Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

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.

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

$$\begin{aligned} w_2(\sigma ) \triangleq \exp \left( -\sum _{\left\{ u,v\right\} \in E}Q(\sigma (u), \sigma (v)) - \sum _{v\in V} h(\sigma (v))\right) . \end{aligned}$$

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

$$\begin{aligned} \beta&= \exp \left( -Q(+,+) + Q(+,-)\right) ;\\ \gamma&= \exp \left( -Q(-,-) + Q(+,-)\right) ;\\ \lambda&= \exp (2h), \end{aligned}$$

which yields

$$\begin{aligned} w(\sigma ) = w_2(\sigma ) \exp \left( Q(+,-)\vert E \vert + h\vert V \vert \right) \end{aligned}$$

for all $\sigma$, and, therefore,

$$\begin{aligned} Z = Z_2\exp \left( Q(+,-)\vert E \vert + h\vert V \vert \right) . \end{aligned}$$

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

$$\begin{aligned} \beta ' = \sqrt{\beta \gamma }, \end{aligned}$$

and a degree-dependent vertex activity given by

$$\begin{aligned} \lambda '_v = \lambda \left( \sqrt{\frac{\gamma }{\beta }}\right) ^{d_v}, \end{aligned}$$

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

$$\begin{aligned} w(\sigma ) = w^\star (\sigma ) \left( \sqrt{\frac{\beta }{\gamma }}\right) ^{\vert E \vert } \end{aligned}$$

and hence

$$\begin{aligned} Z = Z^\star \left( \sqrt{\frac{\beta }{\gamma }}\right) ^{\vert E \vert }. \end{aligned}$$

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

$$\begin{aligned} \log \lambda&> \log \lambda _c(\beta , d)\text {, and}\end{aligned}$$

(19)

$$\begin{aligned} 0 < \log \lambda - \log \beta&< \log \lambda _c(\beta ,d-1) \end{aligned}$$

(20)

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

$$\begin{aligned} \log \lambda _c(\beta ,d) = (d-1)\log \beta - P(d) + Q(\beta ,d), \end{aligned}$$

where

$$\begin{aligned} P(d)&\triangleq d\log d - (d-1)\log (d-1),\text { and}\\ \lim _{\beta \rightarrow \infty }Q(\beta , d)&= 0\text {, for any fixed}\,d. \end{aligned}$$

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

$$\begin{aligned} \log \lambda&> \log \beta ,\end{aligned}$$

(21)

$$\begin{aligned} \log \lambda&> (d-1)\log \beta - P(d) + Q(\beta , d)\text {, and}\end{aligned}$$

(22)

$$\begin{aligned} \log \lambda&< (d-1)\log \beta - P(d-1) + Q(\beta , d - 1). \end{aligned}$$

(23)

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

1.1 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

$$\begin{aligned} \left| g(x) - g(y)\right| \le c|x - y|, \end{aligned}$$

(24)

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

$$\begin{aligned} \left| q_{i+1,1} - q_{i+1, 2}\right|&= \left| g(q_{i,1}) - g(q_{i, 2})\right| \\&\le c\left| q_{i, 1} - q_{i,2}\right| . \end{aligned}$$

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

$$\begin{aligned} \left| p_\rho (\sigma _1, S) - p_\rho (\sigma _2, S)\right|&= \left| \phi ^{-1}(q_{l,1}) - \phi ^{-1}(q_{l,2})\right| \nonumber \\&\le L_2c^l\left| q_{0,1} - q_{0,2}\right| \nonumber \\&\le L_1L_2c^l\nonumber . \end{aligned}$$

1.2 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

$$\begin{aligned} q_i \triangleq \max _{v: \delta (\rho , v) = l-i}\left| \phi (p_v(\sigma _1, S)) - \phi (p_v(\sigma _2, S))\right| . \end{aligned}$$

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

$$\begin{aligned} \left| p_\rho (\sigma _1, S) - p_\rho (\sigma _2, S)\right| \le L_2q_l \le L_2c^lq_0 \le L_1L_2 c^l. \end{aligned}$$