 Z (G ,Φ j +1 ) , Z (G ,Φ j )
+ 1 = O log n + log −1 .
Z (G ) can be expressed as the product Z (G ) = Z (G , Φn+1 )
e − e (− n )n
n p −1 i
−1
p i =1 i
=
Z (G ) Z (G )
n
i =1
1 p− i . Hence,
e ( n )n e .
The complexity of the algorithm for each j in the loop is
O V ( T saw( j ) , j , t j ) = O (d − 1)t j . Thus, the total complexity of the algorithm is
O (1 ) −1 −1 , O (d − 1) O (log n+log( )) = nO (d − 1) O (log n+log( )) = O n O (1) + n −1 j
which completes the proof.
2
5. Conclusions and discussion In this paper, we show that for arbitrary external field, the Gibbs measure exhibits strong correlation decay on a bounded degree tree when J < J d , where J d is the critical point for the uniqueness of Gibbs measures of (anti)ferromagnetic Ising model on an infinite d regular tree [4,12]. This generalizes the recent result by Mossel and Sly [10] to the antiferromagnetic cases. By the strong correlation decay, it is proved that there exists a unique Gibbs measure of the two-state spin system on a bounded degree infinite graph. We tighten Dobrushion’s condition d tanh( J ) < 1 to (d − 1) tanh( J ) < 1 for the uniqueness of Gibbs measure of antiferromagnetic and ferromagnetic Ising models [3,4,12]. This condition turns out to be sharp for the uniqueness of Gibbs measure of Ising model on an bounded degree infinite graph. An interesting investigation could be whether the condition (d − 1) tanh( J ) < 1 is sharp for the general two-state spin system. An FPTAS for the partition functions of two-state spin systems on the bounded degree graphs is also presented. Acknowledgement The authors would like to thank the anonymous referees for their careful reading and helpful comments and suggestions. References [1] A. Bandyopadhyay, D. Gamarnik, Counting without sampling: New algorithms for enumeration problems using statistical physics, Random Structures and Algorithms 33 (2008) 452–479. [2] A. Dembo, A. Montanari, Ising models on locally tree-like graphs, The Annals of Applied Probability 20 (2) (2010) 565–592.
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[3] R.L. Dobrushin, Prescribing a system of random variables by the help of conditional distributions, Theory Probability and Its Application 15 (1970) 469–497. [4] H.O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. [5] L.A. Goldberg, M. Jerrum, M. Paterson, The computational complexity of two-state spin systems, Random Structures and Algorithms 23 (2003) 133–154. [6] M. Jerrum, A. Sinclair, Polynomial-time approximation algorithms for Ising model, SIAM Journal on Computing 22 (5) (1993) 1087–1116. [7] J. Jonasson, Uniqueness of uniform random colorings of regular trees, Statistics and Probability Letters 57 (2002) 243–248. [8] K. Jung, D. Shah, Inference in binary pair-wise Markov random field through self-avoiding walk, http://arxiv.org/abs/cs.AI/0610111v2. [9] R. Lyons, The Ising model and percolation on trees and treelike graphs, Communications in Mathematical Physics 125 (2) (1989) 337–353. [10] E. Mossel, A. Sly, Rapid mixing of Gibbs sampling on graphs that are sparse on average, Random Structures and Algorithms 35 (2) (2009) 250–270. [11] R. Pemantle, Y. Peres, The critical Ising model on trees, concave recursions and nonlinear capacity, The Annals of Probability 38 (1) (2010) 184–206. [12] D. Weitz, Combinatorial criteria for uniqueness of Gibbs measures, Random Structures and Algorithms 27 (2005) 445–475. [13] D. Weitz, Counting independent sets up to the tree threshold, in: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149.
Further reading [14] M. Jerrum, L. Valiant, V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science 43 (1986) 169–188. [15] L.G. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8 (1979) 189–201.
