Correlation Decay and the Absence of Zeros Property of Partition Functions
aa r X i v : . [ m a t h . P R ] D ec Correlation Decay and the Absence of Zeros Property ofPartition Functions
David Gamarnik ∗† December 2, 2020
Abstract
Absence of (complex) zeros property is at the heart of the interpolation method devel-oped by Barvinok [Bar17a] for designing deterministic approximation algorithms for variousgraph counting and computing partition functions problems. Earlier methods for solvingthe same problem include the one based on the correlation decay property. Remarkably, theclasses of graphs for which the two methods apply sometimes coincide or nearly coincide.In this paper we show that this is more than just a coincidence. We establish that if theinterpolation method is valid for a family of graphs satisfying the self-reducibility property,then this family exhibits a form of correlation decay property which is asymptotic StrongSpatial Mixing (SSM) at distances ω (log n ), where n is the number of nodes of the graph.This applies in particular to amenable graphs, such as graphs which are finite subsets oflattices.Our proof is based on a certain graph polynomial representation of the associated par-tition function. This representation is at the heart of the design of the polynomial timealgorithms underlying the interpolation method itself. We conjecture that our result holdsfor all, and not just amenable graphs. The algorithmic question at the heart of this paper is one of designing a polynomial time algo-rithm for solving various graph counting problems such as counting the number of independentsets of a graph, the number of proper colorings of a graph, the number of partial matchings,etc. Generically, the problem is one of computing the partition function Z ( G ) associated with agraph induced possibly with some additional parameters such as the number of colors, list-colors,etc. As the existence of a polynomial time algorithm for computing partition functions amountsto the algorithmic complexity statement P = P , widely believed not to be true, the researchhas focused primarily on the question of designing algorithm for computing partition functionsapproximately [Bar17a],[JS97],[Jer03]. The gold standard for the approximation algorithms isthe existence of a Fully Polynomial Time Approximation Scheme (FPTAS). Randomized FPTAS ∗ Operations research Center and Sloan School of Management, MIT. Email: [email protected] † Support from the NSF grant DMS-2015517 is gratefully acknowledged. + Z ( G ( z )) parametrized bycomplex value z such that when z = 1, Z ( G ( z )) = Z ( G ) and when z = 0 the associated partitionfunction Z ( G (0)) is trivially computable. One then considers the Taylor approximation of thelog-partition function log Z ( G ( z )) and computes its first m -terms for m which is typically loga-rithmic in the number of nodes. This can be done by the brute force method in quasi-polynomialtime n O (log n ) , where n is the number of nodes in a graph, but also in just polynomial time n O (1) in bounded degree graphs, using a certain graph polynomial representation of the partitionfunction. This representation was developed by Patel and Regts [PR17], and it is at the coreof the approach of the present paper. Barvinok’s interpolation method provably works providedthat the model exhibits the ”zero-freeness” property, namely the set of zeros of the interpolatedfunction Z ( G ( z )) is outside a connected region containing 0 and 1. Several families of graphswere the method is effective either coincide or nearly coincide with the families of graph forwhich the correlation decay based method applies. For other families of graphs no correlationdecay counterparts are known or those which are known appear to work in a more restrictedsetting. Examples of the former include the problem of counting partial matchings, see [Bar17a]and [BGK + λ < ( d − d − / ( d − d , (1)where d is the largest degree of a graph. The correlation decay based method in this regimewas developed by Weitz in [Wei06], and the interpolation method up to the same threshold wasdeveloped by Peters and Regts [PR + . d + 1,as shown in Lu and Yin [LY13], whereas the polynomial interpolation method applies under asignificantly weaker assumption, where list sizes are at least roughly 1 . d , as shown in Liu,Sinclair and Srivastava [LSS19]. It is known that the correlation decay in the form of the StrongSpatial Mixing (see below) does apply in this regime as well [GKM15], but turning it into acounting algorithm is only known through the interpolation method, as was done in [LSS19].What is the ultimate power and limit of the interpolation method, and how are those relatedto the correlation decay property? We give a one-sided answer to this question by showing2hat the validity of the interpolation method for a self-reducible class of graphs implies a formof correlation decay property we call the asymptotic Strong Spatial Mixing (SSM). This is ourmain result stated as Corollary 3.2, which is a simple implication of our main technical resultstated in Theorem 3.1. The self-reducibility refers to the property that the model remains in thefamily when some of the nodes have prescribed values. We give several examples of such models,with independent set model on bounded degree graphs being one example, and list-coloring of agraph problem being the second example. Our result is thus stated for two types of interpolationschemes successfully used in the past, which we call Type I and Type II interpolations. The firstis one was used to design an FPTAS for counting independent sets as in [PR17], and the secondis a generalization of the type used for designing FPTAS for counting list-colorings of a graph,as in [LSS19]. Both interpolation types are defined precisely in the body of the paper.We now discuss briefly the SSM property. Its weaker counterpart, the Spatial Mixing (SM)property, is a property which is stated in terms of the Gibbs distribution associated with thepartition function. The SM is widely studied in the statistical physics literature [Geo88] and isdirectly related to the properties of uniqueness of Gibbs measures on infinite graphs. Roughlyspeaking, it is the property that the marginal distribution with respect to the Gibbs measureassociated with a subset of nodes of a graph is asymptotically independent from the conditioningof the boundary of a neighborhood of the set, when the radius of the neighborhood is sufficientlylarge. Typically, such a decay of correlations is upper bounded by a function converging to zeroas radius diverges to infinity, and this function is uniform in the choice of the set and graph sizeitself. The SSM is a strengthened version of the SM which is SM applied to the original graphbeing reduced by setting some subset of the nodes of the graph to some fixed values, similarly tothe self-reducibility property. The asymptotic version of the SSM property that we consider inthis paper is a ”non-uniform” version of the SSM which occurs at radius values that depend onthe graph choice. Specifically, we establish that the zero-freeness property implies the SSM atradius value ω (log n ) where n is the cardinality of the node set. As such the property is applicablein particular to graphs, for which for any fixed node the number of nodes with distance ω (log n )from this node still constitutes the bulk of the graph. The special case includes all subgraphs oflattices, and amenable type graphs in general. However, it does not apply to graph sequenceswhich are expanders, and specifically the graphs where nodes beyond distance ω (log n ) fromany given node simply might not exist. Whether our result extends to the case of expanders isthus left open. We note that the SSM by itself does not render the partition function estimationalgorithms and additional steps are needed such as either the SSM on the associated self-avoidingtree as in [Wei06], or the SSM on the associated computation tree as in [GK12].One could wonder whether the opposite implication is true as well. Also one could wonderwhether our result is implied by some existing results in the non-algorithmic literature. Forrestricted models such as lattices, indeed the equivalence between the zero-freeness and long-rangeindependence has been known for a while as discussed in the classical works of Dobrushin andShlossman [DS87]. Remarkably, however, such an equivalence does not extend to unstructuredgraph sequences, such as for example sequences of all bounded degree graphs, and in fact thelack of zero-freeness can coexist with long-range independence, as we now demonstrate. Indeed,consider any model which violates the zero-freeness property, for example the hard-core modelwhich violates condition (1) above. For this choice of λ and d consider any constant size graphwith degree d (for example a clique on d + 1 nodes) and a disjoint union of n/ ( d + 1) of suchgraphs. The set of zeros of the associated partition function is the set of zeros of one individual3lique and thus violates the zero-freeness property. Yet the model trivially exhibits the long-rangeindependence for distances beyond d including the SSM property.Our result does not rule out the applicability of the interpolation method beyond the SSMregime if some modifications are introduced. For example, Helmuth, Perkins and Regts [HPR20]and Jenssen, Keevash and Perkins [JKP20] apply the method to low-temperature models onlattices and bi-partite graphs in general by taking the advantage of the simple structure ofground states on these models and appropriate redefining of the underlying partition function.The fact the long-range dependence might indicate a barrier for a successful implementationof the interpolation argument should not be entirely surprising in light of some of the hardnessresults implied by the long-range dependence. In particular, Sly [Sly10] has shown that for generalgraphs with degree at most d no FPTAS exists for values λ strictly violating the condition (1),unless P = N P . The argument leverages the fact that bi-partite sparse random graphs exhibita long-range dependence which can be then used as a gadget in a more complicated graphstructure to argue that the existence of an FPTAS for computing the partition function of thisgraph structure implies an approximation algorithm for the MAX-CUT problem, which is knownnot to admit an approximation algorithm unless P=NP. The Sly’s result by itself though doesnot imply our result, as our result is not based on any complexity-theoretic assumptions.The proof of our result draws heavily on the work of Patel and Regts [PR17]. It was shown inthis paper that the interpolated partition function Z ( G ( z )) for many models can be written as theso-called graph polynomial, namely, a polynomial with coefficients expressed in terms of linearcombination of subgraph counts. It is then shown that the coefficients of the Taylor expansion oflog Z ( G ( z )) can be expressed entirely in terms of counts of connected subgraphs. This was usedcrucially to ensure the polynomiality as opposed to just quasi-polynomiality of the running timeof the algorithms. The key fact used in this approach was the fact that counting the number ofconnected graphs of order O (log n ) nodes on bounded degree graphs can be done in time n O (1) as opposed to quasi-polynomial time n O (log n ) . For us, though, this property has a completelydifferent ramification. The conditional marginal distribution of any set S , when conditioning onthe boundary ∂B ( S, R ) of an R -radius neighborhood B ( S, R ) of S can be written as a ratio ofpartition functions of the original and reduced models, using the self-reducibility of the class ofmodels we consider. The success of the interpolation argument with up to O (log n ) terms of theTaylor approximation implies that the Taylor approximation of this ratio involves only connectedgraphs of order O (log n ) which ”touch” either the set S , or the boundary ∂B ( S, R ) or both. Butif R is ω (log n ), no connected graph of size at most O (log n ) can touch both S and ∂B ( S, R ).This implies that the Taylor approximations of the conditional marginal distribution, whichwe call the conditional ”pseudo-marginal”, have the same value as the unconditional pseudo-marginal values, thus implying the long-range independence at distance ω (log n ). Our argumentas implemented in the current version does not seem to be capable of showing the long-rangeindependence at distances c log n for small constants c , which would be needed to extend ourresult to a larger families of graphs, including expanders.The remainder of the paper is structured as follows. The model definition and the review ofthe interpolation method are subject of the next section. In the same section we overview someexamples and introduce the definition of pseudo-marginal distributions. The definition of theSSM and the asymptotic SSM, and the statements of the main results are in Section 3. Somepreliminary technical results are in Section 4. The proof of the main result is found in Section 5.We close this section with some notational convention. For every integer K , [ K ] denotes the4et 1 , . . . , K . This will be typically used as the set of colors in this paper. For every graph H , wewrite V ( H ) and E ( H ) for the set of nodes and the set of edges of H , respectively. Two graphs H and H are disjoint if V ( H ) ∩ V ( H ) = ∅ . By default this means that there are no edgeswith one end in V ( H ) and another end in V ( H ). Given a graph G = ( V, E ) and node u ∈ V , B ( u ) denotes the set of neighbors of u , that is the set of nodes v ∈ V such that ( u, v ) ∈ E . Foreach integer R , B ( u, R ) denotes the set of nodes v accessible from u via paths of length at most R . In particular B ( u,
1) = B ( u ). Let ∂B ( u, R ) = B ( u, R ) \ B ( u, R −
1) be the set of ”boundarynodes” – nodes at distance precisely R from u . The distance d ( u, v ) between nodes u and v is thelength of a shortest path connecting u to v . Namely d ( u, v ) = min t such that there exists nodes u = u, u , . . . , u t = v such that each pair ( u i , u i +1 ) , ≤ i ≤ t − S ⊂ V , B ( S, R ) = ∪ u ∈ S B ( u, R ) and ∂B ( S, R ) = B ( S, R ) \ B ( S, R − u | B ( u ) | . A graph H is connected if ∪ R ≥ B ( u, R ) = V ( H ) for each node u ∈ V ( H ).A graph is disconnected if it is not connected. Suppose G = ( V, E ) is a simple undirected graph on the node set V = V ( G ) and the edge set E = E ( G ). Given a positive integer K , suppose a vector a u ∈ R K + with non-negative entriesis associated with every node u ∈ V of G , and a symmetric matrix A ( u,v ) ∈ R K × K + also withnon-negative entries is associated with every edge ( u, v ) ∈ E of G . Let A be short-hand notationfor the collection a u , u ∈ V, A ( u,v ) , ( u, v ) ∈ E . We will often refer to the elements of [ K ] as colorsand call the collection A list-coloring of G for reasons to be discussed below. Define Z ( G, A ) , X φ : V → [ K ] Y u ∈ V a uφ ( u ) Y ( u,v ) ∈ E A ( u,v ) φ ( u ) ,φ ( v ) . (2)For any φ : V → [ K ] letting w ( φ ) = Y u ∈ V a uφ ( u ) Y ( u,v ) ∈ E A ( u,v ) φ ( u ) ,φ ( v ) , (3)we have Z ( G, A ) = P φ : V → [ K ] w ( φ ). We call this value the ”number” of homomorphisms from G to the collection A . The justification for this definition is the special case when a u is the vectorof ones for all u and A ( u,v ) = A are edge independent with A i,j ∈ { , } for all 1 ≤ i, j ≤ K .In this case we can think of A as an adjacency matrix of a graph H on K nodes. This graph H is allowed to have loops if some of A i,i equal to one. Then Z ( G, A ) counts the number ofhomomorphisms from G into H , namely the number of maps φ : V → V ( H ) such that for every( u, v ) ∈ E it is the case that also ( φ ( u ) , φ ( v )) ∈ E ( H ).Throughout the paper we will be considering graphs G associated with some list-coloring A , so we will use a shorthand notation G for a graph along with list-coloring. Thus G is atriplet ( V, E, A ) and we call G a decorated graph. We use Z ( G ) in place of Z ( G, A ) light of thisnotational change. Z ( G ) is also called the partition function, a term more commonly used in the statisti-cal physics literature. The partition functions naturally factorize over disjoint unions graphs.Namely, suppose G = ( V j , E j , A j ) are two disjoint graphs. Let G be the union of G and G A of color-lists A and A . Then Z ( G ) = Z ( G ) Z ( G ) . (4)Let G denote the set of all decorated graphs ( V, E, A ). The set G is uncountable. Yet we will usethe notation of the form P H ∈G · , which will be well defined when only finitely many terms to besummed are non-zero. For every positive integer i let G i ⊂ G denote the (uncountable) set of all i -node decorated graphs G = ( V, E, A ). Namely | V | = i for each such graph. Let ¯ G i = ∪ j ≤ i G j .Denote by G i, conn the subset of G i consisting of only connected graphs. Let ¯ G i, conn = ∪ j ≤ i G i, conn .Similarly, let G i, edge be the uncountable set of all graphs which ares spanned by i -edges( | E | = i ). Namely, ( V, E, A ) ∈ G i, edge if there exists a subset of edges E ′ ⊂ E, | E ′ | = i such thatthe set of nodes incident to edges in E ′ is the entire set V . We note that the same graph maybelong to sets G i, edge with different values of i as clearly subsets of edges of different cardinalitycan span the same set of nodes. The sets ¯ G i, edge , G i, edge , conn and ¯ G i, edge , conn are defined similarly.Given a graph G = ( V, E, A ), we now introduce the associated Gibbs measure µ on theset of mappings φ : V → [ K ]. The measure is defined as follows: the probability weight µ ( φ )associated with φ is µ ( φ ) = w ( φ ) /Z ( G ) ≥
0. The measure is well defined only when Z ( G ) isstrictly positive. Clearly P φ µ ( φ ) = 1, that is µ is indeed a probability measure.Associated with Gibbs measure µ are marginal probability distributions for each subset ofnodes S ⊂ V . Specifically, for any S ⊂ V and any σ ∈ [ K ] S encoding a coloring assignment σ : S → [ K ], the associated marginal probability denoted by µ ( G, S, σ ) is µ ( G, S, σ ) = Z − ( G ) X φ : φ ( u )= σ ( u ) , ∀ u ∈ S w ( φ ) . (5)Namely µ ( G, S, σ ) is simply the likelihood that φ generated at random according to µ , mapseach u ∈ S into σ ( u ). Naturally, by the total probability law P σ : S → [ K ] µ ( S, σ ) = 1.Given two sets
S, T ⊂ V and colorings σ : S → [ K ] , τ : T → [ K ] we will also write µ ( G, S, σ | T, τ ) for the conditional probability of the event φ ( u ) = σ ( u ) , ∀ u ∈ S when conditionedon the event φ ( v ) = τ ( v ) , ∀ v ∈ T . Thus µ ( G, S, σ | T, τ ) = µ ( G, S ∪ T, σ ∪ τ ) µ ( G, T, τ ) , where σ ∪ τ denotes the implied coloring of the union S ∪ T . This is non-zero only when σ and τ are consistent on the intersection S ∩ T .Next we observe that marginals µ ( G, S, σ ) can be conveniently written in terms of ratio ofpartition functions associated with the original and the reduced model, exhibiting the fundamen-tal property of self-reducibility of our graph homomorphism model. Specifically, given S ⊂ [ V ]and σ : S → [ K ], let A S,σ be the modified decoration of G defined by the same values associatedwith node a S,σ,u = a u , u ∈ V and A S,σ ;( u,v ) i,j = A ( u,v ) i,j ( i = σ ( u )) , (6)for every ( u, v ) such that u ∈ S , and A S,σ ;( u,v ) i,j = A ( u,v ) i,j if both u, v ∈ V \ S . By symmetry thisalso means A S,σ ;( u,v ) i,j = A ( u,v ) i,j ( j = σ ( v )) , u, v ) such that v ∈ S . In particular, the weight of φ : V → [ K ] according to themodified list is zero if φ ( u ) = σ ( u ) for at least one u ∈ S , and it is w ( φ ) otherwise. Consideringthe partition function Z ( G S,σ ) of the modified decorated graph G S,σ , ( V, E, A S,σ ), we obtainthe identity µ ( G, S, σ ) = Z ( G S,σ ) Z ( G ) . (7)Similarly, for any S, σ : S → [ K ] , T, τ : T → [ K ], µ ( G, S, σ | T, τ ) = Z ( G S ∪ T,σ ∪ τ ) Z ( G T,τ ) , (8)with term Z ( G ) cancelled out. We thus note that by definition µ ( G, S, σ | T, τ ) = µ ( G T,τ , S, σ ).While it would be arguably more natural to modify the decoration A by modifying nodevalues to a S,σ ; ui = a ui ( i = σ ( u )), the choice above is dictated by the interpolation constructionto be introduced below associated with the list-coloring problem.We now discuss some common examples of the model above. Independent Sets/Hard-core model
An independent set of a graph G is a subset I ⊂ V of nodes which spans no edges. Namely( u, v ) / ∈ E for all u, v ∈ I . Fix a parameter λ >
0, which is sometimes called fugacity in thestatistical physics literature. The counting object of interest is Z ( G ) , P I λ | I | , where the sum isover all independent sets of G . When λ = 1, this is simply the total number of independent setsof the graph G . Letting i k ( G ) stand for the number of independent sets of G with cardinality k and interpreting i ( G ) as 1, we also have Z ( G ) = X ≤ k ≤| V | i k ( G ) λ k . The model above is a special case of homomorphism counting given by K = 2, and A givenby a u = (1 , λ ) for all u ∈ V , A ( u,v )2 , = 0 and A ( u,v ) i,j = 1 for all other 1 ≤ i, j ≤
2, for all edges( u, v ) ∈ E . Indeed, for any φ : V → { , } such that w ( φ ) >
0, the set I = { u : φ ( u ) = 2 } is anindependent set, since otherwise having ( u, v ) ∈ E for some u, v ∈ I implies A φ ( u ) ,φ ( v ) = A , = 0,namely w ( φ ) = 0. Also for every independent set I and the associated map φ ( u ) = 2 , u ∈ I, φ ( u ) = 1 , u / ∈ I , we have w ( φ ) = λ | I | . Thus indeed this model is a special case of the model(2).We note that the restrictions of the form G → G S,σ does not change the model it in anymeaningful way. Specifically, consider the reduced graph ˜ G obtained by deleting from G allnodes u ∈ S such that σ ( u ) = 1, and deleting all nodes u and the associated neighborhoods B ( u ), for nodes u ∈ S such that σ ( u ) = 2. In other words, ˜ G is obtained by deleting allnodes which are forced not to belong to an independent set by σ , and deleting all nodes whichare actually forced to belong to an independent set by σ along with their neighbors. Then Z ( G S,σ ) = λ k Z ( ˜ G ) where k is the number of nodes u ∈ S with σ ( u ) = 2 which are forced to bea part of an independent set. 7 roper Colorings and Proper List-Colorings models For any positive integer K , let a u be the K -vector of ones for all nodes u , and let A ( u,v ) = A be edge independent and given by A i,j = 1 when 1 ≤ i = j ≤ K and A i,i = 0 , i = 1 , , . . . , K .Then for any φ : V → [ K ], w ( φ ) = 1 when the values φ ( u ) and φ ( v ) are distinct for all edges( u, v ) ∈ E , and w ( φ ) = 0 otherwise. Namely, w ( φ ) = 1 iff φ corresponds to a proper coloring of G with colors 1 , , . . . , K , and Z ( G ) is the total number of proper colorings of G .Turning to the list-coloring problem, suppose each node u is associated with a list of colors C ( u ) ⊂ [ K ]. A mapping φ : V → [ K ] is a proper list-coloring if in addition to the requirement φ ( u ) = φ ( v ) for each each ( u, v ) ∈ E it is also the case that φ ( u ) ∈ C ( u ) for each node u . Thisis again a special case of our model given by the following A . We let again a u be the vector ofones for all u , and let A ( u,v ) i,j = ( i = j, i ∈ C ( u ) , j ∈ C ( v )) , ∀ ( u, v ) ∈ E. The number of proper list-colorings is then simply Z ( G ) as defined per (2). Ising model
Fix K = 2 and h, β >
0. Suppose a = (1 , e h ) , A , = A , = e β , A , = A , = e − β . Then Z ( G ) = X φ : V →{ , } exp h X u ∈ V (2 φ ( u ) −
3) + β X ( u,v ) ∈ E (2 φ ( u ) − φ ( v ) − The parameter h is called the strength of the associated magnetic field and the parameter β iscalled inverse temperature. A more canonical equivalent way to represent this model is in termsof spin assignments σ : V → {− , } , in which case Z ( G ) is simply X σ : V →{− , } exp( h X u σ ( u ) + β X u,v σ ( u ) σ ( v )) . The equivalence is immediate by transformation 2 φ − − β >
0, (respectively β <
0) is called ferromagnetic (respectively anti-ferromagnetic)Ising model. The model is interesting including the case of no magnetic field h = 0. The key idea underlying the interpolation method for computing partition functions Z ( G ) relieson first replacing the target decorated graph G = ( V, E, A ), for which Z ( G ) is hard to compute,by an alternative decoration ˆ A on the same ground graph ( V, E ), for which the partition function Z ( ˆ G ) can be easily evaluated, where ˆ G = ( V, E, ˆ A ). Then one builds a convenient interpolation A ( z ) between A and ˆ A , parametrized by some complex parameter z ∈ C (with understandingthat a u and A ( u,v ) are now complex valued), and rewrites log Z ( G ) as z -variable Taylor expansionaround easy to compute log Z ( ˆ G ). One then computes the polynomial associated with the Taylorexpansion truncated at a sufficiently low degree terms and uses it to approximate Z ( G ). Themethod works provided that the partition function of the interpolated model as a function of8 is zero-free in the region containing the set of interpolating values of z , see [Bar17a] for thetextbook exposition of the method.The main result in this paper concerns two types of interpolation schemes which have beensuccessfully used in some of the earlier results. The first one concerns the independent set modeland the second one concerns the proper list-coloring model. While there are other successfulexamples of interpolation schemes, we will focus on just these two to illustrate the main ideas.The first interpolation type is motivated and easy to describe in terms of the problem ofcounting independent sets (hard-core model). Given G and λ >
0, introduce the following z -variable polynomial Z ( G ( z )) = X I z | I | λ | I | = X ≤ k ≤| V | i k ( G ) z k λ k . (9)where the first sum is again over all independent sets I of G . We see that Z ( G ( z )) is the partitionfunction of the model G ( z ) = ( V, E, A ( z )) where A ( z ) is obtained from A simply by replacing λ with λz . Trivially, Z ( G (0)) = 1 and Z ( G (1)) = Z ( G ). Let f ( z ) = log Z ( G ( z )) (with the branchof logarithm appropriately fixed). Consider the infinite Taylor’s expansion around z = 0: f ( z ) = X k ≥ k ! f ( k ) (0) z k , where f ( k ) is the k -th order derivative of f . The idea of the interpolation method is that for m small enough, typically logarithmic in | V | , the truncated expansion T m ( G, z ) , X ≤ k ≤ m k ! f ( k ) (0) z k (10)is a good approximation of f in a connected region of C containing 0 and 1, provided f ( z ) issubstantially distinct from zero in this region (zero-freeness). Specifically, one proves that forany ǫ > C such that if m = C log | V | then1 − ǫ ≤ exp( T m ( G, Z ( G ) ≤ ǫ. One then proceeds to establishing this zero-freeness property using various properties of thegraph such as degree boundedness. This scheme has been implemented in [PR17] where thezero-freeness was shown for λ satisfying (1) for graphs with degree at most d .As it turns out it is a tractable problem to compute the derivatives f ( k ) (0) in quasi-polynomialtime for graphs with degree bounded by some constant ∆. As an explanation, observe that the k -derivative Z ( k ) ( G,
0) of Z ( G, z ) at z = 0 is simply k ! i k ( G ). When k = O (log | V | ), i k ( G ) canbe computed in quasi-polynomial time by brute-force method in time | V | O (log | V | ) . Then oneobserves that the k -th derivative f ( k ) at z = 0 can be expressed in a recursive way as sum-product of terms Z ( ℓ ) ( G, , ℓ ≤ k , namely the sum-product of terms i ℓ ( G ) , ℓ ≤ k , thus allowingfor a quasi-polynomial computation of T m ( G, z ) at any z . Setting z = 1 one uses T m ( G,
1) asan approximation of Z ( G,
1) = Z ( G ). Importantly, the quasi-polynomiality can be improvedto just polynomiality using a clever method based on representing partition function as graphpolynomials of connected subgraph, as achieved in [PR17], and reducing the problem to counting9ver connected subgraphs only. The key ideas behind this method are in fact used in our paperfor establishing the connection between the interpolation method and the correlation decay,and are represented in Lemmas 5.1, 5.2 and 5.3 below. In particular, the graph polynomialrepresentation allows one to express the approximate marginal probabilities (pseudo-marginalsto be defined below) in terms connected small subgraphs forcing such pseudo-marginals to haveindependence over well-separated sets. In the end, exp( T m ( G, z )) evaluated at z = 1 amountsto a deterministic FPTAS for approximation of Z ( G ) up to any constant level of precision ǫ . Infact one can reach accuracy ǫ which is inverse polynomial in | V | : ǫ = n − Ω(1) by selecting theconstant C in m = C log n value appropriately large.The interpolation construction above concerning the independent set models will be referredto as Type I interpolation scheme below. It is only defined for the independent set model.We now turn to the Type II interpolation model, which concerns models generalizing theproper list-coloring model. Given a (decorated) graph G = ( V, E, A ), we construct the modified z -dependent color-list A ( z ) as follows: a u ( z ) = a u for all z , and A ( u,v ) ( z ) is given by A ( u,v ) ( z ) = J + ( A ( u,v ) − J ) z, where J is the K × K matrix of ones. We denote by G ( z ) the triplet ( V, E, A ( z )). When z = 1we have Z ( G ( z )) = Z ( G ), and when z = 0, Z ( G ( z )) trivializes to Y u ∈ V X ≤ i ≤ k a ui ! , L ( G ) . (11)Then we again let f ( z ) = log Z ( G ( z )) and define T m ( G, z ) by (10). We see that in the specialcase of the list-coloring problem, Z ( G ( z )) = X φ : V → [ K ] z e ( φ ) , where e ( φ ) is the total number of ”color violations” of φ . Namely the total number of nodes u with φ ( u ) / ∈ C ( u ) and the total number of edges ( u, v ) with φ ( u ) = φ ( v ). This interpolationscheme was considered in [PR17] and [LSS19] with the latter leading to the deterministic FPTASfor the counting list-colorings problem. If T m is a good approximation of the log-partition function with a well-controlled error, then itstands to reason that marginal distributions µ ( · ) defined in (5) should also be well approximatedin terms of T m , as marginals can be written as ratios of partition functions per (7). Motivatedby this we now introduce the definition of pseudo-marginals – namely values which intend toapproximate marginal values by means of T m . Suppose we are given a decorated graph G =( V, E, A ). Consider Type I or II interpolation with the interpolating partition function Z ( G ( z )).In particular, Z ( G (1)) is the original partition function Z ( G ). Recall the definition of T m ( G, z ).Given a subset of nodes S ⊂ V along with a coloring σ : S → [ K ], and given an integer m ≥ ν ( S, σ, m, z ) is defined as follows. Consider the partition function10 ( G S,σ ( z )) associated with the interpolation of decorated graph G S,σ = (
V, E, A S,σ ), where A S,σ is defined by (6). Let f ( z ) = log Z ( G S,σ ( z )) and let T m ( G S,σ , z ) = X ≤ k ≤ m k ! f ( k ) (0) z k . Recall from (7) that then the associated marginals satisfy µ ( G, S, σ ) = Z ( G, A S,σ ) Z ( G ) . The associated pseudo-marginals are defined by ν ( G, S, σ, z, m ) = exp ( T m ( G S,σ , z ))exp( T m ( G, z )) . Similarly, for every
S, T ⊂ V and σ : S → [ K ] , τ : T → [ K ] we define the associated conditionalpseudo-marginals as ν ( G, S, σ, z, m | T, τ ) = ν ( G, S ∪ T, σ ∪ T, z, m ) ν ( G, T, τ, z, m )= exp ( T m ( G S ∪ T,σ ∪ τ , z ))exp( T m ( G T,τ , z )) . The interpretation of pseudo-marginals should be clear. If T m ( G, z ) is a good approximationof the log-partition function f ( z ) = log Z ( G ( z )) for large enough m , then presumably the sameshould be true for the reduced log-partition function log Z ( G S,σ , z ), obtained when the valuesof homomorphisms of φ are fixed to σ ( u ) at u ∈ S . Namely, it should be the case that also T m ( G S,σ , z ) ≈ log Z ( G S,σ ( z )). In this case we expect to have Z ( G ( z )) ≈ exp( T m ( G, z )), and Z ( G S,σ ( z )) ≈ exp( T m ( G S,σ , z )), leading to µ ( G, S, σ ) ≈ exp( T m ( G S,σ , T m ( G, ν ( G, S, σ, , m ) . We will prove that the conditional pseudo-marginals ν ( ·|· ) equal to unconditional pseudo-marginalsfor sets S when conditioned on a boundary of a sufficiently deep neighborhood T = ∂B ( S, R ).Namely, the set S and its associated boundary ∂B ( S, R ) are ”pseudo-independent”. This is themain technical result of the paper. Then if the pseudo-marginals provide a good approximationof actual marginals, the same should apply to marginal distributions in some approximationsense. In the remainder of the paper we write ν ( G, S, σ, m ) in place of ν ( G, S, σ, , m ) and ν ( G, S, σ, m | T, τ ) in place of ν ( G, S, σ, , m | T, τ ). In this section we state our main result: if low-degree Taylor approximation T m provides a goodapproximation of the log-partition function log Z ( G ), then the model exhibits a version of thecorrelation decay property known as the Strong Spatial Mixing (SSM), which will be defined11recisely. The main approach is based on showing that the pseudo-marginals ν ( · , m ) associatedwith sufficiently well separated sets always exhibit long range independence property. Thusif T m approximates log Z ( G ) accurately, then ν ( · , m ) approximate accurately the true marginaldistributions µ ( · ), and therefore the latter have to exhibit long range independence as well, whichwe prove to be in the form of asymptotic SSM.We begin by formalizing the notion of SSM. We begin by defining the notion of SpatialMixing (SM)and then observe that due to generality and self-reducibility of our model of dec-orated graphs, SM implies SSM on appropriately reduced graphs. Given a decorated graph G = ( V, E, A ), and given any subset S ⊂ [ V ] and positive integer R let ρ R ( G, S ) = max σ : S → [ K ] ,τ ,τ : ∂B ( S,R ) → [ K ] | µ ( G, S, σ | ∂B ( S, R ) , τ ) − µ ( G, S, σ | ∂B ( S, R ) , τ ) | = max σ : S → [ K ] ,τ ,τ : ∂B ( S,R ) → [ K ] | µ (cid:0) G ∂B ( S,R ) ,τ , S, σ (cid:1) − µ (cid:0) G ∂B ( S,R ) ,τ , S, σ (cid:1) | . (12)Namely, ρ R ( G, S ) denotes the largest sensitivity of the conditional marginal distribution on S with respect to setting the color values at the boundary ∂B ( S, R ). Loosely speaking the modelexhibits the SM when ρ R ( G, S ) ≈ R . Typically, the case considered in the literature iswhen the set of interest S is small, often just a singleton. Formally, consider a family of decoratedgraphs F . We say it exhibits the SM if there exists a function ρ ∗ R , R ∈ Z + which converges tozero as R → ∞ , such that max G ∈F ,S ⊂ V ( G ) ρ R ( G, S ) ≤ ρ ∗ R . In other words R -range dependence in the sense of (12) decays to zero uniformly in R , the graphand the set S choices. The SSM property is the SM property which holds when some of thenodes have prescribed colors. Formally, a family of graphs F exhibits the SSM property if thefamily of graphs G Λ ,ν with G ∈ F , Λ ⊂ V ( G ) , η : Λ → [ K ] exhibits the SM property in the senseabove.In our setting the difference between the SM and the SSM properties is hardly seen, but thedifference can be quite substantial. It is known for example that when F is the family of all d -regular trees, the model exhibits the SM property as soon as K ≥ d + 1,[Jon02], whereas forthe SSM property it has been established only when K ≥ . d , [EGH +
19] and for triangle-freegraphs with degree at most d when K ≥ . d [GKM15]. It is conjectured that it holds assoon as K ≥ d + 1. The results are essentially equivalent to establishing the SM property forthe list-coloring problem. The distinction between SM and SSM is also important in structuredgraphs like lattices. The independent set model is known to exhibit the SSM on graphs withdegree d when λ satisfies (1), as was established in [Wei06], and provably fails to exhibit theSM on d -regular trees, as soon as λ > ( d − d − / ( d − d , which has been known for a whilefrom [Kel85],[Spi75] and [Zac83].In this paper we consider a weaker asymptotic version of the SSM. We consider a sequence ofdecorated graphs G n = ( V n , E n , A n ) and a sequence of distances R n . We say that this sequenceexhibits the asymptotic SSM at distances R n , lim n →∞ R n = ∞ iflim n →∞ max S, Λ n , ⊂ V n ,η n :Λ n → [ K ] ρ R n ( G n, Λ n ,η n , S n ) = 0 . The difference of the asymptotic SSM with the SSM property as defined earlier is the lackof uniformity of the upper bound on ρ ( · ) with respect to the graphs G . To appreciate the12istinction, consider the setting when | V n | = n , in the other words the graph has n nodes, andwhen R n = C log α n for some constants C, α . Incidentally, this is the setting we consider in ourmain result with α = 3. Then the asymptotic SSM means long-range independence at distancesΩ(log α n ). For some graphs, such as lattices or amenable graphs in general this is a meaningfulproperty when say S is singleton, as the number of nodes further than O (log α n ) distance awayfrom a single node still constitute the bulk of the graph. But for some graphs with strongexpansion type properties, the distances beyond O (log n ) simply might not exist and thus theproperty is vacuous. Many lattice models exhibit long range dependence and thus the lack ofasymptotic SSM for some choices of the parameters. For example the hard-core model on Z exhibits long range dependence when λ > . Z d for all sufficiently large d [GKRS15].We conjecture that our main result below extends to the case of SSM as originally defined andthus to all graphs, including expanders, but we are not able to prove this yet.We now state our main technical result. Theorem 3.1.
Given a decorated graph G = ( V, E, A ) , consider either the Type I interpolation(associated with the Independent Set model) or the Type II interpolation. Then for every R , S ⊂ V, σ : S → [ K ] and τ : ∂B ( S, R ) → [ K ] , ν ( G, S, σ, z, (1 / R | ∂B ( S, R ) , τ ) = ν ( G, S, σ, z, R ) . In other words, the ”conditional” pseudo-marginal at S when ”conditioning” on the boundaryof the neighborhood of S at distance R equals to ”unconditional” pseudo-marginal, when thepseudo-marginals are computed using the first R/ / Corollary 3.2.
Consider a sequence of graphs G n = ( V n , E n , A n ) on n = | V n | nodes, such that Z ( G n ) = 0 , for all n . Consider either the Type I or Type II interpolation. Suppose for every ǫ > there exists c ( ǫ ) such that for any sequence Λ n ⊂ V n , η n : Λ n → [ K ](1 − ǫ ) Z ( G n, Λ n ,η n ) ≤ exp ( T m ( G n, Λ n ,η n , ≤ (1 + ǫ ) Z ( G n, Λ n ,η n ) , (13) when m ≥ c ( ǫ ) log n and n is large enough. Suppose R n = ω (log n ) . Then lim n →∞ sup S n , Λ n ⊂ V n ,η n :Λ n → [ K ] ρ R n ( G n, Λ n ,η n , S n ) = 0 . (14) Namely the model exhibits the asymptotic SSM at distances asymptotically larger than log n . The result above rules out the possibility of using the interpolation method for models ex-hibiting the (non-trivial) long-range independence, including, for example, the independent setmodel on the 2-dimensional lattice for λ > . − coloring on the d -dimensional lattice,as discussed earlier.Let’s comment on the assumptions of the theorem, specifically in the context of concretemodels. In the case of independent set model, we have trivially Z ( G n ) = 0. In the case of13raph list-coloring, the equality Z ( G n ) = 0 arises when graph G n is not list-colorable with thelist encoded by A n , and it is not a trivial condition to check (in fact it is NP-hard). Typicallythough, the interpolation method is established for sequences of graphs and color lists for whichit is easily verified that the partition function is distinct from zero, in part because the methoditself is built on identifying a zero free region containing z = 1. An example of such assumptionis the assumption that the size of each list is larger than the degree of the graph. A strongerassumption than this was required typically in most papers on approximate counting of colorings,including [LSS19].The assumption (13) is just a statement regarding the success of the interpolation methodfor approximating the partition function. The subtlety here regards the model being reduced byfixing colors of any set Λ n . In the context of the independent set model this amounts to forcingthe nodes in S n to be in or out of the independent set, effectively reducing the underlying graphby deleting nodes u in S n marked 0 by σ n , and deleting nodes and neighbors of u ∈ S n marked 1by σ n , as we have already observed. The degree of the graph is not increased in this procedureso if the interpolation method was successful for the original graph G n it presumably shouldbe successful for the reduced graph sequences as well, since the assumption regarding successfulapplications of the interpolation method for independent set model are typically stated in termsof upper bounds on the graph degree in terms of λ . We see in particular that (13) holds for theany sequence of degree d bounded graphs and λ satisfying ( ?? ) as was established in [HSV18].Similarly, for the case of the problem of counting list-colorings, forcing the colors of Λ n tobe ones according to η n amounts to deleting nodes in Λ n and deleting colors η n ( u ) from thelists associated with neighbors of u in G n . The assumption used in successful implementation ofthe interpolation method typically include such reductions of the graph. Specifically, since thisprocedure reduces the degree of each neighbor of S n and its list color size by the same amount, thetypical assumptions which take the form ”list-size is at least α times the node degree”, adoptedfor example in [LSS19] is maintained. As mentioned earlier this paper considers the list-coloringmodel of triangle-free graphs with list of each node exceeding the degree of each node by amultiplicative factor approximately 1 . d . A sequence of graphs satisfying this condition thussatisfies (13) as follows from the result in [LSS19].We now prove Corollary 3.2 assuming the validity of Theorem 3.1. Proof of Corollary 3.2.
Consider any sequence of graphs G n = ( V n , E n , A n ) satisfying the as-sumptions of the theorem. In particular Z ( G n ) >
0. Fix any ǫ > S n , Λ n ⊂ V n , η n : Λ n → [ K ]. We write G n for G n, Λ n ,η n for short. Applying (13) and setting m = c ( ǫ ) log n we have for any τ : ∂B ( S n , R n ) → [ K ] µ ( G n , S n , σ n | ∂B ( S n , R n ) , τ n ) = µ ( G n , S n ∪ ∂B ( S n , R n ) , σ n ∪ τ n ) µ ( G n , ∂B ( S n , R n ) , τ n )= Z ( G n,S n ∪ ∂B ( S n ,R n ) ,σ n ∪ τ n ) Z ( G n,∂B ( S n ,R n ) ,τ n ) ≤ ǫ − ǫ exp (cid:0) T m ( G n,S n ∪ ∂B ( S n ,R n ) ,σ n ∪ τ n , (cid:1) exp (cid:0) T m ( G n,∂B ( S n ,R n ) ,τ n , (cid:1) = 1 + ǫ − ǫ ν ( G n , S n , σ n , m | ∂B ( S n , R n ) , τ n ) . R n ≥ m for all sufficiently large n , then Aapplying Theorem 3.1 the last expression is1 + ǫ − ǫ ν ( G n , S n , σ n , m ) , for all large enough n . As µ ( · ) ≤ < (1 + ǫ ) / (1 − ǫ ) we obtain in fact µ ( G n , S n , σ n | ∂B ( S n , R n ) , τ n ) ≤ ǫ − ǫ min ( ν ( G n , S n , σ n , m ) , , for all large n . Similarly, we establish that for all enough large nµ ( G n , S n , σ n | ∂B ( S n , R n ) , τ n ) ≥ − ǫ ǫ ν ( G n , S n , σ n , m ) ≥ − ǫ ǫ min ( ν ( G n , S n , σ n , m ) , . Considering now two boundary assignments τ n, , τ n, : ∂B ( S n , R n ) → [ K ], we obtain | µ ( G n , S n , σ n | ∂B ( S n , R n ) , τ n, ) − µ ( G n , S n , σ n | ∂B ( S n , R n ) , τ n, ) ≤ (cid:18) ǫ − ǫ − − ǫ ǫ (cid:19) min ( ν ( G n , S n , σ n , m ) , ≤ ǫ − ǫ . As the left-hand side does not depend on ǫ , the result follows. In this section we present some simple preliminary results that we need for proving Theorem 3.1Given a complex variable polynomial p ( z ) = c ( p ) + c ( p ) z + · · · + c n ( p ) z n with c ( p ) assumed tobe non-zero, denote its n non-zero complex roots by ζ , . . . , ζ n . Let Roots( p, k ) = P ≤ j ≤ n ζ − kj .The following identity known as Newton identity states kc k ( p ) = − k − X i =0 c i ( p )Roots( p, k − i ) . (15)Here c k ( p ) = 0 are assumed for k > n . Its short derivation is given in [PR17] and is skipped. Inthe special case c ( p ) = 1 this means that Roots( p, k ) can be computed in terms of c ( p ) , . . . , c k ( p )recursively. In fact it can be expressed explicitly due to the formula by Girard (developed in factin 1629, thus before Newton [Tig15], see also [wik]), via the following relationRoots( p, k ) = ( − k k X m m ··· + kmk = km ≥ ,...,mk ≥ ( m + m + · · · + m k − m ! m ! · · · m k ! k Y i =1 ( − c i ( p )) m i . We rewrite this in a more general formRoots( p, k ) = X m m ··· + kmk = km ≥ ,...,mk ≥ α m ,...,m k Y ≤ i ≤ k c m i i ( p ) , (16)15or some coefficients α m ,...,m k . Considering now f ( z ) = log p ( z ) = P ≤ i ≤ n log( z − ζ i ) + log c n ( p )we obtain f ( k ) (0) = X ≤ i ≤ n k !( − k ζ − ki = k !( − k Roots( p, k ) . (17)The m -order Taylor expansion of f around z = 0 is then T m ( p, z ) , X ≤ k ≤ m k ! z k k !( − k Roots( p, k )= X ≤ k ≤ m z k ( − k Roots( p, k ) . (18)Next we observe the following basic additivity property of the function Roots( p, k ) when p is a interpolated partition function G ( z ). Suppose G is a disjoint union of graphs G j , j = 1 , Z ( G ( z )) is the union of roots of Z ( G ( z )) and Z ( G ( z )), and thuscounting multiplicityRoots( Z ( G ( z )) , k ) = Roots( Z ( G ( z )) , k ) + Roots( Z ( G ( z )) , k ) . (19)We now turn to the notion of color-respecting graph isomorphism and color-respecting graphembeddings. Given two decorated graphs F = ( V ( F ) , E ( F ) , A ( F )) and G = ( V ( G ) , E ( G ) , A ( G )),a mapping ψ : V ( F ) → V ( H ) is a color-respecting graph isomorphism if it is a graph isomorphismwith respect to the underlying graphs ( V ( F ) , E ( F )) and ( V ( G ) , E ( G )), if a ψ ( u ) ( H ) = a u ( F )for all u ∈ V ( F ) and A ( ψ ( u ) ,ψ ( v )) ( H ) = A ( u,v ) ( F ) for all ( u, v ) ∈ E ( F ). Here a u ( F ) , u ∈ V ( F ) , A ( u,v ) , ( u, v ) ∈ E ( F ) and a u ( H ) , u ∈ V ( F ) , A ( u,v ) ( H ) , ( u, v ) ∈ E ( F ) are expanded no-tations for A ( F ) and A ( H ), respectively. We have that A ( ψ ( u ) ,ψ ( v )) ( H ) is well defined for every( u, v ) ∈ E ( F ) since by the graph isomorphism property ( ψ ( u ) , ψ ( v )) ∈ E ( H ).Given decorated graphs F = ( V ( F ) , E ( F ) , A ( F )) and G = ( V ( G ) , E ( G ) , A ( G )) a mapping ψ : V ( F ) → V ( H ) is a color-respecting embedding if it is a color-respecting graph isomorphismbetween F and the subgraph of H induced by the image ψ ( V ( F )). We denote by Ind( F, H )the total number of of the subsets of nodes S ⊂ V ( H ) such that there exists color respectinggraph isomorphism between F and the decorated subgraph of H induced by S . Namely, it is thenumber of embeddings of F into H up to isomorphism. Later we will use the notation of theform P F ∈G i Ind(
F, H ) where the sum is over all uncountable collection G i , yet it makes sensesince only finitely many elements of this collection have a non-zero value for Ind( F, H ).Given a connected decorated graph F and another decorated graph H which is a disjointunion of two decorated graphs H and H we naturally have the following identityInd( F, H ) = Ind(
F, H ) + Ind( F, H ) . (20)The following relation for products of the number of embeddings will be useful. This observationwas also used in [PR17]. 16 emma 4.1. There exists a sequence of functions α m : G m → Z + such that for any decoratedgraph H and any sequence of decorated graphs F , . . . , F m Y ≤ ℓ ≤ m Ind ( F i , H ) = X α m +1 ( F , . . . , F m , F ) Ind ( F, H ) , (21) where the sum is over F ∈ ¯ G t with t = P ≤ ℓ ≤ k | V ( F ℓ ) | .Proof. For every m -tuple of color-respecting isomorphic embeddings ψ ℓ : V ( F ℓ ) → V ( H ) , ≤ ℓ ≤ m , consider the subgraph F of H induced by the union ∪ ≤ ℓ ≤ m ψ ( V ( F ℓ )). This graph has atmost P ℓ | V ( F ℓ ) | nodes. We obtain an embedding of this graph F in H . Then we see that (21)holds, where α m +1 ( F , . . . , F m , F ) is the number of m -tuples of embeddings of F , . . . , F m into F which span F .A key property stated in the lemma is that α m depends on the collection F , . . . , F m aloneand not on the target graph H . This section is devoted to the proof of Theorem 3.1. We prove the claim separately for each in-terpolation type. Both developments follow ideas similar to ones in [PR17]. The main distinctionis that our development is geared towards establishing the equality between the conditional andunconditional pseudo-marginals, whereas the goal in [PR17] is developing a method of countingconnected subgraph in order to obtain a polynomial time algorithm for computing T m ( G, z ). Type I interpolation
Fix a graph G = ( V, E ), fugacity λ > Z ( G ( z )) = X ≤ k ≤| V | i k ( G ) z k λ k . We note that the free coefficient of this polynomial i = 1. Applying the identity (16) we haveRoots( Z ( G ( z )) , k ) = X ( m ,...,m k ) ∈ Γ k α m ,...,m k Y ≤ j ≤ k (cid:0) i j ( G ) λ j (cid:1) m j , where Γ k denotes the set of all m , . . . , m k ≥ P ≤ ℓ ≤ k ℓm ℓ = k . Denote by I j an indepen-dent set of size j . Then i j ( G ) is Ind( I j , G ) with respect to trivial coloring a = A = 1 of both G and I j . In other words it is the number of isomorphic embeddings of a size j independent setinto G purely in graph theoretic sense. We then rewrite the above asRoots( Z ( G ( z )) , k ) = X ( m ,...,m k ) ∈ Γ k α m ,...,m k Y ≤ j ≤ k λ jm j (Ind( I j , G )) m j . · ) m i and applying Lemma 4.1 we see that we can write Roots( Z ( G ( z )) , k )in the form Roots( Z ( G ( z )) , k ) = X H ∈ ¯ G k β H,k
Ind(
H, G ) , (22)The bound k on the size appears since the set spanned by a union of m ℓ copies of I ℓ with1 ≤ ℓ ≤ k has size at most k , in light of P ℓ ℓm ℓ = k . A key fact for us is the following lemma. Lemma 5.1.
For every disconnected graph H and every k , β H,k = 0 .Proof.
This fact is established in several places including [CF16] and [PR17]. We reproduce theproof here for convenience.Fix any k . Assume for the purposes of contradiction that there exists a disconnected r -nodegraph H = ( V ( H ) , E ( H )) with β H ,k = 0. Without the loss of generality we may assume that r is the smallest value for which such a graph exists. Applying the identity (22) to G = H wehave Roots( Z ( H ( z )) , k ) = X H ∈ ¯ G k β H,k
Ind(
H, H ) . We expand the right-hand side as X H = H ∈ ¯ G k β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . (23)We will prove that β H ,k = 0, thus arriving at contradiction. Trivially Ind( H, H ) = 0 if | V ( H ) | > | V ( H ) | . Also Ind( H, H ) = 0 if | V ( H ) | = | V ( H ) | , but H = H (up to isomorphism). Thus theright-hand side above is X H ∈ ¯ G k , | V ( H ) | < | V ( H ) | β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . By the assumption of minimality of r = | V ( H ) | we have β H,k = 0 for all disconnected graphs H with | V ( H ) | < | V ( H ) | . ThusRoots( Z ( H ( z )) , k )= X H ∈ ¯ G k, conn , | V ( H ) | < | V ( H ) | β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . (24)Let H ,j , j = 1 , H into any two disconnected parts. For everyconnected graph H we have by (20)Ind( H, H ) = X j =1 , Ind(
H, H ,j ) . Z ( H ( z )) , k )= X j =1 , X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) + β H ,k Ind( H , H ) . (25)Applying (22) to H ,j , j = 1 , Z ( H ,j ( z )) , k ) = X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) . By (19) we have Roots( Z ( H ( z )) , k ) = X j =1 , Roots( Z ( H ,j ( z )) , k ) , and therefore Roots( Z ( H ( z ) , k )) = X j =1 , X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) . Comparing with (25) we conclude β H ,k Ind( H , H ) = 0 . Since Ind( H , H ) trivially has value at least 1, we conclude β H ,k = 0 thus arriving at contra-diction.Applying (22) and Lemma 5.1 we haveRoots( Z ( G ( z )) , k ) = X H ∈ ¯ G k, conn β H,k
Ind(
H, G ) . Now let f ( z ) = log Z ( G ( z )). Applying (17) we have f ( k ) (0) = k !( − k X H ∈ ¯ G k, conn β H,k
Ind(
H, G ) . and from (18) we obtain T m ( G, z ) = X ≤ k ≤ m z k ( − k X H ∈ ¯ G k, conn β H,k
Ind(
H, G ) . Similarly, for every S ⊂ V and σ : S → [ K ], letting f ( z ) = log Z ( G S,σ ( z )) we obtain f ( k ) (0) = k !( − k X H ∈ ¯ G k, conn β H,k
Ind(
H, G
S,σ ) , T m ( G S,σ , z ) = X ≤ k ≤ m z k ( − k X H ∈ ¯ G k, conn β H,k
Ind(
H, G
S,σ ) . We obtain the following representation for the pseudo-marginals: ν ( G, S, σ, z, m ) = exp (cid:16)P ≤ k ≤ m z k ( − k P H ∈ ¯ G k, conn β H,k
Ind(
H, G
S,σ ) (cid:17) exp (cid:16)P ≤ k ≤ m z k ( − k P H ∈ ¯ G k, conn β H,k
Ind(
H, G ) (cid:17) . Letting ∆(
H, S, σ ) = Ind(
H, G ) − Ind(
H, G
S,σ ), this simplifies to ν ( G, S, σ, z, m ) = exp − X ≤ k ≤ m z k ( − k X H ∈ ¯ G k, conn β H,k ∆( H, S, σ ) . Similarly, for any R and the set S ∪ ∂B ( S, R ) with τ : ∂B ( S, R ) → [ K ] we have ν ( G, S ∪ ∂B ( S, R ) , σ ∪ τ, z, m )= exp − X ≤ k ≤ m z k ( − k X H ∈ ¯ G k, conn β H,k ∆( H, S ∪ ∂B ( S, R ) , σ ∪ τ ) , and ν ( G, ∂B ( S, R ) , τ, z, m )= exp − X ≤ k ≤ m z k ( − k X H ∈ ¯ G k, conn β H,k ∆( H, ∂B ( S, R ) , τ ) , A key observation is that ∆(
H, S, σ ) involves only copies of connected graphs H in G with atmost k ≤ m nodes which intersect with S . As a result, when the distance R is sufficiently largethe sets of graphs H intersecting S and intersecting ∂B ( S, R ) are disjoint. Specifically, if R ≥ m then for every H with V ( H ) ∩ S = ∅ we have V ( H ) ∩ ∂B ( S, R ) = ∅ , and vice verse. As a result∆( H, S ∪ ∂B ( S, R ) , σ ∪ τ ) = ∆( H, S, σ ) + ∆(
H, ∂B ( S, R ) , τ ) . Therefore, ν ( G, S, σ, z, R | ∂B ( S, R ) , τ ) = ν ( G, S ∪ ∂B ( S, R ) , σ ∪ τ, z, R ) ν ( G, ∂B ( S, R ) , τ, z, R )= exp − X ≤ k ≤ R z k ( − k X H ∈ ¯ G k, conn β H,k ∆( H, S, σ ) = ν ( G, S, σ, z, R ) . This completes the proof of the theorem for the case of Type I interpolation.20 ype II interpolation
Turning next to the Type II interpolation, fix a decorated graph G = ( V, E, A ) with the deco-ration A = ( a u , u ∈ V, A ( u,v ) , ( u, v ) ∈ E ). Recall the definition of L from (11) and consider theassociated renormalized polynomial¯ Z ( G ( z )) , L − Z ( G ( z ))= L − X φ : V ( G ) → [ K ] Y u ∈ V ( G ) a uφ ( u ) Y ( u,v ) ∈ E ( G ) (cid:16) z (cid:16) A ( u,v ) φ ( u ) ,φ ( v ) − (cid:17)(cid:17) . By construction ¯ Z ( G (0)) = 1. Introduce a modified decoration ¯ A of the underlying graph ( V, E )as follows: ¯ a u = a u P i ∈ [ K ] a ui , u ∈ V, (26)¯ A ( u,v ) = A ( u,v ) − , ( u, v ) ∈ E. (27)We have X i ∈ [ K ] ¯ a ui = 1 , ∀ u ∈ V. (28)Denote by ¯ G the graph ( V, E ) with this modified decoration ¯ A . For any decorated graph H =( V ( H ) , E ( H ) , A ( H )) ∈ G i, edge let Z i ( H ) = X E ′ X φ : V ( H ) → [ K ] Y u ∈ V ( H ) ¯ a H,ui Y ( u,v ) ∈ E ′ ¯ A H, ( u,v ) φ ( u ) ,φ ( v ) , where the outer sum is taken over all subsets of edges E ′ ⊂ E ( H ) which span H and which havecardinality | E ′ | = i . Here a H, ·· and A H, ·· are the decorations associated with A ( H ), and the baroperation is defined for the decoration A ( H ) as per (26) and (27). Z i ( H ) is a partition functiontype object except the products over edges are taken only over spanning subsets of the edges of H with cardinality exactly i .Expanding the product Y ( u,v ) ∈ E ( G ) (cid:16) z (cid:16) A ( u,v ) φ ( u ) ,φ ( v ) − (cid:17)(cid:17) in powers of z , we claim that the following representation holds: Lemma 5.2. ¯ Z ( G ( z )) = X ≤ i ≤| V | z i X H ∈G i, edge Z i ( H ) Ind ( H, ¯ G ) . As noted earlier, the same graph H may appear in summands corresponding to more thanone values of i , as the graph can be spanned by different number of edges. The contributionto the ¯ Z ( G ( z )) though is different for different values of i as those will correspond to differentpowers of z . 21 roof. The coefficient associated with z i in polynomial ¯ Z ( G ( z )) is X φ : V ( G ) → [ K ] X E ′ ⊂ E : | E | = i L − Y u ∈ V ( G ) a uφ ( u ) Y ( u,v ) ∈ E ′ (cid:16) A ( u,v ) φ ( u ) ,φ ( v ) − (cid:17) . (29)Denote by H ( E ′ ) ∈ G i, edge the subgraph of G spanned by edges in E ′ . Then the sum in (29) is= X E ′ ⊂ E : | E | = i X φ : V ( G ) → [ K ] L − Y u ∈ V ( H ( E ′ )) a ui Y ( u,v ) ∈ E ′ (cid:16) A ( u,v ) φ ( u ) ,φ ( v ) − (cid:17) Y u/ ∈ V ( H ( E ′ )) a uφ ( u ) = X E ′ ⊂ E : | E | = i X φ : V ( H ( E ′ )) → [ K ] L − Y u ∈ V ( H ( E ′ )) a ui Y ( u,v ) ∈ E ′ (cid:16) A ( u,v ) φ ( u ) ,φ ( v ) − (cid:17) ×× X φ : V \ V ( H ( E ′ )) → [ K ] Y u ∈ V \ V ( H ( E ′ )) a uφ ( u ) = X E ′ ⊂ E : | E | = i X φ : V ( H ( E ′ )) → [ K ] Y u ∈ V ( H ( E ′ )) ¯ a ui Y ( u,v ) ∈ E ′ ¯ A ( u,v ) φ ( u ) ,φ ( v ) , Here in the second equality the map φ : V ( G ) → [ K ] is partition into its reduction to V ( H ( E ′ ))and its complement, and the product form structure is used. The last equality follows from thedefinition of L and ¯ a u . We recognize the last expression as X H ∈G i, edge Z i ( H )Ind( H, ¯ G ) . Using the representation (16) for the polynomial p ( z ) = ¯ Z ( G ( z )) and since the roots of¯ Z ( G ( z )) and Z ( G ( z )) are identical, we obtainRoots( Z ( G ( z )) , k )= X ( m ,...,m k ) ∈ Γ k α m ,...,m k Y ≤ i ≤ k X H ∈G i, edge Z i ( H )Ind( H, ¯ G ) m i . Next we expand the powers ( · ) m i . Each graph H ∈ G i, edge has at most 2 i nodes. ApplyingLemma 4.1, and using P ℓ ℓm ℓ = k for each ( m , . . . , m k ) ∈ Γ k we obtain a representation forevery k of the form: Roots( Z ( G ( z )) , k ) = X H ∈ ¯ G k β H,k
Ind( H, ¯ G ) , (30)where β H,k depend on the decorated graph H and k only. Note that by (28) we must have β H,k = 0 unless A ( H ) satisfies X i ∈ [ K ] a ui ( H ) = 1 , u ∈ V ( H ) . (31)22 emma 5.3. For every disconnected graph H and every k , β H,k = 0 .Proof.
The proof is similar to the one of Lemma 5.1, but with a minor adaptation required tohandle the case of decorated graph. A similar property for decorated graphs is also found in[PR17] for a different notion of color respecting isomorphisms.Fix any k . Assume for the purposes of contradiction that there exists a disconnected r -nodedecorated graph H = ( V ( H ) , E ( H ) , A ( H )) with β H ,k = 0. Without the loss of generality wemay assume that r is the smallest value for which such a decorated graph exists. Let us constructa coloring A of H such that ¯ A = A ( H ), where the transformation A → ¯ A is obtained by(26) and (27). This is achieved by simply adding 1 to every value A ( u,v ) i,j ( H ) , ( u, v ) ∈ E ( H ) , ≤ i, j ≤ K , and leaving a u ( H ) , u ∈ V ( H ) intact, due to (31). The graph ( V ( H ) , E ( H ) with thisnew coloring A ( H ) is denoted by H ′ . Applying the identity (30) to G = H ′ we haveRoots( Z ( H ′ ( z )) , k ) = X H ∈ ¯ G k β H,k
Ind( H, ¯ H ′ )= X H ∈ ¯ G k β H,k
Ind(
H, H ) , where the second equality is obtained since ¯ H ′ = H . We expand the right-hand side as X H = H ∈ ¯ G k β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . (32)We will prove that β H ,k = 0, thus arriving at contradiction. Trivially Ind( H, H ) = 0 if | V ( H ) | > | V ( H ) | . Also Ind( H, H ) = 0 if | V ( H ) | = | V ( H ) | , but H = H (up to isomorphism). Thus theright-hand side above is X H ∈ ¯ G k , | V ( H ) | < | V ( H ) | β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . By the assumption of minimality of r = | V ( H ) | we have β H,k = 0 for all disconnected graphs H with | V ( H ) | < | V ( H ) | . ThusRoots( Z ( H ′ ( z ) , k )= X H ∈ ¯ G k, conn , | V ( H ) | < | V ( H ) | β H,k
Ind(
H, H ) + β H ,k Ind( H , H ) . (33)Let H ,j , j = 1 , H into any two disconnected parts, with respectivecoloring reductions A ( H ,j ) , j = 1 ,
2. We denote by H ′ ,j , j = 1 , A . For every connected graph H we have by (20).Ind( H, H ) = X j =1 , Ind(
H, H ,j ) . Thus we may rewrite (33) asRoots( Z ( H ′ ( z ) , k )= X j =1 , X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) + β H ,k Ind( H , H ) . (34)23pplying (30) for H ′ ,j , j = 1 , Z ( H ′ ,j ( z ) , k ) = X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) . By (19) we have Roots( Z ( H ′ ( z ) , k ) = X j =1 , Roots( Z ( H ′ ,j ( z ) , k ) , and therefore Roots( Z ( H ′ ( z ) , k ) = X j =1 , X H ∈ ¯ G k, conn , | V ( H ) |≤| V ( H ,j ) | β H,k
Ind(
H, H ,j ) . Comparing with (34) we conclude β H ,k Ind( H , H ) = 0 . Since Ind( H , H ) trivially has value at least 1, we conclude that β H ,k = 0, thus arriving atcontradiction.Applying (30) and Lemma 5.3 we haveRoots( Z ( G ( z )) , k ) = X H ∈ ¯ G k, conn β H,k
Ind( H, ¯ G ) . The remainder of the proof is the same as for the case of Type I interpolation except that thevalue of R has to change to 2 k as opposed to k , since the sum is over connected graphs with atmost 2 k nodes. Acknowledgement
Several insightful conversations with Alexander Barvinok are gratefully acknowledged. The au-thor is very grateful to Guus Regts for suggesting more up to date references and for a suggestionon improving the bound appearing in the main result, Corollary 3.2. Many thanks Yuzhou Guand Yuri Polyanski for making the author aware of the Girard’s formula.
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