## Abstract

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most \(d\), throughout the uniqueness region of the Gibbs measure on the infinite \(d\)-regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree \(d\) for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.

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## Notes

- 1.
The description of the Ising model given here differs slightly from the more popular description in terms of edge and vertex potentials outlined in the first paragraph. However, translating between the two descriptions is easy; see Appendix 1.

- 2.
We remark here that the infinite \((d+1)\)-regular tree (also known as the

*Bethe lattice*) and the infinite \(d\)-ary tree show exactly the same behavior with respect to the uniqueness of the Gibbs measure. This follows immediately from the fact that the \((d+1)\)-regular tree can be viewed as a root attached to the roots of \(d+1\) infinite \(d\)-ary trees. We shall thus move freely between these two objects for ease of exposition throughout the paper. - 3.
To be precise, this condition does not hold on the boundary of the uniqueness region, that is, for \(\left| \log \lambda \right| = \log \lambda _c(\beta ,d)\); at this critical value, the l.h.s of Eq. (3) still decays to \(0\) with \(l\), but not at an exponential rate. We will focus on the interior of this region, and by a slight abuse of terminology refer to it as the “uniqueness region”.

- 4.
We refer to Sect. 2.1 for a formal definition of an FPTAS.

- 5.
- 6.
As stated in the introduction, we exclude the boundary of the uniqueness region here.

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## Acknowledgments

We thank Prasad Tetali for providing a manuscript of [11]. We also thank Colin McQuillan, Dror Weitz, Yitong Yin and two anonymous referees for several helpful comments. Alistair Sinclair was supported in part by United States National Science Foundation (NSF) Grant CCF-1016896. Piyush Srivastava was supported by the Berkeley Fellowship for Graduate Study and by NSF grant CCF-1016896, and performed part of this work while he was a research intern at Microsoft Research India. Marc Thurley was supported in part by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by Marie Curie Intra-European Fellowship 271959, and performed part of this work while he was a postdoctoral scholar at the University of California, Berkeley, and at the Centre de Recerca Mathemàtica, Bellaterra.

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## Additional information

A preliminary version of this paper appeared in the proceedings of the *23rd Annual ACM-SIAM Symposium on Discrete algorithms (SODA)*, pp. 941–953, 2012.

## Appendices

### Appendix 1: Translation Between Various Descriptions of the Ising Model

General two-state spin systems are usually described in terms of (symmetric) energy functions \(Q(+~, +~)\), \(Q(+~, {-}~) = Q({-}~, +~)\) and \(Q({-}~, {-}~)\), and an odd vertex field \(h(+ ) \!=\! -h(-) = h\). For a graph \(G = (V,E)\), the partition function of the system is then \(Z_2 = \sum _{\sigma } w_2(\sigma )\), where the sum is over all states of the system \(\sigma : V \rightarrow \{+,-\}\) and \(w_2(\sigma )\) is defined as

This is in fact equivalent to our formulation of the system given in Sect. 1.1 (see Eqs. (1) and (2)). To see this, define

which yields

for all \(\sigma \), and, therefore,

We call the above spin systems *soft constraint* systems if \(\beta ,\gamma \) and \(\lambda \) are non-zero, or equivalently, if the energy functions and field are finite for all spin values. As we shall now see, every such soft constraint system can be represented in terms of the Ising model (this translation can also be found, e.g., in [6]). Consider a general two-state spin system with parameters \(\beta ,\gamma > 0\) and \(\lambda \). Then the equivalent Ising model has edge activity

and a degree-dependent vertex activity given by

where \(d_v\) denotes the degree of vertex \(v\). Now, denote the weight of a configuration \(\sigma \) in the Ising model just defined by \(w^\star (\sigma )\) and its partition function by \(Z^\star \). Then one calculates straightforwardly that

and hence

Thus we have translated the original spin system with parameters \((\beta , \gamma , \lambda )\) into an Ising model with locally changing field. Note that on regular graphs the resulting field is in fact constant at all vertices. Furthermore, the Ising model is anti-ferromagnetic if and only if \(\beta \gamma < 1\). This justifies our use of the term “anti-ferromagnetic” for general spin systems based on the value of \(\beta \gamma \). We also observe that in the special case of \(d\)-regular trees, this implies that weak (strong) spatial mixing in the original spin system \((\beta , \gamma , \lambda )\) is equivalent to weak (strong) spatial mixing in the Ising model given by the translation. A little thought shows that since all vertices except the root have the same degree in the \(d\)-ary tree, the last observation holds also for \(d\)-ary trees.

### Appendix 2: Weak Spatial Mixing Does Not Imply Strong Spatial Mixing for the Ferromagnetic Ising Model

We construct a counterexample as follows: given a degree \(d \ge 3\), consider the infinite rooted \(d\)-ary tree, with the fixed boundary condition where each vertex in the tree has one of its children fixed to \(+\). Notice that if the original parameters are \(\beta \ge 1\) and \(\lambda \), the effect of this fixed boundary condition can be simulated by changing the vertex field to \(\frac{\lambda }{\beta }\). Therefore, strong spatial mixing on this subgraph of the \(d\)-ary tree with parameters \((\beta , \lambda )\) holds only if weak spatial mixing holds on the \((d-1)\)-ary tree with parameters \((\beta , \frac{\lambda }{\beta })\). It is therefore sufficient to choose \(\beta \) and \(\lambda \) satisfying both the conditions

in order to construct a counterexample. To see that such a choice of parameters is possible, we consider the exact form of \(\lambda _c(\beta , d)\). Translating the results in [4, p. 250] to our notation using Appendix 1, we have

where

We note that \(P(d)\) is an increasing function of \(d\). Thus, the required conditions (19) and (20) become

For a fixed \(d\), \(Q(\beta ,d)\) is \(o_\beta (1)\), and hence inequality (22) implies inequality (21) for \(\beta \) large enough and \(d \ge 3\). Again, since \(Q(\beta , d)\) is \(o_\beta (1)\), and \(P(d)\) is an increasing function, it is possible to find \(\lambda \) satisfying both the inequalities (22) and (23) for \(\beta \) large enough. Thus, for any \(d \ge 3\), we can find \((\beta , \lambda )\) with \(\beta > 1\) such that weak spatial mixing for the \(d\)-ary tree does not imply strong spatial mixing.

### Appendix 3: Contraction and Spatial Mixing

###
**Contraction and Weak Spatial Mixing**

We show in this section that a stepwise contraction in the recurrence for \(\phi (p_v)\) implies weak spatial mixing. As before, we denote \(f^\phi \) by \(g\), and assume that for any \(x\), \(y\) in the range of \(\phi \), it holds that

for some \(c < 1\). To show that this implies weak spatial mixing, we consider boundary conditions \(\sigma _1\) and \(\sigma _2\) on a set \(S\) whose distance from the root \(\rho \) is \(l\). Using the monotonicity of the tree recurrence \(F\) (defined in Eq. (6)) in all its arguments, it can be verified that \(\left| \phi (p_\rho (\sigma _1, S)) - \phi (p_\rho (\sigma _2, S))\right| \) is maximized when \(S\) is the set of all leaves at distance \(l\) from \(\rho \) and \(\sigma _1\) assigns all vertices in \(S\) to \(+\) and \(\sigma _2\) assigns all vertices in \(S\) to \(-\). With this definition of \(\sigma _1\) and \(\sigma _2\), we notice that the tree recurrence for \(\phi (p_v)\) outputs the same value for all vertices \(v\) at the same distance from the root. For a vertex at distance \(l-i\) from the root \(\rho \), we denote by \(q_{i,j}\) the quantity \(\phi (p_v(\sigma _j, S))\), for \(j \in \left\{ 1,2\right\} \). Now, using condition (24), we have

Since both \(\phi \) and \(\phi ^{-1}\) are continuously differentiable functions defined over compact sets, they are Lipschitz continuous, say with parameters \(L_1\) and \(L_2\) respectively. We therefore have weak spatial mixing, since

###
**Contraction and Strong Spatial Mixing**

In this section, we show that contractive spatial mixing, as defined in Definition 6 implies strong spatial mixing. We again consider boundary conditions \(\sigma _1\) and \(\sigma _2\) on a set \(S\) which differ only on a subset \(T\) which is at distance \(l\) from the root \(\rho \). Again, since both \(\phi \) and \(\phi ^{-1}\) are continuously differentiable functions defined over compact sets, they are Lipschitz continuous, say with parameters \(L_1\) and \(L_2\) respectively. We define the quantity \(q_i\) as

Notice that \(q_0 \le \left| \phi (1) - \phi (0)\right| \le L_1\). Also, since \(G\) is the tree recurrence for \(\phi \left( p_v\right) \), contractive spatial mixing for \(G\) implies \(q_{i+1} \le cq_{i}\) for \(c < 1\). Thus, we get strong spatial mixing since

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Sinclair, A., Srivastava, P. & Thurley, M. Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs.
*J Stat Phys* **155, **666–686 (2014). https://doi.org/10.1007/s10955-014-0947-5

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### Keywords

- Phase transitions
- Complexity theory
- Approximation algorithms
- Decay of correlations
- Two-spin systems on trees