Location of zeros for the partition function of the Ising model on bounded degree graphs

10/03/2018
by   Han Peters, et al.
0

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in C. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree d≥ 3, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree d. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function.

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1. Introduction and main result

For a graph , , the partition function of the Ising model is defined as

(1.1)

where denotes the collection of edges with one endpoint in and one endpoint in . If and are clear from the context, we will often just write instead of . In this paper we typically fix and think of as a polynomial in . The case is often referred to as the ferromagnetic case, while the case where is referred to as the anti-ferromagnetic case.

The Ising model is a simple model to study ferromagnetism in statistical physics. In statistical physics the partition function of the Ising model is often written as

(1.2)

where denotes the coupling constant, the external magnetic field and the temperature. Setting***We remark that in the mathematical physics literature one often sets . , and , then up to a factor of , the two partition functions (1.1) and (1.2) are the same.

Lee and Yang [12] proved that for any graph and any all zeros of lie on the unit circle in . Their result attracted enormous attention in the literature, and similar statements have been proved in much more general settings, see for example [18, 21, 3, 2, 5, 6, 19, 13, 7, 14].

It is known that the union of the roots of over all graphs lies dense in the unit circle. So it is natural to wonder for which classes of graphs and choice of parameters there are zero-free regions on the circle. For the class of binary Cayley trees (see Section 2 for a definition) this question has been studied by Barata and Marchetti [3] and Barata and Goldbaum [2]. In the present paper we focus on the collection of graphs of bounded degree, and completely describe the location of the zeros for this class of graphs. For we denote by the collection of graphs with maximum degree at most . By we denote the open unit disk in . Moreover, we will occasionally abuse notation and identify with , the unit circle. Given we write

Our main results are:

Theorem A (ferromagnetic case).

Let and let . Then there exists such that the following holds:

  • for any and any graph we have ;

  • the set is dense in .

We remark that part (ii) has recently been independently proved by Chio, He, Ji, and Roeder [7]. They focus on the class of Cayley trees and obtain a precise description of the limiting behaviour of the zeros of the partition function of the Ising model.

Even though the parameters for which do not need to lie on the unit circle in the anti-ferromagnetic case , our methods give a similar description in this case:

Theorem B (anti-ferromagnetic case).

Let and let . Then there exists such that the following holds:

  • for any , any and any graph we have ;

  • the set accumulates on and .

Another recent related contribution to the Lee-Yang program is due to Liu, Sinclair and Srivastava [14], who showed that for and there exists an open set containing the interval such that for any and , .

1.1. Motivation

The motivation for studying the location of zeros of partition functions traditionally comes from statistical physics. Since this is well known and since many excellent expositions exist, see for example [1, Section 7.4], we choose not discuss the physical background here. However, recently there has also been interest in understanding the location of zeros from the perspective of theoretical computer science, more precisely from the field of approximate counting.

In theoretical computer science it is known that computing partition functions, such as the Ising model, the hardcore model, and the number of proper -colorings of a graph

exactly, generally is an example of a #P-hard problem (i.e., it is as hard as computing the number of satisfying solutions to a SAT-problem). For this reason much effort has been put in designing efficient approximation algorithms. Traditionally such algorithms are randomized and are based on Markov chains, see

[9]. In particular, Jerrum and Sinclair [10] showed that for all and the partition function of the Ising model can be efficiently approximated on any graph . Another approach is based on decay of correlations and was initiated by Weitz [22]. This leads to deterministic approximation algorithms. Using decay of correlations, Sinclair, Srivastava and Thurley [20] gave an efficient deterministic approximation algorithm for computing the Ising partition function on graphs of maximum degree at most when and .

Recently a new approach for obtaining deterministic approximation algorithms was proposed by Barvinok, see [1], based on truncating the Taylor series of the logarithm of the partition functions in regions where the partition function is nonzero. It was shown by Patel and the second author in [16] that this approach in fact yields polynomial time approximation algorithms when restricted to bounded degree graphs. Combining the approach from [16] (cf. [13]) with Theorem A and the original Lee-Yang result, we immediately obtain the following as a direct corollary:

Corollary 1.

Let , let and let , for as in Theorem A. Then for any there exists an algorithm that, given an -vertex graph of maximum degree at most , computes a relative -approximationA relative -approximation to a nonzero complex number is a nonzero complex number such that . to in time polynomial in .

An identical statement holds for , except there it does not follow directly from Theorem B. One also needs that for in a small disk around zero the partition function does not vanish, see Remark 24.

Given the recent progress on understanding the complexity of approximating independence polynomial at nonpositive fugacities based on connections to complex dynamics due to Bezaková, Galanis, Goldberg and Štefankovič [4], a natural question that arises is the following:

Question 2.

Let . Is it NP-hard (or maybe even #P-hard) to approximate the partition model of the Ising model on graphs of maximum degree at most when and ?

1.2. Approach

Our approach to proving our main theorems is to make use of the theory of complex dynamics and combine this with some ideas from the approximate counting literature. The value of the partition function of a Cayley tree can be expressed in terms of the value of the partition function for the Cayley tree with one fewer level, inducing the iteration of a univariate rational function. Understanding the dynamical behaviour of this function leads to understanding of the location of the zeros of the partition function for Cayley trees. The same approach forms the basis of [3, 2, 7, 14]. To prove our result for general bounded degree graphs, we use the tree of self avoiding walks, as defined by Weitz [22], to relate the partition function of a graph to the partition function of a tree with additional boundary conditions. This relationship no longer gives rise to the iteration of a univariate rational function, but with some additional effort we can still transfer the results for the univariate case to this setting.

We remark that a similar approach was used by the authors in [17] to answer a question of Sokal concerning the location of zeros of the independence polynomial, a.k.a., the partition function of the hard-core model.

1.2.1. Organization

This paper is organized as follows. In the next section we will define ratios of partition functions, and prove that for Cayley trees this gives rise to the iteration of a univariate rational function. In Section 3 we employ basic tools from complex dynamics to analyze this iteration. In particular a proof of part (ii) of our main theorems will be given there. Finally, in Section 4 we collect some additional ideas and provide a proof of part (i) of our main theorems.

2. Ratios

At a later stage, in Section 4, it will be convenient to have a multivariate version of the Ising partition function defined for a graph , complex numbers , and as follows:

The two-variable version is obtained from this version by setting all equal. We will often abuse notation and just write for the multivariate version.

Let be a graph and let . We call any map a boundary condition on . Let now be a boundary condition on . We say that is compatible with if for each vertex with we have and for each vertex with we have . We shall write if is compatible with . We define

Fix a vertex of . We let , and respectively, denote the boundary conditions on where is set to , and respectively. In case is contained in , we consider as a multiset to make sure and are well defined. For one element the vertex gets two different values, in which case no set is compatible with and consequently we set .

We denote the extended complex plane, , by . We introduce the ratio , by

(2.1)

If no boundary condition is present, or if it is clear from the context, we just write for the ratio. We have the following trivial, but important, observation:

if
(2.2)

The following lemma show how to express the ratio for trees in terms of ratios of smaller trees.

Lemma 3.

Let be a tree with boundary condition on . Let be the neighbors of in , and let be the components of containing respectively. We just write for the restriction of to for each . For let and denote the respective boundary conditions obtained from on where is set to and respectively. If for each , not both and are zero, then

(2.3)
Proof.

We can write

Let us fix . Suppose first that . Then we can divide the numerator and denominator by to obtain

(2.4)

If , then on the left-hand side of (2.4) we obtain while on the right-hand side, plugging in , we also obtain . Therefore this expression is also valid when . This finishes the proof. ∎

Let us now specialize the previous lemma to a special class of (rooted) trees. Fix . The tree consists of single vertex, its root. For , the tree consists of a root vertex of degree with each edge incident to connected to the root of a copy of . This class of trees is also known as the class of (rooted) Cayley trees. If is clear from the context, we just write instead of .

Define by for ,

(2.5)
Corollary 4.

Let and let . Then the orbit of under avoids if and only if for all .

Proof.

We observe that, as there is no boundary condition, . Now suppose that for all . Since , it follows that for some , So by (2) we see that for all . By Lemma 3 we obtain

(2.6)

and hence the orbit of avoids .

Conversely, suppose that for some while for all . We may assume that for all . Then by Lemma 3 we have (2.6) and hence implies . Now since we have that and are not both equal to . But then (2) implies that . A contradiction. This finishes the proof. ∎

This corollary motivates the study of the complex dynamical behaviour of the map at starting point . We will do this in the next section, returning to general graphs in Section 4

3. Complex dynamics of the map

In this section we study the dynamical behavior of the map for of norm . It is our aim to prove the following result.

Theorem 5.

Let and let . There exists such that

  • for each with there exists an interval that contains and an attracting fixed point of and that does not contain , which is forward invariant under . In particular, the orbit of under avoids the point ;

  • The interval is maximal: When the collection , for which the orbit of under does not avoid , is dense in . When the collection , for which the orbit of under does not avoid accumulates on .

Remark 6.

When , we can provide an explicit formula for as a function of . See Lemma 12 and its proof below.

While this result for was also independently proved in [7] we will provide a proof for it, as certain parts and ideas of our proof will be used in Section 4.

Observe that by Corollary 4, part (ii) of this theorem implies part (ii) of our main theorems. Part (i) of our main theorems, which will be proved in Section 4, will rely upon Theorem 5 (i).

To prove Theorem 5, we start with some observations from (complex) analysis and complex dynamics concerning the map , after which we first prove part (i) and then part (ii).

3.1. Observations from analysis and complex dynamics


3.1.1. Elementary properties of

We start with some basic complex analytic properties of the map . We first of all note that if , the map just equals multiplication by . Therefore we will restrict to . Throughout we assume that is real valued, and we write .

Let us define by for so that . Since is real, it follows that the Möbius transformation preserves . As the same holds for .

The behavior of on the outer disk is conjugate to that on the inner disk :

Lemma 7.

The map is invariant under conjugation by the the anti-holomorphic map

Proof.

First of all, we have . Now since , it follows that , as desired. ∎

Thus, for most purposes it is sufficient to consider only the behavior on and on .

Lemma 8.

Let . For the map induces a -fold covering on . For this covering is orientation preserving, for it is orientation reversing.

Proof.

Since has no critical points on it follows that the map is a -fold covering for any real . If both and are invariant under , hence conformality of near implies that is orientation preserving. If then maps into and vice versa, which implies that is orientation reversing. ∎

From now on we will only consider . The derivative of satisfies:

(3.1)

It follows that is independent of and, since , is strictly increasing with .

Let us define

Note that

from which it follows that when or , when or , and when or .

Recall that a map is said to be expanding if it locally increases distances, and uniformly expanding if distances are locally increased by a multiplicative factor bounded from below by a constant strictly larger than . Our above discussion implies the following.

Lemma 9.

Let .

  1. If or then the covering is uniformly expanding.

  2. If , or if , then the covering is expanding, but not uniformly expanding: .

  3. If , then .

Lemma 10.

Let and . Let be a fixed point of . Then .

Proof.

Let us denote the tangent space at the circle of a point by

; this is spanned by some vector in

. Then since the derivative is a linear map from to , it follows that has to be a real number. ∎

3.1.2. Observations from complex dynamics

We refer to the book  [15] for all necessary background. Throughout we will assume that and .

By Montel’s Theorem the family of iterates is normal on and on . Recall that the set where the family of iterates is locally normal is called the Fatou set, and its complement is the Julia set. Thus, the Julia set of is contained in , and there are two possibilities for the connected components of the Fatou set, i.e. the Fatou components:

Lemma 11.

Either the Fatou set of consists of precisely two Fatou components, and , or there is only a single Fatou component which contains both and . In the latter case the component is necessarily invariant. In the former case the two components are invariant when , and are periodic of order when .

Recall that invariant Fatou components are classified: each invariant Fatou component is either the basin of an attracting or parabolic fixed point, or a rotation domain. An invariant attracting or parabolic basin always contains a critical point, while a rotation domain does not. The critical points of

are , hence it follows that in both of the above cases the Fatou components must be either parabolic or attracting.

If there is only one Fatou component, by Lemma 7 this component must be an attracting or parabolic basin of a fixed point lying in . If there are two Fatou components then they are either both attracting basins, or they are both basins of a single parabolic fixed point in . We emphasize that there can be no other parabolic or attracting cycles.

The parameters for which their exist parabolic fixed points will play a central role in our analysis.

Lemma 12.

Let . Then there exists a unique such that for the function has a (unique) parabolic fixed point. Moreover the following holds:

  • If , then the parabolic fixed point of satisfies and is a solution of the equation

    (3.2)
  • If , then the parabolic fixed point of satisfies and is a solution of the equation

    (3.3)
Proof.

Recall that for fixed the value of is independent of , depends only on , is strictly increasing in , and satisfies and . Thus there exists a unique pair of complex conjugates for which . Hence there exists a unique for which , and by symmetry . Since the action of on the unit circle is orientation preserving for , and orientation reversing for , it follows by Lemma 10 that equals for , and equals for .

Let us first consider the case that . We are then searching for solutions to the two equations

(3.4)
(3.5)

Rewriting equation (3.4) gives

which can be plugged into (3.5) to give

which is equivalent to

For there are two solutions for , a pair of complex conjugates lying on the unit circle. For each of these solutions there exists a unique value of for which equation (3.4) is satisfied. These values of are clearly complex conjugates of each other and, when the two solutions and are distinct, must be distinct as has at most one parabolic fixed point.

If , we need to replace by on the right-hand side of (3.5). Similar to the case, this then leads to equation (3.3), which, when , has two solutions for , a pair of complex conjugates lying on the unit circle. As before, for each of these solutions there exists a unique value of for which equation (3.4) is satisfied. Again these values of are complex conjugates of each other. ∎

We note that in the lemma above when or when there is a double solution at , and hence the corresponding equals . For this map there are two separate parabolic basins: the inner and outer unit disk. When the parabolic fixed point is a double fixed point, and hence has only one parabolic basin. It follows that in this case there is a unique Fatou component, which contains both the inner and outer unit disk, and all orbits approach the parabolic fixed point along a direction tangent to the unit circle. When the inner and outer disk are inverted by , the fact that implies that orbits in these components converge to the parabolic fixed point along the direction normal to the unit circle, while nearby points on the unit circle move away from the parabolic fixed point.

3.2. Proof of Theorem 5 (i) for .

We will now consider the behavior for parameters and for which has an attracting fixed point on , and prove Theorem 5 (i) for these maps. We will consider the case later.

The Julia set, which is nonemptyIn fact, it can be shown that the Julia set is a Cantor set., is contained in the unit circle, and the complement is the unique Fatou component, the (immediate) attracting basin. Its intersection with the unit circle consists of countably many open intervals. We refer to the interval containing the attracting fixed point as the immediate attracting interval. We note that this interval is forward invariant, and the restriction of to the interval is injective. We emphasize that we may indeed talk about the immediate attracting interval, as there are no other parabolic or attracting cycles.

Theorem 13.

Let and let (from Lemma 12). Then for the map has an attracting or parabolic fixed point on if and only if . If then the point lies in the immediate attracting interval, while does not.

Proof.

We will consider the changing behavior of the map as varies, for fixed. By the implicit function theorem the fixed points of , i.e. the solutions of , depend holomorphically on , except when . By Lemma 12 this occurs exactly at two parameters .

Recall that the absolute value of the derivative, , is independent of , strictly decreasing in , and that while . For each there exists a unique for which is fixed, inducing a map , holomorphic in a neighborhood of . Since there can be at most one attracting or parabolic fixed point on , the map is injective on the circular interval . It follows that the image of this interval under equals , and that for outside of this interval the function cannot have a parabolic or attracting fixed point on .

When has an attracting fixed point on , the boundary points of the immediate attracting interval are necessarily fixed points. The fact that there cannot be other attracting or parabolic cycles on implies that the two boundary points are repelling. It follows also that cannot be a boundary point of the immediate attracting interval, and since these boundary points vary holomorphically (with ), and thus in particular continuously, it follows that is always contained in the immediate attracting interval.

It remains to show that cannot lie in the immediate attracting interval, which follows from the fact that cannot be one of the repelling boundary points of the immediate attracting interval. To see this, observe that can only be fixed when . depending on the parity of . When the point does not lie in the immediate attracting basin since this interval is proper, and must be invariant under complex conjugation. When the point is mapped to , which is either fixed or maps back to . In either case does not lie in the immediate attracting interval, hence there is no attracting fixed point. This completes the proof. ∎

Theorem 5 (i) for is a direct corollary. Indeed just observe that and since lies in the attracting interval it follows that the orbit of does as well, and therefore avoids .

3.3. Proof of Theorem 5 (i) for .

Lemma 14.

Let , and let be such that has a parabolic fixed point on . Then does not lie in the shortest closed circular interval bounded by and .

Proof.

Recall that is not equal to or , and that is minimal at and increases monotonically with , and is therefore strictly smaller than on the open circular interval bounded by and . By integrating over this open interval it follows from that . Since is orientation reversing it follows that and have opposite sign. The statement follows from

Lemma 15.

Let , and let be such that has a parabolic fixed point on . The shortest closed circular interval bounded by and cannot be forward invariant.

Proof.

First observe that the parabolic fixed point is not equal to . By the previous lemma it cannot also not be equal to .

Suppose now that for the purpose of contradiction. Then the open interval is also forward invariant, and since neither nor is equal to the parabolic fixed point, it follows that cannot be contained in the parabolic basin. Hence must intersect the Julia set, say in a point . Let be a sufficiently small open disk centered at so that . Since lies in the Julia set, it follows that

This contradicts the assumption that is forward invariant.

Note that we used here that the exceptional set of the rational function is empty, which follows immediately from the fact that there are no attracting periodic cycles. We again refer to [15] for background on complex dynamical systems.

It follows from the above lemma that the situation is different from the orientation preserving case: when has an attracting fixed point on the point does not necessary lie in the immediate attracting interval. If it did, then would be forward invariant on the shortest interval with boundary points and , which cannot happen for close to the parabolic parameter as follows from the lemma above.

Recall that the boundary points of the immediate attracting interval form a repelling periodic cycle. Since for near the point does lie in the immediate attracting interval, while for near the parabolic parameters the point does not, it follows by continuity of the repelling periodic orbit that there must be a parameter for which is one of the boundary points. In fact, it follows quickly from the fact that strictly increases with that there exists a unique (with such that for , is a boundary point of the immediate attracting interval. To see that is unique, suppose there is another such . We may assume that . Set and . Then since and since , it follows that the distance between and is strictly larger than the distance between and , hence cannot equal to .

Theorem 16.

Let and let as discussed above. If and , then the point lies in the immediate attracting interval, while does not.

Proof.

The fact that does lie in the immediate attracting interval follows from the above discussion.

To see that does not lie in the immediate attracting basin, it is sufficient to argue that the imaginary parts of the attracting fixed point and have the same sign, i.e. that the two points either lie both in the upper half circle or in the lower half circle. This is clearly the case when the attracting fixed point lies close to . The fact that it holds for any attracting fixed point follows from the fact that the map is continuous and injective. See the case for the discussion of this map. ∎

Theorem 5 (i) for with is again a direct corollary.

3.4. Proof of Theorem 5 (ii) for

We start with analyzing what happens when .

Proposition 17.

For the parameters for which the orbit of under the map takes on the value is dense in .

Proof.

We consider the orbits for parameters in a small interval , with . The initial values lie in this interval. Since the map

is expanding, this interval is mapped to an interval which is strictly larger. Let us write for , and . By Lemma 8 the covering map is orientation preserving. Noting that it follows that the length satisfies

Hence as we have , counting multiplicity. Thus there must exist and for which . ∎

Proposition 18.

Let and let be such that has no attracting or parabolic fixed point on . Then there are parameters arbitrarily close to and for which .

Proof.

By the assumption that has no attracting or parabolic fixed point on , it follows that both and are attracting basins, and hence the orbits of the two critical points stay bounded away from the Julia set . It follows that is a hyperbolic set, i.e. that there exists a metric on , equivalent to the Euclidean metric, with respect to which is a strict expansion. We will refer to this metric as the hyperbolic metric on .

The proof concludes with an argument similar to the one used in Proposition 17. For a circular interval we denote by the diameter with respect to the hyperbolic metric on . Let be a proper subinterval containing , small enough so that the maps for are all strict expansions with respect to the hyperbolic metric obtained for the parameter . It follows that

where the equality follows from the fact that the maps are all orientation preserving, and the constant is a uniform lower bound on the expansion of the maps for .

By induction it follows that , counting multiplicity. Thus for sufficiently large the interval will contain the unit circle, proving the existence of a parameter for which . ∎

This results completes the proof of Theorem 5 (ii) in the case .

3.5. Proof of Theorem 5 (ii) for

In the case the statement of Theorem 5 (ii) is weaker. One cannot expect parameters with arbitrarily close to any point

as there will be for which the point lies in the attracting basin, just not in the immediate attracting basin. In this case the orbit of will still converge to the attracting fixed point, and, except for at most countably many parameters , will avoid . This is then still the case for sufficiently close to .

Our goal is to show that there exist parameters arbitrarily close to for which the orbit of contains ; the complex conjugate is completely analogues. Recall that , and that this periodic orbit is repelling. It follows that the point cannot be passive, i.e. the family of holomorphic maps

cannot be a normal family in a neighborhood of . To see this, note that for an open set of parameters (for example, those for which does lie in the immediate attracting interval) accumulating on the maps converge to , but the points remain bounded away from . Thus the sequence of maps cannot have a convergent subsequence in any neighborhood of .

Recall that the strong version of Montel’s Theorem says that a family of holomorphic maps into the thrice punctured Riemann sphere is normal. We claim that this implies that the point cannot be avoided for all parameters in a neighborhood of . Of course, if is avoided, then so are all its inverse images. Since the map induces a -fold covering on the unit circle, it is clear that there exist points , distinct from each other as well as from , such that