Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights
aa r X i v : . [ m a t h . C O ] D ec Complex zero-free regions at large | q | for multivariate Tutte polynomials(alias Potts-model partition functions)with general complex edge weights Bill Jackson
School of Mathematical SciencesQueen Mary University of LondonMile End RoadLondon E1 4NS, England
Aldo Procacci
Departamento de Matem´aticaUniversidade Federal de Minas GeraisAv. Antˆonio Carlos, 6627 – Caixa Postal 70230161-970 Belo Horizonte, MG – BRASIL
Alan D. Sokal ∗ Department of PhysicsNew York University4 Washington PlaceNew York, NY 10003 USA
Version 1: October 26, 2008Version 2: November 20, 2009Version 3: September 23, 2011
Abstract
We find zero-free regions in the complex plane at large | q | for the multivari-ate Tutte polynomial (also known in statistical mechanics as the Potts-modelpartition function) Z G ( q, w ) of a graph G with general complex edge weights w = { w e } . This generalizes a result of Sokal [28] that applies only within thecomplex antiferromagnetic regime | w e | ≤
1. Our proof uses the polymer-gasrepresentation of the multivariate Tutte polynomial together with the Penroseidentity.
Key Words:
Graph, chromatic polynomial, multivariate Tutte polynomial, Pottsmodel, Penrose identity, Penrose inequality, Lambert W function. Mathematics Subject Classification (MSC) codes: ∗ Also at Department of Mathematics, University College London, London WC1E 6BT, England. Introduction
A decade ago, Sokal [28] proved that if G = ( V, E ) is a loopless graph of maximumdegree ∆, then all the roots (real or complex) of the chromatic polynomial P G ( q ) liein the disc | q | < C (∆), where C (∆) are semi-explicit constants (given by a variationalformula) satisfying C (∆) ≤ . More generally, Sokal proved a bound onthe zeros of the multivariate Tutte polynomial (also known in statistical mechanicsas the Potts-model partition function, see [30, 26, 34, 35]) Z G ( q, w ) = X A ⊆ E q k ( A ) Y e ∈ A w e (1.1)[here k ( A ) denotes the number of connected components in the subgraph ( V, A )] whenthe edge weights w = { w e } lie in the “complex antiferromagnetic regime” | w e | ≤ Theorem 1.1 [28, Corollary 5.5]
Let G = ( V, E ) be a loopless graph equippedwith complex edge weights w = { w e } e ∈ E satisfying | w e | ≤ for all e . Then all thezeros of Z G ( q, w ) lie in the disc | q | < K ∆( G, w ) , where ∆( G, w ) = max x ∈ V X e ∋ x | w e | (1.2) and K = min ( L : inf α> α − ∞ X n =2 e αn L − ( n − n n − n ! ≤ ) (1.3a)= min a> a + e a log(1 + ae − a ) (1.3b) ≈ .
963 906 075 890 002 502 . . . . (1.3c)
Moreover, we rigorously have K ≤ . . Here the simpler formula (1.3b) for the constant K is due to Borgs [9, Theorem 2.1].The purpose of this paper is to extend Sokal’s bound by removing the conditionthat | w e | ≤ e . More precisely, we shall prove: All graphs in this paper are finite and undirected; furthermore, they are allowed to contain loopsand multiple edges unless we explicitly state otherwise. More recently, Borgs [9] has provided a simpler variational characterization of the constant K = lim ∆ →∞ C (∆) / ∆ ≈ . C (∆). Furthermore, Fern´andez andProcacci [14] have improved the constants C (∆) to smaller constants C ∗ (∆), for which K ∗ =lim ∆ →∞ C ∗ (∆) / ∆ ≈ . A simpler but weaker version of this result can be found in the first and second preprint versionsof this paper ( http://arxiv.org/abs/0810.4703v1 and v2 ). heorem 1.2 Let G = ( V, E ) be a loopless graph equipped with complex edge weights w = { w e } e ∈ E . Then all the zeros of Z G ( q, w ) lie in the disc | q | < b K (Ψ( G, w )) b ∆( G, w ) , (1.4) where b ∆( G, w ) = max x ∈ V X e ∋ xe = xy min (cid:26) | w e | , | w e || w e | (cid:27) Y f ∋ y max { , | w f |} / (1.5)Ψ( G, w ) = max x ∈ V Y e ∋ x max { , | w e |} (1.6) and b K ( ψ ) = min ( L : inf α> ( e α − − ∞ X n =2 e αn ψ / L − ( n − n n − n ! ≤ ) (1.7a)= min 907 651 697 774 449 218 . . . . This explicitformula for the Fern´andez–Procacci [14] constant K ∗ appears to be new.Let us also remark that the upper bound (1.7d) gives precisely the first two termsof the large- ψ asymptotics of b K ( ψ ): see equation (A.29) in the Appendix.Please note that both Ψ( G, w ) and b ∆( G, w ) involve a product over all edgesincident to a given vertex rather than a sum, and hence grow exponentially (ratherthan linearly) with the vertex degree whenever | w e | > 1. The resulting exponentialdependence of the bound on | q | given in Theorem 1.2 is not merely an artifact of ourproof, but is a genuine feature of the regime | w e | > To see this, it suffices to notethat whenever one replaces an edge e by k edges in parallel, the effective couplings w e, eff = (1 + w e ) k − k when | w e | > | w e | ≤ 1. For instance, the graph G = K ( k )2 (a pair of vertices connected by k parallel edges) with all edge weights equal has Z G ( q, w ) = q [ q + (1 + w ) k − See also [28, Remark 2 after Corollary 5.5]. 3e must take | q | > | (1 + w ) k − | to avoid a root. This has roughly (but not exactly)the same dependence in w and k as the bound of Theorem 1.2. See Example 7.3below for details.When all edge weights are equal, the two factors b K (Ψ( G, w )) and b ∆( G, w ) com-bine to produce a bound that grows linearly with Ψ( G, w ) as Ψ( G, w ) → ∞ . If werestrict attention to simple graphs, then with a little more combinatorial work we canobtain a bound that grows only like Ψ( G, w ) / : Theorem 1.3 Let G = ( V, E ) be a simple graph (i.e. no loops or multiple edges)equipped with complex edge weights w = { w e } e ∈ E . Then all the zeros of Z G ( q, w ) liein the disc | q | < K ∗ µ ∆ ∗ ( G, w ) , (1.8) where ∆ ∗ ( G, w ) = max x ∈ V X e ∋ xe = xy min (cid:26) | w e | , | w e || w e | / (cid:27) Y f ∋ y max { , | w f |} / (1.9) and µ = b ∆( G, w ) / ∆ ∗ ( G, w ) and K ∗ µ = min ( L : inf α> ( e α − − ∞ X n =2 e αn L − ( n − [1 + ( n − µ ] n − ( n − ≤ ) (1.10a)= min 1, by contrast, Theorem 1.3 is in most cases a big improvementover Theorem 1.2: this is because K ∗ µ is always order 1 while b K (Ψ( G, w )) is orderΨ( G, w ) / .Note that the bound (1.4) involves a double maximum: once over x ∈ V inΨ( G, w ), and once over x ∈ V in b ∆( G, w ). Such a bound is “unnatural” in the sensethat if G is a disjoint union G = G ⊎ G , then the chromatic roots of G are the unionof those of G and G , and b K (Ψ) and b ∆ are each the maximum of those for G and G , but the product b K (Ψ) b ∆ for G can exceed the maximum of those for G and G because one factor could be maximized for G and the other for G (see Example 7.7below). The bound (1.8) has the virtue of avoiding such a double maximum. It is4n open question whether a bound avoiding a double maximum can be obtained fornon-simple graphs.On the other hand, in the bound (1.8) we do pay a price, compared to (1.4),by having ∆ ∗ ( G, w ) in place of b ∆( G, w ), since as noted above we have ∆ ∗ ( G, w ) ≥ b ∆( G, w ). In fact, the simple example G = K shows that the bound of Theorem 1.3can in some cases be inferior to that of Theorem 1.2, by a factor of up to K ∗ / ≈ . G is a loopless graph with multiple edges, thenits multivariate Tutte polynomial is identical to that of the underlying simple graph b G in which each set of parallel edges e , . . . , e k in G is replaced by a single edge e in b G with weight b w e = Q ki =1 (1 + w e i ) − 1. So one is always free to apply Theorem 1.2 or 1.3to ( b G, b w ) instead of applying Theorem 1.2 to ( G, w ). The following lemma concerningthe behavior of Ψ( G, w ) and b ∆( G, w ) under parallel reduction — which will be provenat the end of Section 6 — implies that the bound we get by applying Theorem 1.2to ( b G, b w ) will never be worse than the bound we get by applying Theorem 1.2 to( G, w ). So we can find our best bound for any given (multi)graph G by constructing( b G, b w ) and then taking the minimum of the bounds we obtain by applying (1.4) and(1.8) to ( b G, b w ). Lemma 1.4 Let w , w ∈ C and put w = (1 + w )(1 + w ) − . Then max { , | w |} ≤ max { , | w |} max { , | w |} (1.11) and min (cid:26) | w | , | w || w | (cid:27) ≤ min (cid:26) | w | , | w || w | (cid:27) + min (cid:26) | w | , | w || w | (cid:27) . (1.12)Sokal’s proof of Theorem 1.1 involved the following steps:1. Write the multivariate Tutte polynomial Z G ( q, w ) as the partition function ofa polymer gas with weights depending on q and w (this is easy: see Section 2below).2. Invoke the Koteck´y–Preiss [21] condition for the nonvanishing of the partitionfunction of a polymer gas.3. Control the polymer weights by bounding sums over connected subgraphs bysums over trees, using the Penrose inequality [25]. This step required | w e | ≤ 1. 5. Bound the total weight of n -vertex trees (or more generally, of connected sub-graphs with m edges) in G that contain a specified vertex x ∈ V .5. Put everything together to prove that Z G ( q, w ) = 0 whenever q lies outside aspecified disc.Here we follow the same outline, but modify step 3 so as to allow arbitrary complexweights w e . In addition, in step 2 we replace the Koteck´y–Preiss condition by themore powerful Gruber–Kunz–Fern´andez–Procacci [16, 13] condition, thereby slightlyimproving the numerical constant along the lines of the work of Fern´andez and Pro-cacci [14] for chromatic polynomials. Finally, we need a slightly strengthened versionof the bound in step 4.The plan of this paper is to treat each of these five steps in successive sections.Thus, in Section 2 we recall how the multivariate Tutte polynomial Z G ( q, w ) can bewritten as the partition function of a polymer gas. In Section 3 we recall the Koteck´y–Preiss and Gruber–Kunz–Fern´andez–Procacci conditions for the nonvanishing of thepartition function of a polymer gas. In Section 4 we recall the Penrose identity[25] and show how to use it to bound the polymer weights without assuming that | w e | ≤ 1; this is our main new contribution. In Section 5 we prove a bound on thetotal weight of connected m -edge subgraphs in G that contain a specified vertex x ;this strengthens the bound of [28, 17] by taking specific account of the edges incidenton x and by introducing vertex weights. In Section 6 we put everything togetherto prove Theorems 1.2 and 1.3; we also prove Lemma 1.4. Finally, in Section 7 weexamine some examples that shed light on the extent to which Theorems 1.2 and 1.3are sharp or non-sharp. In an Appendix we prove Lemma 6.1 and some related facts. Z G ( q, w ) In statistical mechanics, an abstract polymer gas is a triple ( P, ξ, R ) where P isa finite set (whose elements are called “polymers”), ξ is a complex-valued functiondefined on P (the value ξ ( p ) is called the “activity” or “fugacity” or “weight” ofthe polymer p ∈ P ), and R ⊆ P × P is a symmetric and reflexive relation (calledthe “incompatibility relation”). Note that, since R is supposed reflexive, we have( p, p ) ∈ R for all p ∈ P . Then the partition function of the polymer gas ( P, ξ, R ) — akey quantity from which all thermodynamic properties of the system can in principlebe derived — is defined byΞ( ξ ) = ∞ X n =0 X { p ,...,p n }⊆ P ( p i ,p j ) / ∈R ∀ i = j ξ ( p ) · · · ξ ( p n ) (2.1)where the sum runs over unordered collections { p , . . . , p n } of mutually compatibleelements of P , and the n = 0 term in the sum is understood to contribute 1.6n this section we recall how to rewrite the multivariate Tutte polynomial Z G ( q, w )of a graph G = ( V, E ) as the partition function of a polymer gas living on the vertexset of G , i.e. an abstract polymer gas whose polymers are nonempty subsets of V .This easy result is due to Sokal and Kupiainen [28, Proposition 2.1].First, some notation: If H = ( V , E ) is a graph equipped with edge weights w = { w e } e ∈ E , we denote by C H ( w ) the generating polynomial of connected span-ning subgraphs of H , i.e. C H ( w ) = X A ⊆ E ( V ,A ) connected Y e ∈ A w e . (2.2)Note that C H ( w ) ≡ H is disconnected.If G = ( V, E ) is a graph and S ⊆ V , we denote by G [ S ] the induced subgraph of G on S , i.e. G [ S ] is the graph whose vertex set is S and whose edges consist of allthe edges of G both of whose endpoints lie in S . Proposition 2.1 (polymer representation of the multivariate Tutte polynomial) Let G = ( V, E ) be a loopless graph equipped with edge weights w = { w e } e ∈ E . Then q −| V | Z G ( q, w ) = ∞ X N =0 X { S ,...,S N } disjoint n Y i =1 ξ ( S i ) , (2.3) where the sum runs over unordered collections { S , . . . , S N } of disjoint nonemptysubsets of V , and the weights ξ ( S ) are given by ξ ( S ) = ( q − ( | S |− C G [ S ] ( w ) if | S | ≥ if | S | = 1 (2.4) [The N = 0 term in the sum (2.3) is understood to contribute .] The identity (2.3) thus represents q −| V | Z G ( q, w ) as the partition function of apolymer gas given by the triple ( P, ξ, R ) with the polymer space P being the set ofall nonempty subsets of V , the activity ξ being the function defined in (2.4), and theincompatibility relation R being nonempty intersection, i.e. ( S, S ′ ) ∈ R if and only if S ∩ S ′ = ∅ . Note that, since the weight ξ ( S ) vanishes for sets of cardinality 1 and alsovanishes whenever the induced subgraph G [ S ] is disconnected, we can equivalentlyrestrict our polymer set P to be the set of all subsets S ⊆ V of cardinality at least 2and for which G [ S ] is connected.Hereafter we will refer to a polymer gas in which polymers are subsets of a givenset V and the incompatibility relation is nonempty intersection as “a gas of nonover-lapping polymers living on V ”. Proof of Proposition 2.1. Starting from the definition (1.1) of Z G ( q, w ), let usseparate the terms in the sum according to the number k of connected components7i.e. k ( A ) = k ] and according to the partition { S , . . . , S k } of V that is induced by thevertex sets of those connected components; we will then sum over all ways of choosingedges within those vertex sets S i so as to connect those vertices. We thus have Z G ( q, w ) = q | V | X k ≥ X { S ,...,S k } V = U S i k Y i =1 q − ( | S i |− C G [ S i ] ( w ) , (2.5)where the sum runs over all unordered partitions { S , . . . , S k } of V into nonemptysubsets, and we have used | V | = P ki =1 | S i | . Note now that any set S i of cardinality 1gets weight q − ( | S i |− C G [ S i ] ( w ) = 1 (here we have used the fact that G is loopless).So let us define { S ′ , . . . , S ′ N } to be the subcollection of { S , . . . , S k } consisting ofthe sets of cardinality ≥ 2; and let us note that there is a one-to-one correspondencebetween unordered partitions { S , . . . , S k } of V into nonempty subsets and unorderedcollections { S ′ , . . . , S ′ N } of disjoint subsets of V of cardinality at least 2 (which neednot cover all of V : indeed, the points not covered correspond to the singleton sets S i in the original partition). Passing to { S ′ , . . . , S ′ N } and dropping the primes, we have(2.3)/(2.4). (cid:3) Let V be a finite set, and let { ρ ( S ) } ∅ = S ⊆ V be a collection of complex weightsassociated to the nonempty subsets of V . Consider now a gas of nonoverlappingpolymers living on V , with weights ρ ( S ): the partition function of such a polymergas is, by definition, Ξ = ∞ X N =0 X { S ,...,S N } disjoint N Y i =1 ρ ( S i ) , (3.1)where the sum runs over unordered collections { S , . . . , S N } of disjoint nonemptysubsets of V , and the N = 0 term in (3.1) is understood to contribute 1. The followingproposition — essentially proven almost four decades ago by Gruber and Kunz [16,Section 4, cf. eq. (33)] but largely forgotten, and then rediscovered very recently byFern´andez and Procacci [13, eq. (3.17)] with a new proof — gives a sufficient conditionfor the nonvanishing of a polymer-gas partition function: Proposition 3.1 (Gruber–Kunz–Fern´andez–Procacci condition) Let V be afinite set, and let { ρ ( S ) } ∅ = S ⊆ V be complex weights associated to the nonempty subsetsof V . Suppose that there exists a number α > such that sup x ∈ V X S ∋ x e α | S | | ρ ( S ) | ≤ e α − . (3.2)8 hen Ξ ≡ ∞ X N =0 X { S ,...,S N } disjoint n Y i =1 ρ ( S i ) = 0 . (3.3)See also [6] for an extremely simple proof of Proposition 3.1 by induction on V .In the slightly less powerful Koteck´y–Preiss [21] condition, the term e α − α . Remark. Suppose that (as happens in all nontrivial cases) there exists a set S with | S | ≥ ρ ( S ) = 0. Then the hypothesis that there exists α > α> ( e α − − sup x ∈ V X S ∋ x e α | S | | ρ ( S ) | ≤ , (3.4)since in this case the infimum on the left-hand side of (3.4) will always be attainedat some α > We will use the Gruber–Kunz–Fern´andez–Procacci condition in theform (3.4). C H ( w ) via the Penrose identity In this section we recall the Penrose identity [25] and show how it can be used tobound a sum over connected subgraphs by a sum over trees even in the absence ofthe hypothesis | w e | ≤ H = ( V , E ) be a graph. Recall that C H ( w ) denotes the generating polynomialof connected spanning subgraphs of H : C H ( w ) = X A ⊆ E ( V ,A ) connected Y e ∈ A w e . (4.1)We denote by T H ( w ) the generating polynomial of spanning trees in H : T H ( w ) = X A ⊆ E ( V ,A ) tree Y e ∈ A w e . (4.2) If there exists a set S with | S | ≥ ρ ( S ) = 0, then the function f ( α ) being minimized onthe left-hand side of (3.4) is a continuous function that tends to + ∞ as α ↓ α ↑ ∞ , henceits minimum is attained.There is one exceptional case in which (3.4) holds but there does not exist α > ρ ( S ) = 0 whenever | S | ≥ x ∈ V | ρ ( { x } ) | = 1. Indeed, if ρ ( S ) = 0 for | S | ≥ 2, we have Ξ = Q x ∈ V [1 + ρ ( { x } )], which vanishes when at least one ρ ( { x } ) equals − 1; so (3.4) fails (barely) to imply Ξ = 0 in this case. C (resp. T ) be the set of subsets A ⊆ E such that ( V , A ) is connected (resp.is a tree). Clearly C is an increasing family of subsets of E with respect to set-theoretic inclusion, and the minimal elements of C are precisely those of T (i.e. thespanning trees). It is a nontrivial combinatorial fact — apparently first discoveredby Penrose [25] — that the (anti-)complex C is partitionable : that is, there exists amap R : T → C such that R ( T ) ⊇ T for all T ∈ T and C = U T ∈T [ T, R ( T )] (disjointunion), where [ E , E ] denotes the Boolean interval { A : E ⊆ A ⊆ E } . We call anysuch map R a partition scheme . In fact, many alternative choices of R are available ,and most of our arguments will not depend on any specific choice of R . An immediateconsequence of the existence of R is the following simple but fundamental identity: Proposition 4.1 (Penrose identity [25]) Let R : T → C be any partition scheme.Then C H ( w ) = X T ⊆ E ( V ,T ) tree Y e ∈ T w e X T ⊆ A ⊆ R ( T ) Y e ∈ A \ T w e (4.3a)= X T ⊆ E ( V ,T ) tree Y e ∈ T w e Y e ∈ R ( T ) \ T (1 + w e ) . (4.3b)If | w e | ≤ e , then it is obvious that we can take absolute valueseverywhere in (4.3b) and drop the factors | w e | , yielding: Proposition 4.2 (Penrose inequality [25]) Let H = ( V , E ) be a graph equippedwith complex edge weights w = { w e } e ∈ E satisfying | w e | ≤ for all e . Then | C H ( w ) | ≤ T H ( | w | ) . (4.4) Remark. By using a specific choice of the map R (namely, that of Penrose [25]),Fern´andez and Procacci [13] have recently shown how to improve Proposition 4.2 when w e ∈ {− , } for all e ; and this improvement plays a key role in their proof of theGruber–Kunz–Fern´andez–Procacci condition (Proposition 3.1) for polymer gases withhard-core repulsive interactions. See also Fern´andez et al. [12] for a generalizationto − ≤ w e ≤ 0, which leads to an improved convergence criterion for the Mayerexpansion in lattice gases with soft repulsive interactions. (cid:3) Let us now show what can be done without the hypothesis | w e | ≤ 1. Given avertex x in a graph H = ( V , E ), we denote by E ( x ) the set of edges of H incident on x . For any subset A ⊆ E , let us write A + = { e ∈ A : | w e | > } (4.5a) A − = { e ∈ A : | w e | ≤ } (4.5b) See for example [25], [7, Sections 7.2 and 7.3], [37, Section 8.3], [15, Sections 2 and 6], [5,Proposition 13.7 et seq.], [28, Proposition 4.1] and [27, Lemma 2.2]. roposition 4.3 (extended Penrose inequality) Let H = ( V , E ) be a looplessgraph equipped with complex edge weights w = { w e } e ∈ E . Then | C H ( w ) | ≤ T H ( | w ′ | ) Y e ∈ E max { , | w e |} (4.6a)= T H ( | w ′ | ) Y y ∈ V Y e ∈ E ( y ) max { , | w e |} / (4.6b) where w ′ e = w e if | w e | ≤ w e w e if | w e | > | w e | ≤ e , then w ′ = w and max { , | w e |} = 1 for all e ,so Proposition 4.3 is a genuine extension of Proposition 4.2. Proof of Proposition 4.3. In the Penrose identity (4.3b), multiply and dividethe summand by Q e ∈ T + (1 + w e ): this yields C H ( w ) = X T ⊆ E ( V ,T ) tree Y e ∈ T w ′ e Y e ∈ ( R ( T ) \ T ) ∪ T + (1 + w e ) . (4.8)Taking absolute values and using the trivial bound Y e ∈ ( R ( T ) \ T ) ∪ T + | w e | ≤ Y e ∈ E max { , | w e |} , (4.9)we obtain (4.6a). Then (4.6b) follows by observing that each edge e ∈ E is incidenton precisely two vertices (since H is loopless). (cid:3) Remark. Quite a lot has been thrown away in (4.9). Can we do better in ausable way? (cid:3) If we assume that the graph H is simple (i.e. has no loops or multiple edges), thenwe can get a slightly better bound: Proposition 4.4 (extended Penrose inequality for simple graphs) Let H =( V , E ) be a simple graph (i.e. no loops or multiple edges) equipped with complex edgeweights w = { w e } e ∈ E . Then, for any vertex x ∈ V , we have | C H ( w ) | ≤ T H ( | w [ x ] | ) Y e ∈ E \ E ( x ) max { , | w e |} (4.10a) ≤ T H ( | e w [ x ] | ) Y y ∈ V \{ x } Y e ∈ E ( y ) max { , | w e |} / (4.10b)11 here w [ x ] e = w e if | w e | ≤ e ∈ E ( x ) w e w e if | w e | > e ∈ E \ E ( x ) (4.11) and e w [ x ] e = w e if | w e | ≤ w e | w e | / if | w e | > e ∈ E ( x ) w e | w e | if | w e | > e ∈ E \ E ( x ) (4.12)Please note that (4.10b) is indeed an improvement of (4.6b), because the product Q e ∈ E ( x ) max { , | w e |} / more than compensates the factors | e w [ x ] e /w ′ e | = max { , | w e |} / for the subset of edges in E ( x ) that happen to lie in any given spanning tree T .The proof of Proposition 4.4 will be based on the following key combinatorial fact(to be proven later): Lemma 4.5 Let H = ( V , E ) be a simple graph and let x ∈ V be any vertex. Thenthere exists a partition scheme R with the property that R ( T ) \ T does not containany edge incident on x . Proof of Proposition 4.4, assuming Lemma 4.5. In the Penrose identity(4.3b), multiply and divide the summand by Q e ∈ [ T \ E ( x )] + (1 + w e ): this yields C H ( w ) = X T ⊆ E ( V ,T ) tree Y e ∈ T w [ x ] e Y e ∈ [ R ( T ) \ T ] ∪ [ T \ E ( x )] + (1 + w e ) . (4.13)Choosing the partition scheme as in Lemma 4.5, we have R ( T ) \ T ⊆ E \ E ( x ) andhence Y e ∈ [ R ( T ) \ T ] ∪ [ T \ E ( x )] + | w e | ≤ Y e ∈ E \ E ( x ) max { , | w e |} . (4.14)Taking absolute values in (4.13) and using (4.14), we obtain | C H ( w ) | ≤ X T ⊆ E ( V ,T ) tree Y e ∈ T | w [ x ] e | Y e ∈ E \ E ( x ) max { , | w e |} , (4.15)which is (4.10a). 12ow observe that Y e ∈ E \ E ( x ) max { , | w e |} = Q y ∈ V \ x Q e ∈ E ( y ) max { , | w e |} / Q e ∈ E ( x ) max { , | w e |} / (4.16)since the numerator of (4.16) counts every edge in E \ E ( x ) twice and every edge in E ( x ) once. If in the denominator of (4.16) we replace the product over e ∈ E ( x ) bythe smaller product over e ∈ E ( x ) ∩ T , we get an upper bound; inserting this into(4.15) yields (4.10b). (cid:3) Let us conclude this section by proving Lemma 4.5. This proof — unlike all thepreceding results in this section — depends on a specific choice of the map R , namelythe one used by Penrose in his original paper [25]. Let us briefly recall Penrose’sconstruction (see [13, 12] for more details). We assume that H = ( V , E ) is a simple graph, and we choose (arbitrarily) an ordering of the vertex set V by numbering thevertices 1 , , . . . , n (where n = | V | ). We consider the vertex 1 to be the root, anddenote it by r . If T ⊆ E is the edge set of a spanning tree in H [that is, ( V , T ) isa tree], then for each x ∈ V we denote by dist T ( x ) the graph-theoretic distance inthe tree ( V , T ) from the root r to the vertex x . Given T , the vertex set V is thuspartitioned into “generations”, defined as the sets of vertices at a given distance fromthe root r .The Penrose map R : T R ( T ) is then defined as follows. For any tree T ⊆ E ,the edge set R ( T ) ⊇ T is obtained from T by adjoining all edges e ∈ E that either(a) connect two vertices in the same generation [i.e. at equal distance from theroot r in the tree ( V , T ) — note that no such edge can belong to T ], or(b) connect a vertex x to a vertex x ′ in the preceding generation [i.e. with dist T ( x ′ ) =dist T ( x ) − 1] that is higher-numbered than the parent of x [here the parent of x is the unique vertex y with dist T ( y ) = dist T ( x ) − xy ∈ T ].It can be shown [25, 13, 12] that R is indeed a partitioning map in the sense that C isthe disjoint union of Boolean intervals [ T, R ( T )]. Furthermore, it follows immediatelyfrom this construction that R ( T ) \ T cannot contain any edge incident on the root r ;that is, R ( T ) \ T ⊆ E \ E ( r ). Since any vertex could have been chosen as the root,Lemma 4.5 is proven. We remark that this would no longer be the case in a generalization to the Penrose constructionto non-simple graphs. In such a generalization, we would also order the edges connecting each pairof vertices, and we would add to the definition of R ( T ) a third case:(c) connect a vertex x to its parent y by any edge that is higher-numbered than the edge con-necting x to y in T .We would then no longer be able to guarantee that R ( T ) \ T contains no edges incident on theroot r ; rather, we could assert only that R ( T ) \ T cannot contain any edge incident on the root r that is the lowest-numbered among its set of parallel edges. emark. Lemma 4.5 suggests the following combinatorial question: Let H =( V , E ) be a graph (simple or not). For which subsets S ⊆ E does there exist apartition scheme R with the property that R ( T ) \ T ⊆ E \ S for all T ? The samequestion can also be posed for matroids. (cid:3) m -edge subgraphs con-taining a specified vertex In this section consider a loopless graph G = ( V, E ) equipped with nonnegativereal edge weights { w e } e ∈ E and nonnegative real vertex weights { w v } v ∈ V . Let usdefine the weighted sum over connected subgraphs G ′ = ( V ′ , E ′ ) ⊆ G that contain aspecified vertex x and have exactly m edges: c m ( x ; G, w ) = X G ′ =( V ′ ,E ′ ) ⊆ GG ′ connected V ′ ∋ x | E ′ | = m Y e ∈ E ′ w e Y v ∈ V ′ w v , (5.1)where we write w = { w e } e ∈ E ∪ { w v } v ∈ V . We will abbreviate c m ( x ; G, w ) to c m ( x )when it is obvious which weighted graph ( G, w ) we are referring to. Now define theweighted degree at x by d ( x ; G, w ) = X e = xy ∈ E w e w y (5.2)(note that this contains a factor w y for each edge e = xy incident to x but not afactor w x ), and define the maximum weighted degree by∆( G, w ) = max x ∈ V d ( x ; G, w ) . (5.3)The following bound on c m ( x ) extends an earlier result of the third author [28,Proposition 4.5], which is obtained by putting w v = 1 for all v ∈ V and using thefact that both d ( x ; G, w ) and ∆( G − x, w | G − x ) are bounded above by ∆( G, w ). Proposition 5.1 Let G = ( V, E ) be a loopless graph equipped with nonnegative realweights w = { w e } e ∈ E ∪ { w v } v ∈ V , and let x ∈ V . Suppose that either w v ≥ for all v ∈ V or G is simple. Then c m ( x ) ≤ w x d ( x ; G, w ) [ d ( x ; G, w ) + m ∆( G − x, w | G − x )] m − m ! (5.4) for all m ≥ . 14e remark that the bound (5.4) need not hold if we remove the hypothesis thateither w v ≥ v ∈ V or G is simple. Consider, for instance, the graph G = K ( m )2 consisting of two vertices x, y joined by m ≥ w x = w y = w and w e = 1 for all e ∈ E . Then c m ( x ) = w , while the right-hand side of (5.4) is m m w m +1 /m !, which is less than c m ( x ) when w is small enough.In the proof of Proposition 5.1 it will be convenient to employ the quantities C ( m, κ ) = ( κ ( m + κ ) m − /m ! for m ≥ 11 for m = 0 (5.5)defined for integer m ≥ κ . Then (5.4) can be rewritten in the form c m ( x ) ≤ w x C ( m, d/ ∆) ∆ m (5.6)where d = d ( x ; G, w ) and ∆ = ∆( G − x, w | G − x ).Our proof of Proposition 5.1 uses induction on m , and is similar to the first proofof [17, Proposition 7.1]. It relies on the following properties of C ( m, κ ):(a) For each integer m ≥ C ( m, κ ) is a polynomial of degree m in κ , with nonneg-ative coefficients. In particular, C ( m, κ ) is an increasing function of κ for real κ ≥ C ( z ) solves the equation C ( z ) = e z C ( z ) , (5.7)then C ( z ) κ = ∞ X m =0 C ( m, κ ) z m (5.8)for all real κ ; this follows from the Lagrange inversion formula. Moreover, theseries (5.8) is absolutely convergent for | z | ≤ /e and satisfies C (1 /e ) = e .(c) For integer k ≥ C ( m, k ) = X m ,...,m k ≥ m + ··· + m k = m k Y i =1 C ( m i , . (5.9)This is an immediate consequence of (5.8).(d) For all real κ and z , C ( m, κ ) = m X f =0 z f f ! C ( m − f, κ − z + f ) . (5.10)See [17, eq. (7.7)]. 15or any subset F ⊆ E , we use the notation w ( F ) = Q e ∈ F w e . Also, for any F ⊆ E ( x ), we denote by Y F the set of vertices of V − x that are incident with edgesin F , and we write j ( F ) = | Y F | for the number of such vertices. Please observe that j ( F ) ≤ | F | ; and if the graph G is simple, then j ( F ) = | F | .Our proof of Proposition 5.1 will be based on the following two lemmas: Lemma 5.2 Let G = ( V, E ) be a loopless graph equipped with nonnegative realweights w = { w e } e ∈ E ∪ { w v } v ∈ V , and let x ∈ V . For each F ⊆ E ( x ) , let Y F = { x F , x F , . . . , x Fj ( F ) } be a labeling of the vertices of V − x that are incident with edgesin F . Then, for all m ≥ , c m ( x ; G, w ) ≤ w x X ∅ = F ⊆ E ( x ) w ( F ) X m ,...,m j ( F ) ≥ m + ··· + m j ( F ) = m −| F | j ( F ) Y i =1 c m i ( x Fi ; G − x, w | G − x ) . (5.11) Proof. Similar to that given for Facts 1 and 2 in [17, Section 7]. (cid:3) Lemma 5.3 [17, Lemma 7.2] Let S be a set in which each element e ∈ S is givena nonnegative real weight w e . Then, for each integer f ≥ , we have X F ⊆ S | F | = f w ( F ) ≤ f ! X e ∈ S w e ! f . (5.12) Proof of Proposition 5.1. Let d = d ( x ; G, w ) and ∆ = ∆( G − x, w | G − x ). Wewill prove (5.4)/(5.6) by induction on m . The statement holds trivially when m = 0,so let us assume that m ≥ 1. By Lemma 5.2, c m ( x ) ≤ w x X ∅ = F ⊆ E ( x ) w ( F ) X m ,...,m j ( F ) ≥ m + ··· + m j ( F ) = m −| F | j ( F ) Y i =1 c m i ( x Fi ; G − x, w | G − x ) ≤ w x X ∅ = F ⊆ E ( x ) w ( F ) X m ,...,m j ( F ) ≥ m + ··· + m j ( F ) = m −| F | j ( F ) Y i =1 w x Fi C ( m i , 1) ∆ m i = w x X ∅ = F ⊆ E ( x ) C ( m − | F | , j ( F )) ∆ m −| F | w ( F ) j ( F ) Y i =1 w x Fi ≤ w x m X f =1 C ( m − f, f ) ∆ m − f X F ⊆ E ( x ) | F | = f Y e = xx Fi ∈ F w e w x Fi (5.13)16here the second line used the induction hypothesis (5.4) applied to the graph G − x (note that m i < m ) and the fact that d ( v ; G − x, w | G − x ) ≤ ∆ for all v ∈ V − x ; thethird line used the identity (5.9); and the last line used j ( F ) ≤ | F | , the fact that C ( m, k ) is an increasing function of k , and the hypothesis that either w x Fi ≥ ≤ i ≤ j ( F ) or G is simple. Using Lemma 5.3, we have c m ( x ) ≤ w x ∆ m m X f =1 ( d/ ∆) f f ! C ( m − f, f )= w x ∆ m m X f =0 ( d/ ∆) f f ! C ( m − f, f )= w x C ( m, d/ ∆) ∆ m , (5.14)where the second line used C ( m, 0) = 0 for m ≥ 1, and the last line used identity(5.10) with κ = z = d/ ∆. This proves (5.6). (cid:3) We now combine Proposition 5.1 with the extended Penrose inequalities fromSection 4: Proposition 5.4 Let G = ( V, E ) be a loopless graph equipped with complex edgeweights w = { w e } e ∈ E . Let x ∈ V and let n be a positive integer. Then X S ∋ xS ⊆ V | S | = n | C G [ S ] ( w ) | ≤ n n − n ! b ∆( G, w ) n − Y e ∈ E ( x ) max { , | w e |} / (5.15) where b ∆( G, w ) is defined in (1.5). Furthermore, if G is simple, then X S ∋ xS ⊆ V | S | = n | C G [ S ] ( w ) | ≤ ∆ ∗ ( G, w )( n − h ∆ ∗ ( G, w ) + ( n − b ∆( G, w ) i n − (5.16) where ∆ ∗ ( G, w ) is defined in (1.9). Proof. We first prove (5.15). Construct a nonnegative real weight function b w on V ∪ E by putting b w y = Q e ∈ E ( y ) max { , | w e |} / for all y ∈ V , and b w e = | w ′ e | forall e ∈ E , where w ′ e is defined in (4.7). For y ∈ S ⊆ V let E ( y ; G [ S ]) denote the setof edges of G [ S ] incident on y . By bound (4.6b) of Proposition 4.3, we have X S ∋ xS ⊆ V | S | = n | C G [ S ] ( w ) | ≤ X S ∋ xS ⊆ V | S | = n T G [ S ] ( | w ′ | ) Y y ∈ S Y e ∈ E ( y ; G [ S ]) max { , | w e |} / (5.17a) ≤ c n − ( x ; G, b w ) (5.17b)17ince the n -vertex trees are a subset of the connected graphs with n − E ( y ; G [ S ]) ⊆ E ( y ). Inequality (5.15) now follows by applying Proposition 5.1, usingthe fact that d ( x ; G, b w ) and ∆( G − x, b w | G − x ) are both bounded above by ∆( G, b w ) = b ∆( G, w ).We next prove (5.16). Construct a weight function w ∗ on V ∪ E by putting w ∗ x = 1, w ∗ y = Q e ∈ E ( y ) max { , | w e |} / for all y ∈ V \ { x } , and w ∗ e = | e w [ x ] e | for all e ∈ E ,where e w [ x ] e is defined in (4.12). By bound (4.10b) of Proposition 4.4, we have X S ∋ xS ⊆ V | S | = n | C G [ S ] ( w ) | ≤ X S ∋ xS ⊆ V | S | = n T G [ S ] ( | e w [ x ] | ) Y y ∈ S \{ x } Y e ∈ E ( y ; G [ S ]) max { , | w e |} / (5.18a) ≤ c n − ( x ; G, w ∗ ) (5.18b)by the same reasoning as before. Inequality (5.16) now follows by applying Propo-sition 5.1, using the facts that d ( x ; G, w ∗ ) ≤ ∆ ∗ ( G, w ) and ∆( G − x, w ∗ | G − x ) ≤ b ∆( G, w ). (cid:3) We can now put together the results of the preceding sections to prove Theo-rems 1.2 and 1.3. At the end of this section we will also prove Lemma 1.4.We begin by stating an analytic lemma that will be needed in proving the equiv-alence between the various versions (1.7a–d) and (1.10a–c) of our bounds. To avoiddisrupting the flow of the argument, the proof of this lemma is deferred to an Ap-pendix. Lemma 6.1 For λ ≥ and β > , define the function F λ ( β ) = min ( L : inf α> ( e α − − ∞ X n =2 e αn L − ( n − [1 + ( n − λ ] n − ( n − ≤ β ) . (6.1) Then F λ ( β ) = min We modify the proof of Theorem 1.2 by using (5.16) inplace of (5.15).Since G is simple, it follows from (5.16) that for each x ∈ V and each n ≥ X S ∋ xS ⊆ V | S | = n | C G [ S ] ( w ) | ≤ ∆ ∗ ( G, w )( n − h ∆ ∗ ( G, w ) + ( n − b ∆( G, w ) i n − = ∆ ∗ ( G, w ) n − [1 + ( n − µ ] n − ( n − µ = b ∆( G, w ) / ∆ ∗ ( G, w ). Therefore, the condition (3.4) for the weights (2.4)/(6.5)is verified as soon asinf α> ( e α − − X n ≥ e αn [ | q | − ∆ ∗ ( G, w )] n − [1 + ( n − µ ] n − ( n − ≤ . (6.9)If we set L = | q | ∆ ∗ ( G, w ) − , this is precisely the inequality contained in the right-hand side of (1.10a). So Z G ( q, w ) = 0 whenever L ≥ K ∗ µ , i.e. whenever | q | ≥ ∗ µ ∆ ∗ ( G, w ), where K ∗ µ = F µ (1) is defined by (1.10a). The equivalence of (1.10a)with (1.10b) and the inequality (1.10c) then follow from Lemma 6.1. (cid:3) Discussion. 1. We can now understand why the apparently minor improvementfrom (4.6b) to (4.10b) leads to the significant improvement (in most cases) of the finalbound from Theorem 1.2 to Theorem 1.3, namely, replacing a growth ∼ Ψ( G, w ) / by 1. Indeed, we can see using Lemma 6.1 that whenever we have a bound of theform X S ∋ x | S | = n | C G [ S ] ( w ) | ≤ [1 + λ ( n − n − ( n − D n − Ψ b , (6.10)we will obtain a bound on the roots of Z G ( q, w ) of the form | q | < D F λ (Ψ − b ) . (6.11)The bound (4.6b) gives rise to inequality (5.15), which in turn allows us to deduceTheorem 1.2 by taking D = b ∆, λ = 1 and b = 1 / 2. On the other hand, the bound(4.10b) gives inequality (5.16), which allows us to deduce Theorem 1.3 by taking D = ∆ ∗ , λ = b ∆ / ∆ ∗ and b = 0.2. Let us compare the bounds provided by Theorems 1.2 and 1.3:Theorem 1.2: b K (Ψ( G, w )) b ∆( G, w ) (6.12a)Theorem 1.3: K ∗ µ ∆ ∗ ( G, w ) (6.12b)where µ = b ∆( G, w ) / ∆ ∗ ( G, w ) ∈ (0 , K ∗ µ µ b K (Ψ( G, w )) = F µ (1) µ F (Ψ( G, w ) − / ) . (6.13)Now, it is not difficult to see that ∆ ∗ ( G, w ) ≤ b ∆( G, w ) Ψ( G, w ) / , or in other wordsΨ( G, w ) − / ≤ µ . Since F ( β ) is a decreasing function of β (see Proposition A.1(a)in the Appendix), we have F (Ψ( G, w ) − / ) ≥ F ( µ ) and henceTheorem 1.3Theorem 1.2 ≤ F µ (1) µ F ( µ ) ≡ g ( µ ) . (6.14) Proof. For each edge e = xy we havemin (cid:26) | w e | , | w e || w e | / (cid:27) = min (cid:26) | w e | , | w e || w e | (cid:27) × max { , | w e |} / ≤ min (cid:26) | w e | , | w e || w e | (cid:27) × Ψ( G, w ) / . Multiplying this by Q f ∋ y max { , | w f |} / , summing over e ∋ x , and taking the maximum over x ∈ V , we obtain the desired inequality. F µ (1) and µ F ( µ ) are increasing functions of µ [see Proposition A.1(a,b)], buttheir ratio g ( µ ) does not have any obvious monotonicity. Numerically we find that g ( µ ) decreases from the value K ∗ / ≈ . µ = 0 to a minimum value ≈ . µ ≈ . µ → ∞ . We have not succeeded in proving that g ( µ ) ≤ g (0) for µ ∈ [0 , ≈ . g ( µ ) ≤ F (1)lim µ → µ F ( µ ) = K ∗ ≈ . µ ∈ [0 , . (6.15)We shall see in Examples 7.1 and 7.2 that Theorem 1.3 can indeed be up to a factor ≈ . T H ( | e w [ x ] | ) T H ( | w ′ | ) Q e ∈ E ( x ) max { , | w e |} / (6.16)is the product of a a “good” factor Q e ∈ E ( x ) max { , | w e |} − / and a “bad” factor T H ( | e w [ x ] | ) /T H ( w ′ ). Now, the “bad” factor T H ( | e w [ x ] | ) /T H ( | w ′ | ) is always bounded by Q e ∈ E ( x ) max { , | w e |} / — which is why (4.10b) is always better than (4.6b) —so it follows that P S ∋ x, | S | = n T G [ S ] ( | e w [ x ] | ) P S ∋ x, | S | = n T G [ S ] ( | w ′ | ) ≤ Y e ∈ E ( x ) max { , | w e |} / ≤ Ψ( G, w ) / . (6.17)But there is no guarantee that the upper bounds on the numerator and denominatorof (6.17), obtained by applying respectively the bounds (6.6) and (6.8), will also havea ratio ≤ Ψ( G, w ) / . Indeed, it can happen that this fails (see Examples 7.1 and7.2).It is, nevertheless, somewhat disconcerting that Theorem 1.3 is not always betterthan Theorem 1.2. It would be nice to find a single natural bound that simultaneouslyimproves Theorems 1.2 and 1.3. (cid:3) Finally, let us prove Lemma 1.4 concerning the behavior of Ψ( G, w ) and b ∆( G, w )under parallel reduction: Proof of Lemma 1.4. Inequality (1.11) follows immediately from the fact that(1 + w )(1 + w ) = 1 + w . To prove (1.12), let us consider the following cases:21 ase 1 : | w | ≤ | w | ≤ 1. Then min n | w i | , | w i || w i | o = | w i | for 1 ≤ i ≤ | w | ≤ | w | + | w | . Since w = w + w + w w , we have | w | = | w + w + w w | = | w + w (1 + w ) | ≤ | w | + | w (1 + w ) | = | w | + | w | | w | ≤ | w | + | w | (6.18)since | w | ≤ Case 2 : | w | ≥ | w | ≥ 1. Then min n | w i | , | w i || w i | o = | w i || w i | for1 ≤ i ≤ 3. Let w ′ i = − w i w i for 1 ≤ i ≤ 3, so that 1 + w ′ i = (1 + w i ) − for 1 ≤ i ≤ w ′ )(1 + w ′ ) = 1 + w ′ . Since | w ′ | ≤ | w ′ | ≤ 1, we mayapply Case 1 to w ′ , w ′ , w ′ to deduce that | w ′ | ≤ | w ′ | + | w ′ | , as required. Case 3 : | w | ≤ | w | ≥ | w | | w | ≤ 1. Then min n | w i | , | w i || w i | o = | w i | for i ∈ { , } , and min n | w | , | w || w | o = | w || w | . By hypothesis we have | w | ≤| w | − . Hence | w | = | w + w (1 + w ) | ≤ | w | + | w | | w | ≤ | w | + | w || w | , (6.19)as required. Case 4 : | w | ≤ | w | ≥ | w | | w | ≥ 1. Then min n | w | , | w || w | o = | w | , and min n | w i | , | w i || w i | o = | w i || w i | for i ∈ { , } . Let w ′ i = − w i w i for 1 ≤ i ≤ | w ′ | ≥ | w ′ | ≤ | w ′ | | w ′ | ≤ 1, so we may apply Case 3(with indices 1 and 2 interchanged) to deduce that | w ′ | ≤ | w ′ || w ′ | + | w ′ | = | w | + | w ′ | ,as required. (cid:3) Remark. We suspect that the transformation w ′ = − w w (6.20)employed in Cases 2 and 4, which satisfies (1 + w ′ ) = (1 + w ) − and hence preservesthe parallel-connection law (1 + w )(1 + w ) = 1 + w , may have other applications inthe study of the multivariate Tutte polynomial. This transformation is involutive [i.e.( w ′ ) ′ = w ], maps the complex antiferromagnetic regime | w | ≤ | w ′ | ≥ − ≤ w ≤ ≤ w ′ ≤ + ∞ and vice versa.In the physicists’ notation w = e J − J is the Potts-model coupling, thetransformation (6.20) takes the simple form J ′ = − J , which makes its propertiesobvious. 22 Examples In this section we examine some examples that shed light on the extent to whichTheorems 1.2 and 1.3 are sharp or non-sharp. For each weighted graph ( G, w ), weattempt to compute or estimate the quantity Q max ( G, w ) = max {| q | : Z G ( q, w ) = 0 } (7.1)and compare it to the upper bounds given by Theorem 1.2 and Theorem 1.3. In whatfollows we abbreviate b ∆( G, w ), ∆ ∗ ( G, w ), Ψ( G, w ), Q max ( G, w ) by b ∆, ∆ ∗ , Ψ, Q max . Example 7.1 Let G = K , where the single edge has weight w . Then Z K ( q, w ) = q ( q + w ), so that Q max = | w | . On the other hand, if | w | ≥ b ∆ = | w | / | w | / , ∆ ∗ = | w | , Ψ = | w | and µ = b ∆ / ∆ ∗ = 1 / | w | / . Theorem 1.2gives the bound | q | < b K (Ψ) b ∆, which behaves like 4 | w | as | w | → ∞ , while Theorem 1.3gives the bound | q | < K ∗ µ ∆ ∗ , which behaves like K ∗ | w | ≈ . | w | as | w | → ∞ .So Theorem 1.2 is off by a factor of 4 from the truth, while Theorem 1.3 is off bya factor of ≈ . K ∗ / ≈ . G = K , the convergence conditions (6.7) and (6.9), whichwere used in the proofs of Theorems 1.2 and 1.3, respectively, becomeinf α> ( e α − − e α | q | − b ∆( G, w ) Ψ( G, w ) / ≤ α> ( e α − − e α | q | − ∆ ∗ ( G, w ) ≤ K has size n = 2. Since b ∆( G, w ) Ψ( G, w ) / =∆ ∗ ( G, w ) = | w | , we have(7.2) ⇐⇒ (7.3) ⇐⇒ | q | ≥ | w | , (7.4)which differs from the truth Q max = | w | by a factor of 4. We can understand thisbehavior as follows:1) The lost factor of 4 comes from the fact that, for a polymer gas consistingof a single polymer S of cardinality | S | = 2, the Gruber–Kunz–Fern´andez–Procaccicondition (Proposition 3.1) gives Ξ = 0 whenever | ρ ( S ) | ≤ / 4, whereas the truth isthat Ξ = 0 whenever | ρ ( S ) | < P ∞ n =2 , the terms for n > | w | → ∞ because | q | − ( n − b ∆( G, w ) n − Ψ( G, w ) / = ( | w | / | q | ) n − | w | − ( n − / , (7.5)which tends to zero as | w | → ∞ whenever | q | ≥ const × | w | and n > 2. That is whyTheorem 1.2 is off from the truth by the same factor 4 that we see in (7.4), despite23he fact that its proof allows for arbitrarily large polymers that do not occur when G = K .3) By contrast, in the convergence condition (6.9), the terms with n > not disappear in the limit | w | → ∞ with | q | of order | w | , because[ | q | − ∆ ∗ ( G, w )] n − = ( | w | / | q | ) n − (7.6)is of order 1 for all n . This is why Theorem 1.3 is off from the truth by more thanthe factor 4 that we see in (7.4); we lose an additional factor K ∗ / ≈ . (cid:3) Example 7.2 In any simple graph G with at least one edge, we can choose weights w such that Theorem 1.2 beats Theorem 1.3 by a factor arbitrarily close to K ∗ / ≈ . w e = w (with | w | ≥ 1) on all the edges of a nonemptymatching, and w e = w on all other edges; then as w → Q max → | w | , b ∆ → | w | / | w | / , ∆ ∗ → | w | , Ψ → | w | and µ = b ∆ / ∆ ∗ → / | w | / . Sothe comparison of the bounds is the same as for G = K , and Theorem 1.2 beatsTheorem 1.3 by a factor tending to K ∗ / ≈ . | w | → ∞ .For instance, let G be the n -cycle C n with n ≥ 3, taking w e = w for exactly oneedge and w e = w for all other edges. Then Z G ( q, w ) = ( q + w )( q + w ) n − + ww n − ( q − | w | → ∞ at fixed n and w , we have Q max ( G, w ) = | w | + o ( | w | ). On the otherhand, if | w | ≥ | w | ≫ | w | we have b ∆( G, w ) = | w | + | w | / | w | / | w | / ,∆ ∗ ( G, w ) = | w || w | / + | w | / | w | and Ψ( G, w ) = | w | | w | . Therefore,as | w | → ∞ the bounds of Theorems 1.2 and 1.3 are 4 | w || w | + O ( | w | / ) and K ∗ | w | / | w | + O (1), respectively, where K ∗ ≈ . | w | → ∞ at fixed n and w , but are off bya constant factor (4 | w | or K ∗ | w | / , respectively). The bound given byTheorem 1.2 is better than that given by Theorem 1.3 when | w | is small, andworse when | w | is large. (cid:3) Example 7.3 Let G = K ( k )2 (a pair of vertices connected by k parallel edges) with w e = w for all e . Then Z G ( q, w ) = q [ q + (1 + w ) k − Q max ( G, w ) = | (1 + w ) k − | .Now, if | w | ≥ b ∆( G, w ) = k | w || w | k − and Ψ( G, w ) = | w | k .Therefore, as | w | → ∞ at fixed k , the bound of Theorem 1.2 is a factor 4 k from beingsharp.On the other hand, we may first apply parallel reduction to yield a simple graph b G = K with weight b w = (1 + w ) k − b G, b w ). The resulting bound is then (as | w | → ∞ ) a factor 4 or ≈ . (cid:3) xample 7.4 Let G be the n -cycle C n (which is simple for n ≥ w e = w for all e . Then Z G ( q, w ) = ( q + w ) n + ( q − w n . As | w | → ∞ at fixed n , wehave Q max ( G, w ) = | w | n/ ( n − + O ( | w | ). On the other hand, if | w | ≥ b ∆ = 2 | w | , ∆ ∗ = 2 | w | | w | / and Ψ = | w | . Therefore, as | w | → ∞ the boundsof Theorems 1.2 and 1.3 are 8 | w | + O ( | w | ) and 2 K ∗ | w | / + O ( | w | ), respectively(here 2 K ∗ ≈ . | w | → ∞ at fixed n ≥ 4, but the bound given by Theorem 1.3 is a significantimprovement over that given by Theorem 1.2. (cid:3) Example 7.5 Let G be the complete graph K n . Take w e = w > e , with w fixed independent of n (unlike the usual [8] scaling w = λ/n ). Then Janson [18] hasvery recently proven thatlim n →∞ n log Z K n ( e αn , w ) = max[ log(1 + w ) , α ] for α ≥ . (7.7)[This is because the sum (1.1) is dominated by two contributions: the terms with( V, A ) connected, which together contribute e αn (1 + w )( n )[1 + o (1)], and the term A = ∅ , which contributes e αn .] It then follows from the Yang–Lee [36] theory of phasetransitions (see e.g. [29, Theorem 3.1]) that Z K n ( e αn , w ) must have complex roots α n that converge to α ⋆ = log(1 + w ) as n → ∞ . Hence Q max ( K n , w ) ≥ (1 + w ) n/ o ( n ) (and this is presumably the actual order of magnitude). On the other hand, we have∆ ∗ ( K n , w ) = ( n − w (1 + w ) n/ − , so that the upper bound given by Theorem 1.3is nearly sharp when n → ∞ at fixed w > e o ( n ) even though both the truth and the bound are growing exponentially in n ].By contrast, b ∆ = ( n − w (1 + w ) ( n − / and Ψ = (1 + w ) n − , so the bound ofTheorem 1.2 is much worse because of its growth as (1+ w ) n − rather than (1+ w ) n/ − . (cid:3) Example 7.6 Let G be a large finite piece of the simple hypercubic lattice Z d (forsome fixed d ≥ 2) with nearest-neighbor edges, and take w e = w > e .For real q > w t lies at w t ( q ) = q /d + O (1) . (7.8)It then follows from the Yang–Lee [36] theory of phase transitions that there will becomplex zeros of the partition function arbitrarily close (as G grows) to the phase-transition point ( q, w t ( q )); so as w ↑ ∞ (for fixed d ≥ 2) we will have asymptotically Q max ( G, w ) ≥ w d [1+ O (1 /w )] (and this is presumably the actual order of magnitude).Since ∆ ∗ ( G, w ) = 2 dw (1 + w ) d − / , the upper bound given by Theorem 1.3 is off byat most a factor of order w / (i.e. it grows as w d +1 / instead of w d ). By contrast, b ∆ = 2 dw (1 + w ) d − and Ψ = (1 + w ) d , so the bound of Theorem 1.2 is again muchworse, because it grows as w d rather than w d +1 / . (cid:3) xample 7.7 Let G be a disjoint union G = G ⊎ G . Then Q max ( G ) = max { Q max ( G ) ,Q max ( G ) } , b ∆( G ) = max { b ∆( G ) , b ∆( G ) } and Ψ( G ) = max { Ψ( G ) , Ψ( G ) } . But theproduct b K (Ψ) b ∆ for G can exceed the maximum of those for G and G because onefactor could be maximized for G and the other for G . For instance, for i = 1 , G i be an r i -regular graph with all edge weights equal to w i , where | w i | ≥ 1. Then b ∆( G i ) = r i | w i | | w i | r i / − (7.9a)Ψ( G i ) = | w i | r i (7.9b)Now choose (for instance) r = ρ ≫ r = 3, w = 1, w ≫ 1. Then b ∆( G ) b ∆( G ) = ρ ρ/ − w (1 + w ) / ≈ ρ ρ/ w / (7.10)while Ψ( G )Ψ( G ) = (1 + w ) ρ ≈ w ρ . (7.11)So if we choose ρ ρ ≫ w ≫ ρ (7.12)we will have b ∆( G ) ≫ b ∆( G ) but Ψ( G ) ≫ Ψ( G ). (cid:3) A Appendix: Proof of Lemma 6.1 and related facts In this appendix we prove Lemma 6.1. Actually, we prove much more: thoughonly parts (e,f,h) of Proposition A.1 below actually arise in Lemma 6.1 and hencein the proofs of Theorems 1.2 and 1.3, we think it worthwhile to collect here someadditional properties of the function F λ ( β ) defined by (A.1). Some of these propertieswill be invoked in the Discussion after the proof of Theorem 1.3, while others mayend up playing a role in future work. Proposition A.1 For λ ≥ and β > , define the function F λ ( β ) = min ( L : inf α> ( e α − − ∞ X n =2 e αn L − ( n − [1 + ( n − λ ] n − ( n − ≤ β ) . (A.1) Then:(a) F λ ( β ) is an increasing function of λ and a decreasing function of β .(b) βF λ ( β ) is an increasing function of both λ and β .(c) F λ ( µ/λ ) /λ is a decreasing function of both λ and µ ( > . In particular, F λ ( β ) /λ is a decreasing function of both λ and β . d) log F λ ( β ) is a convex function of log β .(e) We have F λ ( β ) = min 1) (A.9)27or i = 1 , 2. Now let κ ∈ [0 , 1] and define¯ α = κα + (1 − κ ) α (A.10a)¯ L = L κ L − κ (A.10b)¯ β = β κ β − κ (A.10c)Then H¨older’s inequality with p = 1 /κ and q = 1 / (1 − κ ) yields ∞ X n =2 e ¯ αn ¯ L − ( n − [1 + ( n − λ ] n − ( n − ≤ ¯ β ( e α − κ ( e α − − κ . (A.11)And since the function α log( e α − 1) is concave on (0 , ∞ ), we have ( e α − κ ( e α − − κ ≤ e ¯ α − 1. This proves (d).(e) The proof that (A.1) is equivalent to (A.2) will be modelled on an argumentof Borgs [9, eq. (4.22) ff.], who proved a related result.Note first that c ce − c maps the interval [0 , 1] strictly monotonically onto theinterval [0 , /e ]; and recall [32, p. 28] that its inverse map is the tree function T ( x ) = ∞ X n =1 n n − n ! x n , (A.12)which is convergent and monotonically increasing for 0 ≤ x ≤ /e and satisfies T ( ce − c ) = c for 0 ≤ c ≤ 1. Moreover, it is well known (see e.g. [11, eq. (2.36)]) thatfor all real κ > (cid:18) T ( z ) z (cid:19) κ = ∞ X m =0 κ ( m + κ ) m − m ! z m (A.13)(this is an easy consequence of the Lagrange inversion formula). Writing for conve-nience U ( z ) = T ( z ) /z , we therefore have ∞ X n =1 [1 + ( n − λ ] n − ( n − z n = z U ( λz ) /λ (A.14)for all real λ > λe α /L ≤ /e (otherwise the sum would be divergent) and e α U ( λe α /L ) /λ − e α ≤ β ( e α − . (A.15)Eliminating L in favor of a new variable c defined by λe α /L = ce − c with 0 ≤ c ≤ U ( ce − c ) = e c , we see that the inequality on the right-handside of (A.1) is equivalent to c ≤ min n , λ log[1 + β (1 − e − α )] o . (A.16)28ince L = λe α / ( ce − c ), and ce − c increases monotonically with c for 0 ≤ c ≤ 1, wededuce that (A.16) is equivalent to L ≥ e α [1 + β (1 − e − α )] λ log[1 + β (1 − e − α )] if β (1 − e − α ) ≤ e /λ − λe α +1 if β (1 − e − α ) ≥ e /λ − α to y = 1 + β (1 − e − α ), we can rewrite this as L ≥ βy λ (1 + β − y ) log y if 1 < y < min( e /λ , β ) λβe β − y if e /λ ≤ y < β (A.18)Now we can optimize over y : the minimum will always be found in the interval1 < y ≤ e /λ , so we have F λ ( β ) = min 0; andthe case λ = 0 follows by taking limits (or by an easy direct proof).(f) For λ = 0, simple calculus shows that the minimum in (A.2) is attained at y = (1 + β ) /W ((1 + β ) e ), so that F ( β ) is given by (A.3). Likewise, for λ = 1, simplecalculus shows that the minimum in (A.2) is attained at y = (1 + β ) W ( e/ (1 + β )),so that F ( β ) is given by (A.4).(g) To prove the comparison inequality (A.5), it suffices to observe that whenever0 ≤ λ ≤ λ ′ and n ≥ (cid:18) n − λ n − λ ′ (cid:19) n − ≤ (cid:18) λ λ ′ (cid:19) n − (A.20)(just consider n = 2 and n ≥ y = 1 + x in (A.2) and use the inequal-ities 1log(1 + x ) ≤ x + 12 (A.21)(1 + x ) λ ≤ λx (A.22)29hich are valid for all x > ≤ λ ≤ Therefore, βy λ (1 + β − y ) log y ≤ β (1 + λx ) (cid:0) x + (cid:1) β − x . (A.23)The latter function is minimized at x = ( − p λ ) β + 2 λβ ) / [1+(2+ β ) λ ] ∈ (0 , β ), with minimum value12 + λ + 2 β + 2 β p (1 + β/ λβ ) . (A.24)This, in turn, is bounded above by 4 β − + (1 + 2 λ ) on the entire interval 0 < β < ∞ . [Alternatively, it suffices to make this proof for λ = 1 and then invoke (A.5) to deducethe result for 0 ≤ λ < (cid:3) Remarks. 1. The proof of Proposition A.1(e) becomes a bit simpler for β ≤ e /λ − 1, since we then always have β (1 − e − α ) ≤ e /λ − λ ≤ β ≤ 1, which covers what is needed in the proofs of bothTheorem 1.2 ( λ = 1, β = ψ − / ≤ 1) and Theorem 1.3 (0 < λ ≤ β = 1).2. We can compute the small- β asymptotics of F λ ( β ) by expanding (A.2) in powersof y − 1: the minimum is located at y = 1 + 12 β − λ β + 5 + 12 λ β − 43 + 122 λ + 12 λ − λ β + . . . (A.25)and we have F λ ( β ) = 4 β − + (1 + 2 λ ) − λ − λ β + 11 + 26 λ − λ − λ β + . . . . (A.26)For λ = 0 , F ( β ) = 4 β − + 1 − β + 11192 β − β + 60736864 β − . . . (A.27) F ( β ) = 4 β − + 3 − β + 17192 β − β + 8113184320 β − . . . (A.28) Proof of (A.21): Write t = log(1 + x ) > 0; then (A.21) states that 1 /t ≤ / ( e t − 1) + 1 / t ≥ 2; and for 0 < t < e t − ≤ t/ (1 − t/ Proof: We have p (1 + c β )(1 + c β ) ≤ c + c β for all c , c , β ≥ 0, as is easily seen by squaring both sides and using the arithmetic-geometric-meaninequality √ c c ≤ ( c + c ) / ψ asymptotics of b K ( ψ ) = F ( ψ − / ) is b K ( ψ ) = 4 ψ / + 3 − ψ − / + 17192 ψ − − ψ − / + 8113184320 ψ − − . . . . (A.29)3. In the preprint version of this paper , we conjectured (based on plots of F and its derivatives) that F ( β ) is a completely monotone function of β on (0 , ∞ ),i.e. ( − k d k F ( β ) /dβ k ≥ β > k ≥ 0, and indeed that G ( β ) = F ( β ) − /β is completely monotone, which is stronger. Even more strongly,we conjectured (based on computations for Im β > 0) that G λ ( β ) = F λ ( β ) − /β is a Stieltjes function for λ = 0 and λ = 1, i.e. it can be written in the form f ( β ) = C + Z [0 , ∞ ) dρ ( t ) β + t (A.30)where C ≥ ρ is a positive measure on [0 , ∞ ). This latter conjecture has nowbeen proven by Kalugin, Jeffrey and Corless [19]. It is even possible that G λ is aStieltjes function also for 0 < λ < 1, but a different method of proof will be needed. (cid:3) Acknowledgments We are extremely grateful to Svante Janson for answering our query about thebehavior of Z K n ( q, w ) when w > n [cf. (7.7)]. We are alsoindebted to an anonymous referee, whose incisive comments on a previous version ofour paper led us to obtain some notable improvements.We wish to thank the Isaac Newton Institute for Mathematical Sciences, Univer-sity of Cambridge, for generous support during the programme on Combinatorics andStatistical Mechanics (January–June 2008), where this work was begun. One of us(A.D.S.) also thanks the Institut Henri Poincar´e – Centre Emile Borel for hospitalityduring the programmes on Interacting Particle Systems, Statistical Mechanics andProbability Theory (September–December 2008) and Statistical Physics, Combina-torics and Probability (September–December 2009), and the Laboratoire de PhysiqueTh´eorique at the ´Ecole Normale Sup´erieure for hospitality during April–June 2011. http://arxiv.org/abs/0810.4703v2 See e.g. [33] for the theory of completely monotone functions on (0 , ∞ ) — in particular theBernstein–Hausdorff–Widder theorem, which states that a function is completely monotone on (0 , ∞ )if and only if it is the Laplace transform of a positive measure supported on [0 , ∞ ). More information on Stieltjes functions can be found in [33] [1, pp. 126–128] [2, 3, 4, 31] and thereferences cited therein. 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