Contraction: a Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

We study complex zeros of the partition function of 2-spin systems, viewed as a multivariate polynomial in terms of the edge interaction parameters and the uniform external field. We obtain new zero-free regions in which all these parameters are complex-valued. Crucially based on the zero-freeness, we show the existence of correlation decay in these regions. As a consequence, we obtain an FPTAS for computing the partition function of 2-spin systems on graphs of bounded degree for these parameter settings. We introduce the contraction property as a unified sufficient condition to devise FPTAS via either Weitz's algorithm or Barvinok's algorithm. Our main technical contribution is a very simple but general approach to extend any real parameter of which the 2-spin system exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. This result formally establishes the inherent connection between two distinct notions of phase transition for 2-spin systems: the existence of correlation decay and the zero-freeness of the partition function via a unified perspective, contraction.


Introduction
Spin systems originated from statistical physics to model interactions between neighbors on graphs.In this paper, we focus on 2-state spin (2-spin) systems.Such a system is specified by two edge interaction parameters β and γ, and a uniform external field λ.An instance is a graph G = (V, E).A configuration σ is a mapping σ : V → {+, −} which assigns one of the two spins + and − to each vertex in V .The weight w(σ) of a configuration σ is given by w(σ) = β m + (σ) γ m − (σ) λ n + (σ) , where m + (σ) denotes the number of (+, +) edges under the configuration σ, m − (σ) denotes the number of (−, −) edges, and n + (σ) denotes the number of vertices assigned to spin +.The partition function Z G (β, γ, λ) of the system parameterized by (β, γ, λ) is defined to be the sum of weights over all configurations, i.e., Z G (β, γ, λ) It is a sum-of-product computation.If a 2-spin system is restricted to graphs of degree bounded by ∆, we say such a system is ∆-bounded.
In classical statistical mechanics the parameters (β, γ, λ) are usually non-negative real numbers, and such 2-spin systems are divided into ferromagnetic case (βγ > 1) and antiferromagnetic case (βγ < 1).The case βγ = 1 is degenerate.When (β, γ, λ) are non-negative numbers and they are not all zero, the partition function can be viewed as the normalizing factor of the Gibbs distribution, which is the distribution where a configuration σ is drawn with probability Pr G;β,γ,λ (σ) = w(σ) Z G (β,γ,λ) .However, it is meaningful to consider parameters of complex values.By analyzing the location of complex zeros of the partition function, the phenomenon of phase transitions was defined by physicists.One of the first and also the best known result is the Lee-Yang theorem [21] for the Ising model, a special case of 2-spin systems.This result was later extended to more general models by several people [1,33,36,29,24].In this paper, we view the partition function Z G (β, γ, λ) as a multivariate polynomial over these three complex parameters (β, γ, λ).We study the zeros of this polynomial and the relation to the approximation of the partition function.
Partition functions encode rich information about the macroscopic properties of 2-spin systems.They are not only of significance in statistical physics, but also are well-studied in computer science.Computing the partition function of 2-spin systems given an input graph G can be viewed as the most basic case of Counting Graph Homomorphisms (#GH) [11,6,14,8] and Counting Constraint Satisfaction Problems (#CSP) [10,9,5,12,7], which are two very well studied frameworks for counting problems.Many natural combinatorial problems can be formulated as 2-spin systems.For example, when β = γ, such a system is the famous Ising model.And when β = 0 and γ = 1, Z G (0, 1, λ) is the independence polynomial of the graph G (also known as the hard-core model in statistical physics); it counts the number of independent sets of the graph G when λ = 1.

Related work
For exact computation of Z G (β, γ, λ), the problem is proved to be #P-hard for all complex valued parameters but a few very restricted trivial settings [2,8,9].So the main focus is to approximate Z G (β, γ, λ).This is an area of active research, and many inspiring algorithms are developed.The pioneering algorithm developed by Jerrum and Sinclair gives a fully polynomial-time randomized approximation scheme (FPRAS) for the ferromagnetic Ising model [19].This FPRAS is based on the Markov Chain Monte Carlo (MCMC) method which devises approximation counting algorithms via random sampling.Later, it was extended to general ferromagnetic 2-spin systems [15,26].The MCMC method can only handle non-negative parameters as it is based on probabilistic sampling.
The correlation decay method developed by Weitz [42] was originally used to devise deterministic fully polynomial-time approximation schemes (FPTAS) for the hardcore model up to the uniqueness threshold.It turns out to be a very powerful tool for devising FPTAS for antiferromagnetic 2spin systems [43,22,23,38].Combining with hardness results [39,13], an exact threshold of computational complexity transition of antiferromagnetic 2-spin systems is identified and the only remaining case is at the critical point.On the other hand, for ferromagnetic 2-spin systems, limited results [43,17] have been obtained via the correlation decay method.Although correlation decay is usually analyzed in 2-spin systems of non-negative parameters, it can be adapted to complex parameters.An FPTAS was obtained for the hard-core model in the Shearer's region (a disc in the complex plane) via correlation decay in [18].
Recently, a new method developed by Barvinok [3], and extended by Patel and Regts [30] is the Taylor polynomial interpolation method that turns complex zero-free regions of the partition function into FPTAS of corresponding complex parameters.Suppose that the partition function Z G (β, γ, λ) has no zero in a complex region containing an easy computing point, e.g., λ = 0.It turns out that, probably after a change of coordinates, log Z G (β, γ, λ) is well approximated in a slightly smaller region by a low degree Taylor polynomials which can be efficiently computed.This method connects the long-standing study of complex zeros to algorithmic studies of the partition function of physical systems.Motivated by this, more recently some complex zero-free regions have been obtained for hard-core models [4,31], Ising models [27], and general 2-spin systems [16].

Our contribution
In this paper, we obtain new zero-free regions of the partition function of 2-spin systems.Crucially based on the zero-freeness, we show the existence of correlation decay in these complex regions.As a consequence, we obtain an FPTAS for computing the partition function of bounded 2-spin systems for these parameter settings.Our result gives the first zero-free regions in which all three parameters (β, γ, λ) are complex-valued and new correlation decay results for bounded ferromagnetic 2-spin systems.Our main technical contribution is a very simple but general approach to extend any real parameter of which the bounded 2-spin system exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists.We show that for bounded 2-spin systems, the real contraction1 property that ensures correlation decay exists for certain real parameters directly implies the zero-freeness and the existence of correlation decay of corresponding complex neighborhoods.
We formally describe our main result.We use ζ ζ ζ ∈ C 3 to denote the parameter vector (β, γ, λ).Since the case β = γ = 0 is trivial, by symmetry we always assume γ = 0 in this paper.This result formally establishes the inherent connection between two distinct notions of phase transition for bounded 2-spin systems: the existence of correlation decay and the zero-freeness of the partition function, via a unified perspective, contraction.The connection from the existence of correlation decay of real parameters to the zero-freeness of corresponding complex neighborhoods was already observed for the hard-core model [31] and the Ising model without external field [27].
In this paper, we extend it to general 2-spin systems, and furthermore we establish the connection from the zero-freeness of complex neighborhoods back to the existence of correlation decay of such complex regions.Now, we give our zero-free regions.We first identify the sets of real parameters of which bounded 2-spin systems exhibit correlation decay.Definition 1.2.Fix integer ∆ ≥ 3. We have the following four sets where correlation decay exists.
When context is clear, we omit the superscript ∆.
The set S ∆ 1 was given in [43] and S ∆ 2 was given in [23].To our best knowledge, S ∆ 1 and S ∆ 2 cover all non-negative parameters of which bounded 2-spin systems are known to exhibit correlation decay.The sets S ∆ 3 and S ∆ 4 are obtained in this paper.They give new correlation decay results and hence FPTAS for bounded ferromagnetic 2-spin systems 1 .

Organization
This paper is organized as follows.In Section 2, we briefly describe Weitz's algorithm [42].We introduce real contraction as a sufficient condition for the existence of correlation decay of real parameters, and we show sets S ∆ i (i ∈ [4]) satisfy it.In Section 3, we briefly describe Barvinok's algorithm [3].We introduce complex contraction as a generalization of real contraction, and we show that it gives a unified sufficient condition for both the zero-freeness of the partition function and the existence of correlation decay of complex parameters.Finally, in Section 4, we prove our main result that real contraction implies complex contraction.This finishes the proof of Theorem 1.3.We use the following diagram (Figure 1) to summarize our approach to establish the connection between correlation decay and zero-freeness.We expect it to be further explored for other models.

FPTAS Correlation Decay
Weitz's Algorithm 7 7 Figure 1: The structure of our approach 1 When β < γ and λ is sufficiently large, it is known that approximating the partition function of ferromagnetic 2-spin systems over general graphs is #BIS-hard [26].Our result S ∆ 4 shows that there is an FPTAS for such a problem when restricted to graphs of bounded degree.When β < 1 < γ, the FPTAS obtained from S ∆ 3 is covered by [17].

Independent work
After a preliminary version [35] of this manuscript was posted, we learned that based on similar ideas, Liu simplified the proofs of [31] and [27] and generalized them to antiferromagnetic Ising models (β = γ < 1) in chapter 3 of his Ph.D. thesis [25], where similar zero-freeness results (a complex neighborhood of S ∆ 2 restricted to β = γ) were obtained.We mention that by using the unique analytic continuation and the inverse function theorem, our main technical result (Theorem 4.4) is generic; it does not rely on a particularly chosen potential function.Thus, in our approach we can work with any existing potential function based arguments for correlation decay even if the potential function does not have an explicit expression, for instance, the one used in [23] when β = γ.Furthermore, we mention also that based on the zero-freeness, we obtain new correlation decay result for complex parameters (Lemma 3.4).Note that Barvinok's algorithm requires an entire region in which the partition function is zero-free and there is an easy computing point.However, our correlation decay result shows that one can always devise an FPTAS for these parameter settings via Weitz's algorithm, even if Barvinok's algorithm fails.

Weitz's Algorithm
In this section, we describe Weitz's algorithm.We first consider positive parameters any graph G.This is true even if G contains arbitrary number of vertices pinned to spin + or −.Then, the partition function can be viewed as the normalizing factor of the Gibbs distribution.Let σ Λ ∈ {0, 1} Λ be a configuration of some subset Λ ⊆ V .We allow Λ to be the empty set.We call vertices in Λ pinned and other vertices free.We use p σ Λ v (ζ ζ ζ) to denote the marginal probability of a free vertex v (v / ∈ Λ) being assigned to spin + conditioning on the configuration

Notations and definitions
is the weight over all configurations where vertices in Λ are pinned by the configuration σ Λ , and be the ratio between the two probabilities that the free vertex v is assigned to spin + and −, while imposing some condition σ for convenience.Since computing the partition function of 2-spin systems is self-reducible, if one can compute p v for any vertex v, then the partition function can be computed via telescoping [20].The goal of Weitz's algorithm is to estimate p σ Λ v , which is equivalent to estimating R σ Λ G,v .For the case that the graph is a tree T , R σ Λ T,v can be computed by recursion.Suppose that a free vertex v has d children, and s 1 of them are pinned to +, s 2 are pinned to −, and k are free (s 1 + s 2 + k = d).We denote these k free vertices by v i (i ∈ [k]) and let T i be the corresponding subtree rooted at v i .We use σ i Λ to denote the configuration σ Λ restricted to T i .Since all subtrees are independent, it is easy to get the following recurrence relation, , Remark.Every recursion function is analytic on its domain.
For a general graph G, Weitz reduced computing R σ Λ G,v to that in a tree T , called the self-avoiding walk (SAW) tree, and Weitz's theorem states that [42].(See the appendix for more details.)We want to generalize Weitz's theorem to complex parameters v no longer has a probabilistic meaning.It is just a ratio of two complex numbers.However, one can easily observe that for some special parameters, there are trivial configurations such that We will rule these cases out as they are infeasible.
• σ Λ does not assign any vertex in G to spin + if λ = 0, and • σ Λ does not assign any two adjacent vertices in G both to spin + if β = 0.
Remark.Let σ Λ be a feasible configuration.If we further pin one vertex v / ∈ Λ to spin −, and get the configuration σ Λ on Λ = Λ ∪ {v}, then σ Λ is still a feasible configuration.Thus, given for any graph G and any arbitrary feasible configuration σ Λ on G, then both we can still compute it by recursion via SAW tree.We first consider the case that λ = 0. Let σ Λ be a feasible configuration.It is easy to verify that the corresponding configuration on the SAW tree is also feasible and Weitz's theorem still holds.For the case that λ = 0, it is obvious that R σ Λ G,v ≡ 0 for any graph G, any free vertex v and any feasible configuration σ Λ .This is equal to the value of recursion functions G,v can be computed by recursion functions when λ = 0, although Weitz's theorem does not hold for this case.For the case that β = 0, we have R σ Λ G,v = 0 if one of the children of v is pinned to +.Then, we may view v as it is pinned to −.Thus, for β = 0, we only consider recursion functions F s where s 1 = 0.

Correlation decay and real contraction
The SAW tree may be exponentially large in size of G.In order to get a polynomial time approximation algorithm, we may run the tree recursion at logarithmic depth and hence in polynomial time, and plug in some arbitrary values at the truncated boundary.We have the following notion of strong spatial mixing (SSM) to bound the error caused by arbitrary guesses.It was originally introduced for non-negative parameters.Here, we extend it to complex parameters.
where S ⊆ Λ 1 ∪ Λ 2 is the subset on which σ Λ 1 and τ Λ 2 differ1 , and dist G (v, S) is the shortest distance from v to any vertex in S.
+ , condition 1 is always satisfied.Condition 2 is a stronger form of SSM of real parameters (see Definition 5 of [23]).For real values, by monotonicity one can restrict to the case that Λ 1 = Λ 2 (the two configurations are on the same set of vertices).Here, we allow Λ 1 = Λ 2 .
In statistical physics, SSM is called correlation decay.If SSM holds, then the error caused by arbitrary boundary guesses at logarithmic depth of the SAW tree is polynomially small.Hence, Weitz's algorithm gives an FPTAS.A main technique that has been widely used to establish SSM is the potential method [32,22,23,37,17].Instead of bounding the rate of decay of recursion functions directly, we use a potential function ϕ(x) to map the original recursion to a new domain (See Figure 2 for the commutative diagram).Let F s (ζ ζ ζ, y) be a recursion function.We use We say ϕ defined on J is a good potential function for ζ ζ ζ.
Remark.Since ϕ is analytic and ϕ (x) = 0 for x ∈ J, we have ϕ is invertible and the inverse ϕ −1 : I → J is also analytic by the inverse function theorem.Also for every s with is well-defined on I k .We know I is also a real compact interval since J is a real compact interval and ϕ is a real analytic function.

Note that since
that in a tree of degree at most ∆, only the root node may have ∆ many children, while other nodes have at most ∆ − 1 many children.Proof.The proof directly follows from the argument of the potential method, see [23,17].The FPTAS follows from Weitz's algorithm.Now, we give the sets of non-negative parameters which satisfy real contraction.
where xd is the unique positive fixed point of the function ) be the correlation decay sets defined in Definition 1.2.The set S ∆ 1 was given in [43] and S ∆ 2 was given in [23].Directly following their proofs, it is easy to verify that both sets satisfy real contraction.The sets S ∆ 3 and S ∆ 4 are obtained in this paper, and we show that they also satisfy real contraction.We will give a proof in the appendix for every ), it satisfies real contraction for ∆.In order to generalize the correlation decay technique to complex parameters, we need to ensure that the partition function is zero-free.Now, let us first take a detour to Barvinok's algorithm which crucially relies on the zero-free regions of the partition function.After we carve out our new zero-free regions, we will come back to the existence of correlation decay of complex parameters.

Barvinok's Algorithm
In this section, we describe Barvinok's algorithm.Let I = [0, t] be a closed real interval.We define the δ-strip of I to be {z ∈ C | |z − z 0 | < δ, z 0 ∈ I}, denoted by I δ .It is a complex neighborhood of I. Suppose a graph polynomial P (z) = n i=0 a i z i of degree n is zero-free in I δ .Barvinok's method [3] roughly states that for any z ∈ I δ , P (z) can be (1 ± ε)-approximated using coefficients a 0 , . . ., a k for some k = O(e Θ(1/δ) log(n/ε)), via truncating the Taylor expansion of the logarithm of the polynomial.For the partition function of 2-spin systems, these coefficients can be computed in polynomial-time [30,28].For the purpose of obtaining FPTAS, we will view the partition function as a univariate polynomial Z G;β,γ (λ) in λ and fix β and γ.The following result is known.Proof.This lemma is a generalization of Lemma 4 in [16], where β and γ are both real.The generalization to complex valued parameters directly follows from the argument in [28].

Zero-freeness and complex contraction
With Lemma 3.1 in hand, the main effort is to obtain zero-free regions of the partition function.For this purpose, we will still view Z G (ζ ζ ζ) as a multivariate polynomial in (β, γ, λ).A main and widely-used approach to obtain zero-free regions is the recursion method [40,34,4,31,27].This method is related to the correlation decay method.
As pointed above, the ratio R G,v can be computed by recursion via the SAW tree in which v is the root.Roughly speaking, the key idea of the recursion method is to construct a contraction region Q ⊆ C where λ ∈ Q and −1 / ∈ Q such that for all recursion functions This condition guarantees that with the initial value R G,v = λ where v is a free leaf node in the SAW tree of which the degree is bounded by ∆, the recursion will never achieve −1.Remark.Similar to the remark of Definition 2.4, we have is well-defined and analytic on P k .Here, we directly assume that the inverse ϕ −1 is analytic instead of ϕ (x) = 0 for the sake of simplicity of our proof.Please see the appendix for the proof.Such a proof only uses condition 1 of complex contraction.However, condition 2 combining with the zero-freeness result of Lemma 3.3 gives a sufficient condition for bounded 2-spin systems of complex parameters exhibiting correlation decay.This is a generalization of Lemma 2.5.Also, we will give the proof in the appendix.

From Real Contraction to Complex Contraction
In this section, we will prove our main result.We first give some preliminaries in complex analysis.The main tools are the unique analytic continuation and the inverse function theorem.Here, we slightly modify the statements to fit for our settings.Please refer to [41] for the proofs.Theorem 4.1 (Unique analytic continuation).Let f (x) be a (real) analytic function defined on a compact real interval I ⊆ R.Then, there exists a complex neighborhood I ⊆ C of I, and a (complex) analytic function f (x) defined on I such that f (x) ≡ f (x) for all x ∈ I.Moreover, if there is another (complex) analytic function g(x) also defined on I such that g(x) ≡ f (x) for all x ∈ I and the measure m(I) = 0, then g(x) ≡ f (x) for all x ∈ I.We call f (x) the unique analytic continuation of f (x) on I. Theorem 4.2 (Inverse function theorem).Let ϕ be a (complex) analytic function defined on U ⊆ C, and ϕ (z) = 0 for some z ∈ U .Then there exists a complex neighborhood D of z such that ϕ is invertible on D and the inverse is also analytic.
Combining the above theorems, we have the following result.Lemma 4.3.Let ϕ : J → I be a real analytic function, and ϕ (x) = 0 for all x ∈ J where J and I are both real compact intervals.Then, there exists an analytic continuation ϕ on a complex neighborhood J of J such that ϕ is invertible on J and the inverse ϕ −1 is also analytic.
Proof.If m(J) = 0, i.e., J = {x}, then by Theorem 4.2, there exists an analytic continuation ϕ of ϕ defined on a neighborhood of x on which ϕ is invertible and the inverse ϕ −1 is analytic.
Otherwise, m(J) = 0. Since ϕ(x) is analytic and ϕ (x) = 0 for all x ∈ J, we have ϕ is invertible and by Theorem 4.2, the inverse ϕ −1 : I → J is analytic on I.By Theorem 4.1, there exists an analytic continuation ϕ −1 of ϕ −1 defined on a neighborhood I 1 of I. Similarly, there exists an analytic continuation ϕ of ϕ defined on a neighborhood J of J.We use I to denote the image ϕ( J).Since ϕ is analytic and by the open mapping theorem, we know I is an open set in the complex plane.Clearly, we have ϕ(J) = I ⊆ I.We can pick J small enough while still keeping J ⊆ J such that the image I = ϕ( J) ⊆ I 1 and still I ⊆ I. Thus, the composition ϕ −1 • ϕ is a well-defined analytic function on J. Clearly, we have Since m(J) = 0, by Theorem 4.1, we have ϕ −1 • ϕ(x) ≡ x for all x ∈ J.
Thus, ϕ is invertible on J and the inverse ϕ −1 = ϕ −1 is analytic.Now, we are ready to prove our main result.We first show that we can pick a pair of (δ 1 , ε 1 ) such that for every s with s 1 ≤ ∆ − 1, the composition Given some s with s 1 ≤ ∆ − 1, we consider the function We know that it is analytic on a neighborhood of {ζ ζ ζ 0 } × J k and by real contraction we have Then, we can pick some δ s and a neighborhood J s of J that are small enough such that Since there is only a finite number of s with s 1 ≤ ∆ − 1, we know δ 1 > 0, and J 1 is open and it is a neighborhood of J.We have F s (B δ 1 , J 1 ) ⊆ J for every s with s 1 ≤ ∆ − 1.Since ϕ −1 is analytic on I and ϕ −1 (I) = J, similarly we can pick a small enough neighborhood I 1 of I where I 1 ⊆ I such that ϕ −1 ( I 1 ) ⊆ J 1 .For every z 0 ∈ I, we can pick an ε z 0 such that the disc Recall that I is a compact real interval, by the finite cover theorem, we can uniformly pick a ε 1 such that I ⊆ I ε 1 ⊆ I 1 .Thus, we have ) is well-defined and analytic on B δ 1 × I k ε 1 for every s with s 1 ≤ ∆ − 1.In fact, F ϕ s is a (multivariate) analytic continuation of F ϕ s .Since I is a compact interval, in the following when we pick a neighborhood I of I, without loss of generality, we may always pick I as an ε-strip I ε of I.
Then, we show that we can pick a pair of (δ 2 , ε 2 ) where δ 2 < δ 1 and ε 2 < ε 1 , a constant M > 0 and a constant η > 0 such that for every s with s 1 ≤ ∆ − 1, we have 1 − η for every s with s 1 ≤ ∆ − 1 and all x ∈ I k .Given some s with , by continuity we can pick some δ s < δ 1 and In addition, let and we know M s < +∞ since F ϕ is analytic on B δs × I k εs which is close and bounded.Finally, let {ε s }, and M = max These choices will satisfy our requirement.

A Appendix
A.1 Self-Avoiding Walk Tree We adapt the description of Weitz's self-avoiding walk (SAW) tree construction from [17] with slight modifications.Given a graph G = (V, E) and a vertex v ∈ V , the SAW tree of G at v denoted by T SAW (G, v), is a tree with root v that enumerates all paths originating from v in G.
Additional vertices closing cycles of G are added as leaves of the tree (see Figure 3 for an example).
Each vertex in V of G is mapped to some vertices in V SAW T SAW (G, v).For leaves in V SAW that close cycles, a boundary condition is imposed.The imposed spin of such a leaf depends on whether the orientation of the cycle is from a lower indexed vertex to a higher indexed vertex or conversely, where the order of indices is arbitrarily chosen in G. Vertex sets S ⊆ Λ ⊆ V are mapped to S SAW ⊆ Λ SAW ⊆ V SAW respectively, and any configuration Here is the key result (Theorem 3.1 of Weitz [42]) for the SAW tree construction.
∈ Λ, and S ⊆ V .Then, we have Moreover, dist G (v, S) = dist T (v, S SAW ), the maximum degree of T is equal to the maximum degree of G, and the neighborhood of any vertex in V SAW can be constructed in time proportional to the size of the neighborhood of the corresponding vertex in V .
We first consider a trivial case that λ = 0, in which F ζ ζ ζ,s (x) ≡ 0. We pick J = [0, 1] and the potential function ϕ(x) = x.Clearly, ϕ is analytic on J and ϕ (x) = 1 = 0 for all x ∈ J. Also, we know λ = 0 ∈ J, −γ / ∈ J and −1 / ∈ J.Moreover, for every s ∈ N 3 and all x ∈ J k , we have Thus, the function ϕ defined on J is a good potential function for ζ ζ ζ.
We characterize the point x at which h d (x) achieves its maximum.Recall that we define . Consider the derivative of h d (x), we have where and its derivative We want to solve h d (x) = 0. Since 1 − βγ > 0, it is equivalent to solve the equation Note that as x increases from 0 to +∞, the function γ−βx 2 d(1−βγ)x strictly decreases from +∞ to −∞.On the other hand, the function (βf d (x)+1)(f d (x)+γ) strictly increases since f d (x) strictly decreases as x increases.Therefore equation (1) has a unique solution in (0, +∞), denoted by x d .Furthermore, we have Clearly the sign of h d (x) is the same as that of g d (x).Hence h d (x) achieves its maximum when x = x d .Then for any x > 0, we have Here, we substitute (βf according to (1).Consider the function .
Then, we have h d (x) ≤ p d (x d ) for any x > 0. Now, we claim that for any 1 ≤ d ≤ ∆ − 1, where xd is the unique positive fixed point of f d (x).To prove the above claim, we only need to show that p d (x) is decreasing if xd ≤ x d and increasing if xd > x d .
• Combining (3) and (4), we have for all x > 0, since we further pinned one vertex of G to spin −.Thus, the ratio is well-defined and it can be computed by recursion via SAW tree.Let T be the corresponding SAW tree where v is the root.There exists an s with s 1 ≤ ∆ such that where v 1 , . . ., v k are free vertices of the children of v and T 1 , . . ., T k are the corresponding subtrees rooted at them.Note that in T , only v may have ∆ many children, while other nodes have at most ∆ − 1 many children.Therefore, for any node v = v and the subtree rooted at v , the ratio R σ Λ T ,v can be computed by some recursion function Clearly, for any free vertex v at the leaf of T , we have R σ Λ T ,v = λ ∈ Q, where T is a tree of only one vertex v .By complex contraction, we have A.4 Proof of Lemma 3.4 Proof.By Lemma 3.3, we know condition 1 of SSM (Definition 2.3) is satisfied.We only need to show that condition 2 is satisfied.If λ = 0, then we have p σ Λ v ≡ 0 for any feasible configuration σ Λ and SSM holds trivially.Thus, we assume that λ = 0.By Weitz's SAW tree construction, we only need to show that the 2-spin systems on trees of degree at most ∆ exhibits SSM.
Let ϕ : P → Q be a good potential function for ζ ζ ζ.Let T = (V, E) be a tree of degree at most ∆ and v be the root of T .Consider two feasible configurations σ Λ 1 and τ Λ 2 on Λ 1 ⊆ V and Λ 2 ⊆ V respectively where v / ∈ Λ 1 ∪ Λ 2 .We want to show that p , where S ⊆ Λ 1 ∪ Λ 2 is the subset on which σ Λ 1 and τ Λ 2 differ.Note that all vertices in T except the root v have at most ∆ − 1 many children.We first consider the case that v has at most ∆ − 1 many children.Let t = dist T (v, S).We will show ϕ R for some constant C, η > 0 by induction on t. ≤ C(1 − η) t−1 for t ≤ n where n is a positive integer.We consider t = n + 1.Since t > 1, the configurations of all children of v are the same in both σ Λ 1 and τ Λ 2 .Suppose that v has d children, and in both configurations σ Λ 1 and τ Λ 2 , s 1 of them are pinned to +, s 2 are pinned to −, and k are free.We denote these k free vertices by v i (i ∈ [k]).Let T i be the corresponding subtree rooted at v i , and σ i Λ 1 and τ i Λ 2 denote the configurations σ Λ 1 and τ Λ 2 restricted on subtree T i respectively.Let T i ,v i .Since ζ ζ ζ satisfies complex contraction, same as we showed in the proof of Lemma 3.3, we have R T i ,v i ∈ Q and hence x i , y i ∈ P .Let S i = S ∩ T i .Clearly, we have dist T i (v i , S i ) ≥ dist T (v, S) − 1 = t − 1 = n.By induction hypothesis, we have where the first inequality is due to the fact that P is convex.We are going to bound R Since there is only a finite number of s such that s 1 ≤ ∆, we have M < +∞ and Q is compact, and hence C < +∞.Let N = max{C , CM }.We show that R If t = 1, then there exist s 1 and s 2 where s 1 1 , s 2 1 ≤ ∆ such that R T,v ∈ Q , and hence Otherwise t > 1.The configurations of all children of v are the same in both σ Λ 1 and τ Λ 2 .Again, T i ,v i and y i = ϕ R T i ,v i , where v i , T i , σ i Λ 1 , τ i Λ 2 and S i (i ∈ [k]) are all defined the same as in the above induction proof.Since in the subtree T i , the root v i has at most ∆ − 1 many children, we have Then, similarly as we did in the above induction proof, we have ≤ sup Finally, we bound p T,v .Let K = inf z∈Q |1 + z|.By the complex contraction property, −1 / ∈ Q s for every s 1 ≤ ∆ and thus −1 / ∈ Q .Also, since Q is compact, we have K > 0. Hence,

Definition 2 . 3 (
Strong spatial mixing).A 2-spin system specified by ζ ζ ζ ∈ C 3 on a family G of graphs is said to exhibit strong spatial mixing if for any graph

1 ≤ 1 −
Hence, we have Z G (ζ ζ ζ) = 0 by induction.Again, we may use a potential function ϕ : Q → P to change the domain, and we prove F ϕ ζ ζ ζ,s (P k ) ⊆ P .Now, we introduce the following complex contraction property as a generalization of real contraction.This property gives a sufficient condition for the zero-freeness of the partition function.Definition 3.2 (Complex contraction).Fix ∆ ∈ N. We say ζ ζ ζ ∈ C 3 satisfies complex contraction for ∆ if there is a closed and bounded complex region Q ⊆ C where λ ∈ Q, −γ / ∈ Q and −1 / ∈ Q, and an analytic and invertible function ϕ : Q → P where the inverse ϕ −1 : P → Q is also analytic and P is convex, such that 1. F ζ ζ ζ,s (Q k ) ⊆ Q for every s with s 1 ≤ ∆ − 1 and −1 / ∈ F ζ ζ ζ,s (Q k ) for every s with s 1 = ∆; 2. there exists η > 0 s.t.∇F ϕ ζ ζ ζ,s (x) η for every s with s 1 ≤ ∆ − 1 and all x ∈ P k .