Decay of correlations and zeros for the hard-core model

DECA Y OF CORRELA TIONS AND ZER OS F OR THE HARD-CORE MODEL HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS Abstract. In a recen t pap er the last author pro ved that absence of complex zeros of the partition function of the hard-core model near a parameter λ > 0 implies a form of correlation deca y called strong spacial mixing. In this paper we inv estigate the rev erse implication. W e introduce a strengthening of strong spatial mixing that we call very str ong sp atial mixing (VSSM). Our main result is that if VSSM holds at a parameter λ > 0 for a family of graphs, this implies that the partition function has no zeros near that parameter for eac h graph in the family . W e also demonstrate that a closely related v ariant of very strong spatial mixing do es not imply zero-freeness. As a consequence of our main result, w e moreov er obtain that VSSM implies sp ectral independence. Our pro of relies on transforming the problem to the analysis of an induced non- autonomous dynamical system given by M¨ obius transformations. 1. Introduction The har d-c or e mo del at fugacity λ > 0 on a finite graph G = ( V , E ) is the probabilit y measure on the collection of indep endent sets I G of G (i.e. I G := { I ⊆ V | for all u, v ∈ I : { u, v } / ∈ E } ) defined by µ G,λ ( I ) := λ | I | P J ∈I G λ | J | , where w e often just write µ instead of µ G,λ in case λ and G are clear from the con text. Here Z G ( λ ) := X J ∈I G λ | J | , is the p artition function of the hard-core mo del, also known as the indep endence p olyno- mial of G . The hard-core mo del and its partition function hav e b een studied and used from a v ariety of angles. In statistical ph ysics, the hard-core mo del or hard-core lattice gas mo del, is a discrete mo del for the hard-sphere mo del, in whic h it is assumed that each atom of a gas takes up a fixed space in a given v olume, a c or e , which cannot intersect the core of any other atom. This is for example used to mo del adsorption of molecules on to a crystal surface, see e.g. [ TWP + 85 ]. In extremal combinatorics the hard-core mo del has b een used to study regular graphs with the most indep enden t sets [ DJPR17 , DJPR18 ] and is related to the Lov´ asz Lo cal Lemma [ SS05 ]. The question of the existence of efficien t algorithms to (appro ximately) compute Z G ( λ ) has received m uc h attention in theoretical computer science [ W ei06 , SS14 , GvV16 , SS ˇ SY17 , Bar16 , PR17 , BGG ˇ S20 , dBBG + 24 ]. Tw o of the main approaches in the design of efficient deterministic 1 algorithms for appro ximating the partition function Z G ( λ ) are based on seemingly distinct prop erties of the hard-core mo del and connecting these forms the main motiv ation for the presen t pap er. 1 There are also efficient randomized algorithms based on Mark ov chains, see Subsection 1.2 of the presen t pap er. 1 2 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS One is the correlation decay approach pioneered b y W eitz [ W ei06 ] ab out t w ent y y ears ago, and allo ws to design efficien t algorithms pro vided the model satisfies a strong form of correlation decay , whic h roughly says that the influence on a giv en vertex of v ertices far a wa y in the graphs decays exp onentially fast. W e will mak e this precise b elo w. The other approac h is the in terp olation metho d of Barvinok [ Bar16 ] com bined with an algorithm due to Patel and the second author [ PR17 ]. This approach yields efficient algorithms pro vided the complex zeros of the p olynomial Z G ( x ) do not come arbitrarily close to the in terv al [0 , λ ]. Notably b oth approaches are inspired by statistical physics and op erate in the regime where the mo del does not exhibit phase transitions, in the sense of uniqueness of the Gibbs measure for the first approach and in the Lee-Y ang [ YL52 ] sense for the second approach. Additionally , they yield the same algorithmic results for the family G ∆ of all graphs of a given maximum degree ∆ ∈ N . More precisely , assuming ∆ ≥ 3, for 0 ≤ λ < λ c (∆), where λ c (∆) := (∆ − 1) ∆ − 1 (∆ − 2) ∆ , b oth approac hes yield a fully p olynomial time approximation algorithm to compute a relativ e ε -approximation to Z G ( λ ) for G ∈ G ∆ while for for λ > λ c (∆) the problem of even computing a relative 2-appro ximation to Z G ( λ ) is NP -hard [ SS14 , GvV16 ]. It is therefore natural question to ask how the underlying prop erties of correlation de- ca y and absence of complex zeros relate. F or certain structured families of graphs such as lattices this question has b een answered in a v ery strong sense by Dobrushin and Sc hlossman [ DS85 , DS87 ] around fort y y ears ago. They pro vided more than a dozen prop- erties of the mo del and show ed that they are equiv alen t at a giv en λ > 0. Moreo ver, their work is not restricted to the hard-core mo del, but applies to many local interaction mo dels. Ho wev er, their pro of crucially relied on the structural asp ect of lattices that in these graphs the collection of vertices at distance r of a giv en vertex only grows p olyno- mially in r . Since for a t ypical graph in G ∆ this growth is exp onen tial, the question of the connection b et ween absence of zeros and correlation decay for the hard-core mo del is not answ ered by the work of Dobrushin and Schlossman [ DS85 , DS87 ]. The broader question of the connection b etw een correlation decay and absence of zeros on more general families of graphs (of b ounded degree) has gained interest in the past few y ears. F or example, Liu, Sinclair, and Sriv astav a [ LSS25 ] recognized that a successful approac h for proving decay of correlation for mo dels such as the hard-core mo del at a parameter could also b e used to sho w absence of zeros of the corresp onding partition function near the parameter. Recen tly , the second author [ Reg23 ], impro ving on work of Gamarnik [ Gam23 ], show ed that absence of complex zeros near an interv al of the form [0 , λ ] implies a strong form of correlation deca y , called str ong sp atial mixing , for parameters in that interv al (for an y given family of b ounded degree graphs). In the presen t pap er, w e contribute to this line of research b y introducing a strengthening of the notion of strong spatial mixing that w e coin very str ong sp atial mixing (VSSM) and sho w that this prop erty implies zero-freeness of the partition function near the parameter λ . Informally , a family of graphs that is closed under taking induced subgraphs satisfies VSSM at λ > 0 if the family of self-a v oiding walk trees of the graphs in the family satisfies w eak spatial mixing at λ . See Definition 4 b elo w for the precise definition. W e note that for λ > 0 strong spatial mixing on the infinite ∆-regular tree is equiv alent to VSSM on for the class of graphs of maxim um degree at most ∆, linking our definition of VSSM to the more standard notion of strong spatial mixing. Our main result is the follo wing: Main Theorem. L et G b e a family of b ounde d de gr e e gr aphs close d under taking induc e d sub gr aphs and let λ ⋆ > 0 . If the har d-c or e mo del on G satisfies VSSM at λ ⋆ , then ther e exists an op en set U ⊆ C c ontaining λ ⋆ such that Z G ( λ )  = 0 for al l λ ∈ U and any gr aph G ∈ G . DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 3 W e note that for the family of graphs of maximum degree at most ∆ our main res- ult combined with the result from [ Reg23 ] provides a clear explanation as to why the algorithmic approac hes of W eitz and Barvinok essentially yield the same results. (In fact, in [ BR25 ] it w as observed that also the underlying algorithms are nearly the same.) In the following Section 1.1 , we formally in tro duce the notion of very strong spatial mixing, state a multi-v ariate version of the main result, and discuss results demonstrating that the main result no longer holds under certain relaxations of the very strong spatial mixing assumption. In Section 1.2 , w e deriv e from our main result that v ery strong spatial mixing implies sp ectral indep endence, a prop ert y relev ant in the analysis of the mixing time of the Glaub er dynamics for the hard-core mo del. Finally , in Section 1.3 we giv e a rough outline of our pro of of the main result and an o verview of the remainder of the pap er. 1.1. Definitions and main result. W e start b y in tro ducing some standard terminology . F or a graph G = ( V , E ) and a vertex v 0 ∈ V w e hav e Z v 0 in G ( λ ) : = X I ∈I G v 0 ∈ I λ | I | = λZ G \ N ( v 0 ) ( λ ) and Z v 0 out G ( λ ) : = X I ∈I G v 0 / ∈ I λ | I | = Z G − v 0 ( λ ) , where N ( v 0 ) denotes the set of v 0 and all its neighbours, G \ V ′ denotes the induced subgraph of G with vertex set V \ V ′ , and G − v the induced subgraph with vertex set V \ { v } . This allows to write Z G ( λ ) = λZ G \ N ( v 0 ) ( λ ) + Z G − v 0 ( λ ) . (1) W e define the rational function R G,v 0 ( λ ) : = Z v 0 in G ( λ ) Z v 0 out G ( λ ) and refer to this as the o c cup ation r atio at v 0 (w e will often write ratio for short). W e note that in case λ > 0 we can interpret the ratio as R G,v 0 ( λ ) = P µ [ v 0 ∈ I ] P µ [ v 0 / ∈ I ] . (2) Let G = ( V , E ) b e a graph. A b oundary c ondition on G for the hard-core mo del is a map σ : V σ → { 0 , 1 } on a subset V σ ⊂ V such that V σ, in := σ − 1 (1) ⊆ V is an indep endent set of G . W e mostly w ork with b oundary conditions at a fixed distance of a given vertex v . T o this end let N denote the set of positive integers { 1 , 2 , . . . } . F or a graph G = ( V , E ), a vertex v ∈ V , and ℓ ∈ N , let S G ( v , ℓ ) denote the collection of all vertices of G at distance ℓ from v in G . F or a b oundary condition σ on G , we define the conditional partition function Z ( G,σ ) ( λ ) as Z ( G,σ ) ( λ ) := X I ⊆ V \ V σ I ∪ V σ, in ∈I G λ | I | . F or v / ∈ V σ w e subsequently define the conditional ratio R ( G,σ ) ,v ( λ ) : = Z v in ( G,σ ) ( λ ) Z v out ( G,σ ) ( λ ) , 4 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS where Z v in ( G,σ ) ( λ ) = X v ∈ I ⊆ V \ V σ I ∪ V σ, in ∈I G λ | I | and Z v out ( G,σ ) ( λ ) = X v / ∈ I ⊆ V \ V σ I ∪ V σ, in ∈I G λ | I | , and note that as in ( 2 ) for λ > 0 we ha ve R ( G,σ ) ,v ( λ ) = P µ [ v 0 ∈ I | I ∼ σ ] P µ [ v 0 / ∈ I | I ∼ σ ] , (3) where the notation I ∼ σ stands for the ev ent that the random indep enden t set I agrees with V σ, in on V σ . T o define v ery strong spatial mixing we will need the concept of the tree of self a v oiding w alks, as introduced by Scott and Sok al [ SS05 ] and W eitz [ W ei06 ]. See also Bencs [ Ben18 ]. Informally , the v ertices of this tree correspond to paths starting at a given vertex v of the graph and eac h path is connected with an edge to its single extensions. The following equiv alent and more formal definition will b e useful in later pro ofs. Definition 1 (The tree of self a v oiding w alks) . Given a graph G = ( V , E ) and a v ertex v , the tr e e of self avoiding walks is the tree T SA W( G,v ) obtained from ( G, v ) as follows. Fix a total ordering of the vertices of G . W e define the tree T SA W( G,v ) recursiv ely . If the comp onen t of G containing v consists of a single vertex v , then T SA W( G,v ) = { v } . In case the comp onen t of G containing v consists of more than 1 v ertex, let v 1 , . . . , v d denote the neigh b ours of v in G . Let for i = 1 . . . d , G i b e the graph obtained from G b y remo ving v , v 1 , . . . , v i − 1 from G . Denote for i = 1 , . . . , d , T i = T SA W( G i ,v i ) . Then T SA W( G,v ) is defined as the tree obtained from the disjoin t union of the T i adding a v ertex v connecting it to eac h of the v 1 , . . . , v d . R emark 2 . The tree of self av oiding walks as defined ab o v e is closer to Bencs’s defini- tion [ Ben18 ] of the stable path tree and is somewhat more compact than W eitz’s original definition [ W ei06 ], which inv olv es b oundary conditions. The use of b oundary conditions is necessary in case one wan ts to w ork with other mo dels than the hard-core mo del such as the Ising mo del. Bencs [ Ben18 ] moreov er gives other v arian ts of the stable path tree. Each of these w ould suffices for our purp oses as we will become clear later. F or clarity we just stick to the definition ab ov e. The next lemma indicates the usefulness of the tree of self av oiding w alks. Lemma 3 (W eitz [ W ei06 ]) . L et G b e a gr aph with vertex v . Then R G,v ( λ ) = R T SA W( G,v ) ,v ( λ ) . W e will need a slightly stronger v ersion of this result that we pro ve in Lemma 12 . W e can now define the notion of decay of correlations that we need. Definition 4. The hard-core model on a family G of b ounded degree graphs satisfies very str ong sp atial mixing (VSSM) at a parameter λ > 0 if there exist constan ts α ∈ [0 , 1) and C > 0 such that for all G = ( V , E ) ∈ G , an y induced subgraph H of G and v ∈ V ( H ), and for all b oundary c onditions σ, τ : S T SA W( H,v ) ( v , ℓ ) → { 0 , 1 } we ha v e | R ( T SA W( H,v ) ,σ,v ( λ ) − R ( T SA W( H,v ) ,τ ,v ( λ ) | ≤ C · α ℓ for all ℓ ∈ N . (4) W e refer to α as the r ate of VSSM . Some remarks are in order. R emark 5 . W e note that in [ NT07 ] the term very strong spatial mixing w as used to describ e a different notion. W e trust, how ever, that there will b e no confusion. DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 5 R emark 6 . T ypically notions of correlation deca y are form ulated in terms of the sensitivit y of P µ [ v ∈ I | I ∼ σ ] on the boundary condition σ . F or a graph of maxim um degree at most ∆ we ha v e P µ [ v 0 ∈ I ] ≥ λ λ + (1 + λ ) ∆+1 , (5) as follows b y conditioning on the vertices of distance 2 from v . This implies that if the maxim um degree ∆ of the family G is fixed in adv ance, replacing the conditional ratios in ( 4 ) by the conditional probabilities do es not affect the definition of VSSM. R emark 7 . Let us recall that the hard-core mo del on a family G of bounded degree graphs satisfies str ong sp atial mixing (SSM) at a parameter λ > 0 if there exist constants α ∈ [0 , 1) and C > 0 suc h that for all G = ( V , E ) ∈ G , an y induced subgraph H of G and v ∈ V ( H ), and for all b oundary conditions σ, τ : S H ( v , ℓ ) → { 0 , 1 } we ha v e | R ( H,σ ) ,v ( λ ) − R ( H,τ ) ,v ( λ ) | ≤ C · α ℓ for all ℓ ∈ N . (6) With the aid of Lemma 3 and Corollary 23 b elow it can b e shown that VSSM implies SSM and hence VSSM is indeed a strengthening of SSM. It is not clear whether the other implication also holds; cf. Question 2 . W e note that in case G consists only of trees, the tw o notions are clearly equiv alent since the tree of self a voiding w alks of a ro oted tree is just the tree itself. Giv en a graph G = ( V , E ) and a v ector of v ariables ( λ v ) v ∈ V w e define the multiv ariate indep endence p olynomial by Z G ( λ v ) := X I ∈I G Y v ∈ I λ v . The follo wing m ultiv ariate statement implies the main result, and is the result that w e will actually prov e. Theorem 8. L et ∆ ∈ N and let λ ⋆ > 0 . L et G b e a family of gr aphs of maximum de gr e e at most ∆ . If the har d-c or e mo del on G satisfies very str ong sp atial mixing at λ ⋆ , then ther e exists ε > 0 such that for e ach G = ( V , E ) ∈ G and  λ ∈ B ( λ ⋆ , ε ) V we have Z G (  λ )  = 0 . Giv en the results in [ Reg23 ] it is natural to ask whether a conv ers e to Theorem 8 holds. W e come back to this question in Section 6 . W e no w fo cus on a natural relaxation of the notion of VSSM. Recall that W eitz’s algorithmic approach w orks well on a family of b ounded degree graphs provided the hard-core mo del satisfies VSSM. In fact, for algorithmic purposes one can do with the following sligh tly weak er notion, which w as utilized in [ SS ˇ SY17 ]. Definition 9. Let φ : N → R + b e a function with p ositive v alues. The hard-core mo del on a family G of b ounded degree graphs satisfies very str ong sp atial mixing fr om distanc e φ ( φ -VSSM) at a parameter λ > 0 if there exist constan ts α ∈ [0 , 1) , C > 0 suc h that for all G = ( V , E ) ∈ G and an y induced subgraph H of G and v ∈ V ( H ), and for all b oundary conditions σ, τ : S T SA W( H,v ) ( v , ℓ ) → { 0 , 1 } we ha v e | R ( T SA W( H,v ) ,σ ) ,v ( λ ) − R ( T SA W( H,v ) ,τ ) ,v ( λ ) | ≤ C · α ℓ for all ℓ ≥ φ ( | V | ) . (7) F or families of graphs of b ounded maximum degree it is not hard to see that φ - VSSM with φ ( n ) = log( n ) is sufficien t for algorithmic purp oses using W eitz’ approac h. See [ SS ˇ SY17 ] for extensions to graphs of un b ounded degree under additional conditions on the so-called connectiv e constant. Our next result sho ws that VSSM at distance φ do es no longer necessarily imply zero- freeness provided φ is un b ounded. This suggests that Barvinok’s approach may not b e applicable in all settings where W eitz’s approach can b e successfully applied (suc h as [ SS ˇ SY17 ]). 6 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS Theorem 10. F or e ach incr e asing unb ounde d function φ : N → [0 , ∞ ) , e ach ∆ ∈ N ≥ 3 and λ ⋆ > λ c (∆) , ther e exists a family of gr aphs G of maximum de gr e e at most ∆ such that G satisfies VSSM at distanc e φ for every λ ∈ [0 , λ ⋆ ] and such that ther e exists a se quenc e of gr aphs ( G n ) n in G and a se quenc e ( λ n ) n in C such that Z G n ( λ n ) = 0 and λ n → λ c (∆) . 1.2. VSSM implies sp ectral indep endence. Sp ectral indep endence is an imp ortant prop ert y in the analysis of the mixing time of the Glaub er dynamics for the hard-core mo del on b ounded degree graphs. Anari, Liu and Oveis Gharan [ ALOG20 ] sho wed in a breakthrough result that sp ectral indep endence implies rapid mixing of the Glaub er dynamics and that the hard-core model on the family of all graphs of maxim um degree at most ∆ is spectrally indep endent at all fixed λ < λ c (∆), thereby matching applicability of Mark ov chain based randomized algorithms with that of the deterministic algorithms based on the approaches of W eitz [ W ei06 ] and Barvinok [ Bar16 ]. W e refer to [ CvV25 ] for a surv ey co vering recen t dev elopments and improv emen ts in this rapidly evolving direction of research. Let us p oin t out one recent in teresting developmen t. In [ L WYY25 ] it w as sho wn that a certain pro of approac h for sho wing rapid mixing of the Glaub er dynamics of the hard-core model at a parameter can b e used to sho w zero-freeness in a neighbourho o d of that parameter, thereb y further linking zero-freeness to other notions in the field. F or completeness w e will formally define the notion of sp ectral indep endence follo w- ing [ CvV25 ]. Let λ > 0. F or a graph G = ( V , E ) and distinct vertices u, v ∈ V we define the influenc e of u on v as Ψ G,λ ( u → v ) = P [ v ∈ I | u ∈ I ] − P [ v ∈ I | u / ∈ I ] . Define the V × V matrix I G,λ b y I G,λ ( u, v ) = Ψ G,λ ( u → v ) if u  = v and I G,λ ( u, v ) = 0 otherwise. The hard-core mo del at fugacity λ is called sp e ctr al ly indep endent on a family of graphs G if there exists a constant η > 0 suc h that for each G ∈ G and eac h induced subgraph H of G the largest eigenv alue 2 of I H,λ is b ounded by η . (Note that our definition slightly differs from [ CvV25 ] since we work with induced subgraphs rather than pinnings/b oundary conditions.) Anari, Liu and Oveis Gharan [ ALOG20 ] show ed that, for the family of all graphs of maxim um degree ∆, the hard-core mo del satisfies sp ectral indep endence at all fugacities λ ∈ [0 , λ c (∆)) building on W eitz’ pro of [ W ei06 ] of strong spatial mixing for hard-core mo del on the infinite tree. See also [ CL V23 ] for other usages of pro of techniques for strong spatial mixing leading to pro ofs of spectral indep endence for other models. It is therefore natural to w onder whether notions such as strong spatial mixing itself implies sp ectral indep endence. This question w as explicitly asked by ˇ Stefank ovi ˇ c in a lecture series at TU Dortm und in 2022 3 . Theorem 8 in com bination with [ CL V24 , Theorem 11] (which essentially says that m ultiv ariate zero-freeness implies sp ectral indep endence) directly implies the follo wing corollary , thereby giving a p ositive answ er to this question. Corollary 11. L et ∆ ∈ N and let λ ⋆ > 0 . L et G b e a family of gr aphs of maximum de gr e e at most ∆ . If the har d-c or e mo del on G satisfies very str ong sp atial mixing at λ ⋆ , then the har d-c or e distribution at λ ⋆ on G is sp e ctr al ly indep endent. 1.3. Pro of approach. As is well kno wn, the ratio allows to transform transfer informa- tion ab out the partition function b eing zero to the ratio taking the v alue − 1. More concretely , let G b e a graph with vertex v 0 . Then if Z G − v 0 ( λ )  = 0, then Z G ( λ ) = 0 if and only if R G,v 0 ( λ ) = − 1 . (8) (Note that the implication R G,v 0 ( λ ) = − 1 implies Z G ( λ ) = 0 also holds without the assumption that Z G − v 0 ( λ )  = 0.) 2 It can b e sho wn that all eigenv alues of I G,λ are nonnegativ e, see [ CvV25 , Section 2.1]. 3 See https://eac.cs.tu-dortmund.de/storages/eac-cs/r/summerschool-2022/Stefankovic-II.pdf DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 7 T o pro v e Theorem 8 we make use of this simple observ ation in combination with the recursiv e nature of (rooted) trees by writing the ratios as a comp osition of maps enco ding the contribution of fixed depth subtrees. W e aim to show that these tree ratio maps are con tracting for small p erturbations of λ and use this to inductiv ely sho w that the ratios sta y close to the positive real axis and hence can never reach − 1. This takes inspiration from [ LSS25 ]. T o sho w that the tree ratio maps are con tracting, w e write them as comp ositions of maps of the form z 7→ f λ ( z ) := λ 1+ z for v arying v alues of λ . This tak es inspiration from [ PR19 , dBBG + 24 ]. Here we think of these v arying v alues of λ as functions of a b oundary condition. W e use VSSM to see that the asso ciated sequences of λ for the tw o extremal boundary conditions (all leav es are either ‘in’ or ‘out’) tend to get close exp onentially fast. W e view the sequence of compositions of the maps f λ as a non-autonomous dynamical system and use a carefully chosen non-autonomous c hange of co ordinates to transform the M¨ obius transformations f λ to affine maps. W e then use the assumption of VSSM once more to sho w that the tree ratio maps map the right half plane in to a small disk. This is then com bined with the change of co ordinates to show that the tree ratio maps are indeed con tracting. T o prov e Theorem 10 w e construct a sequence of trees for which w e sho w b y hand that they satisfy φ -VSSM. T o pro v e the statemen t about the zeros, w e use the dynamical prop erties of the rational function z 7→ λ (1+ z ) ∆ − 1 as prov ed in [ PR19 ] in combination with Mon tel’s theorem along similar lines as was done in [ Buy21 , dBBG + 24 ]. In the next section, w e collect some basic prop erties of the recursiv e computation of ratios of trees and show how to write the tree ratio map as a composition of the maps f λ . In Section 3 w e study these comp ositions in more detail, describe the change of co ordinates, and pro v e sev eral consequences of this p ersp ective under VSSM-like assumptions. In Section 4 w e collect all the ingredients to prov e Theorem 8 along the lines explained ab o v e. Section 5 con tains a pro of of Theorem 10 . In Section 6 w e conclude with some questions that arise naturally and some final remarks. Ac kno wledgemen ts. This publication is part of the pro ject Phase tr ansitions, c ompu- tational c omplexity and chaotic dynamic al systems with file n umber OCENW.M.22.155 of the research programme Op en Comp etitie ENW M22-2 which is (partly) financed b y the Dutc h Researc h Council (NW O) under the grant https://www.nwo.nl/en/projects/ ocenwm22155 . Guus Regts is partially funded by the Netherlands Organisation of Scien tific Researc h (NWO): VI.Vidi.193.068. 2. Preliminaries on occup a tion ra tios on trees W e collect here some preliminaries that will b e used throughout. Most of the results that w e state here are w ell kno wn, but w e provide proofs for completeness and occasionally to b e able to expand on these pro ofs. W e will start with a slight extension of the result of W eitz [ W ei06 ] we recalled in Lemma 3 . Lemma 12. L et G = ( V , E ) b e a gr aph, let v b e a vertex of G and let ( λ u ) u ∈ V b e a ve ctor of variables. Then ther e exists a ve ctor of variables ( ˜ λ w ) w ∈ V ( T SA W( G,v ) ) with ˜ λ w ∈ { λ u | u ∈ V } for e ach vertex w of T SA W( G,v ) such that R T SA W( G,v ) ,v ( ˜ λ w ) = R G,v ( λ u ) . (9) Mor e over, for any vertex u of T SA W( G,v ) , the induc e d subtr e e ( T , u ) of T SA W( G,v ) , with those vertic es w such that u is on the unique p ath fr om w to v , is the self avoiding walk tr e e of an induc e d sub gr aph of G . 8 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS Pr o of. W e will pro ve ( 9 ) as w ell as the claim ab out the subtrees of T SA W( G,v ) b y induction on the num b er of vertices of G . Fix an ordering of the v ertices of V . W e note that if the comp onen t of G con taining v consists of one v ertex, then T SA W( G,v ) is just a s ingle vertex, w e set ˜ λ v = λ v and b oth statements clearly hold. Next, assume that v has at least one neigh b our and let v 1 , . . . , v d b e its neighbours. Recall from Definition 1 that for i = 1 . . . d , we denote by G i the induced subgraph obtained by deleting the vertices v , v 1 , . . . , v i − 1 from G . Denote for i = 1 , . . . , d , T i = T SA W( G i ,v i ) ,v i . Let ˜ λ i u b e the associated v ectors of v ariables. By definition T SA W( G,v ) is the tree obtained from the disjoint union of the T i b y adding a vertex v and connecting it to eac h of the v 1 , . . . , v d . By induction w e ha v e that for eac h i , that there exists ( ˜ λ i w ) w ∈ V ( T i ) with ˜ λ i w ∈ { λ u | u ∈ V − v } for each i such that R G i ,v i ( λ u ) = R T i ,v i ( ˜ λ i w ) (10) No w let let ˜ λ w b e defined b y ˜ λ v = λ v and b y letting ˜ λ w = ˜ λ i w in case w app ears in T i . W e claim that this implies ( 9 ). Indeed for the left-hand side of ( 10 ) we hav e, b y a telescoping argument, using ( 1 ) for each of the G i R G,v = λ v Z G \ N [ v ] ( λ u Z G − v ( λ u ) = λ v Z G d − v d ( λ u ) Z G 1 ( λ u ) = λ v d Y i =1 Z G i − v i ( λ u ) Z G i ( λ u ) = λ v Q d i =1 (1 + R G i ,v i ( λ u )) While for the right-hand side of ( 10 ) we ha v e, using that the indep endence p olynomial is m ultiplicative o v er the comp onents of T SA W( G,v ) − v , R T SA W( G,v ) ,v ( ˜ λ w ) = Z v in T SA W( G,v ) ( ˜ λ w ) Z v out T SA W( G,v ) ( ˜ λ w ) = ˜ λ v d Y i =1 Z T i − v i ( ˜ λ i w ) Z T i ( ˜ λ i w ) = λ v Q d i =1 (1 + R T i ,v i ( ˜ λ w )) , as desired. No w for the second part of the statemen t, in case the vertex u has distance 1 to v in T SA W( G,v ) , then the tree ( T , u ) is equal to one of the T i and we are done by construction. In case the distance of u to v is larger than 1, u is contained in one of the T i and the claim follo ws by induction as the tree ( T , u ) is the self av oiding walk tree of an induced subgraph of G i , and hence of an induced subgraph of G . This finishes the pro of. □ Our strategy to pro v e Theorem 8 is to sho w that the ratios av oid − 1. Lemma 12 ab ov e indicates that it suffices to show this for ratios of trees. W e therefore introduce some terminology related to ratios of trees. Consider a ro oted tree ( T , v ) and λ > 0. As seen in the pro of of Lemma 12 , the ratio R T ,v = R T ,v ( λ ) = Z v in T ( λ ) Z v out T ( λ ) can b e recursively computed as follows. Let v 1 , . . . , v d b e the neighbours of v and let T 1 , . . . , T d b e the comp onents of T − v containing v 1 , . . . , v d resp ectiv ely . Then R T ,v = λ Q d i =1 (1 + R T i ,v i ) . (11) F or n ∈ N recall that S T ( v , n ) denotes the collection of v ertices at distance n from v in T and let T n b e the subtree of T induced by the vertices of distance at most n from v . F or eac h u ∈ S T ( v , n ) let R u denote the ratio at u in the comp onen t of T \ S T ( v , n − 1) con taining u . By rep eating ( 11 ), we can express the ratio R T ,v as a rational function F T ,n : C S T ( v ,n ) → C (12) DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 9 u v w 1 w i w n − 1 P S ∗ T ( v , n ) T n,i T n Figure 1. The comp onen ts of the tree T . applied to the v ector ( R u ) u ∈ S T ( v ,n ) . Next, denote b y S ∗ T ( v , n ) the subset of vertices in S T ( v , n ) that are not leav es. W e will consider the restriction of F T ,n to C S ∗ T ( v ,n ) , noting that for each u ∈ S T ( v , n ) \ S ∗ T ( v , n ) we ha v e R u = λ , which w e consider fixed. Let f λ denote the M¨ obius transformation defined by z 7→ λ 1 + z . W e wish to express the function F T ,n as a comp osition of the M¨ obius transformations f λ with v arying v alues of λ when we fix all but one of the inputs of F T ,n . T o this end fix a vertex u ∈ S ∗ ( v , n ) and fix v alues ( r u ′ ) u ′  = u , r u ′ ∈ [0 , λ ]. Consider the function F T ,n ; u defined by z 7→ F T ,n ( z , ( r u ′ )). F or i = 1 , . . . , n let w i b e the vertex at distance i from u on the unique path P from u to v = w n . Let T n,i b e the component of T n \ ( P − w i ) con taining w i (see Figure 1 ). F or ∆ ∈ N and λ > 0 we denote ℓ ∆ ( λ ) := λ (1 + λ ) ∆ − 1 . (13) Lemma 13. L et ( T , v ) b e a r o ote d tr e e of maximum de gr e e at most ∆ ∈ N . L et n ∈ N and let u ∈ S ∗ T ( v , n ) . The function F T ,n ; u as define d ab ove c an b e written as f λ n ◦ · · · f λ 1 , wher e λ i is e qual to the r atio R T n,i ,w i of the r o ote d tr e e ( T n,i , w i ) for i = 1 , . . . , n and is c ontaine d in [ ℓ ∆ ( λ ) , λ ] . Pr o of. W e prov e this by induction on n . In case n = 1, the statement follo ws directly from ( 11 ). Indeed, denoting the neighbours of v by v 1 , . . . , v d w e ma y assume u = v 1 in whic h case w e hav e F T , 1 ( z , r v 2 , . . . , r v d ) = λ (1 + z ) Q d i =2 (1 + r v i ) = λ Q d i =2 (1+ r v i ) 1 + z = λ 1 1 + z = f λ 1 ( z ) , with, λ 1 := λ Q d i =2 (1+ r v i ) = R T 1 , 1 ,w 1 . Clearly λ 1 ∈ [ λ (1+ λ ) d − 1 , λ ] ⊆ [ ℓ ∆ ( λ ) , λ ] pro ving the base case. No w let n ≥ 2. Let v 1 , . . . , v d b e the neigh b ours of v . As ab ov e, denote b y T i the comp onen t of T − v containing v i . Assume that v 1 is on the unique path from v to u . Then by ( 11 ) we ha v e F T ,n ( z , ( r u ′ )) = 1 1 + F T 1 ,n − 1 ( z , ( r 1 u ′ )) · λ Q d i =2 (1 + F T i ,n − 1 ( r i u ′ )) = R T n,n ,w n , 10 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS where we denote by ( r i u ′ ) the vector of those r u ′ for which u ′ are in the tree T i . By induction we ha v e F T 1 ,n − 1; u ( z , ( r 1 u ′ )) = f λ n − 1 ◦ · · · ◦ f λ 1 ( z ) , with λ i ∈ [ ℓ ∆ ( λ ) , λ ]. Setting λ n := λ Q d i =2 (1 + F T i ,n − 1 ( r i u ′ )) , w e see that F T ,n ( z , ( r u ′ )) = f λ n ◦ f λ n − 1 ◦ · · · ◦ f λ 1 ( z ), as desired. Since clearly λ n ≤ λ it remains to prov e that λ n ≥ ℓ ∆ ( λ ). Again by induction we ha v e that for each i = 2 , . . . , d fixing some u i ∈ S T i ( v i , n − 1) that there exist λ i 1 , . . . , λ i n − 1 ∈ [ ℓ ∆ ( λ ) , λ ] such that F T i ,n − 1 (( r i u ′ )) = f λ i n − 1 ◦ · · · ◦ f λ i 1 ( r i u i ). In particular F T i ,n − 1 (( r i u ′ )) ≤ λ , implying that λ n ≥ λ (1+ λ ) ∆ − 1 , completing the pro of. □ W e next utilize part of this lemma to get an upper b ound on the ratios for trees. F or λ > 0 and d ∈ N denote r ∆ ( λ ) := λ 1 + ℓ ∆ ( λ ) = f λ ( ℓ ∆ ( λ )) . Lemma 14. L et λ > 0 and ∆ ∈ N . Then for any r o ote d tr e e ( T , v ) of maximum de gr e e at most ∆ , c onsisting of at le ast two vertic es, we have R T ,v ( λ ) ≤ r ∆ ( λ ) . Pr o of. Let v 1 , . . . v d b e the neigh bours of v and let T i b e the components of T − v containing the v i . Note that d ≥ 1 . Then we ha v e R T ,v ( λ ) := λ Q d i =1 (1 + R T i ,v i ( λ )) ≤ λ 1 + R T 1 ,v 1 ( λ ) ≤ λ 1 + ℓ ∆ ( λ ) , where the last inequalit y is due to Lemma 13 ab ov e. □ In the next section we will study maps of the form F ( z ) = f λ n ◦ · · · ◦ f λ 1 ( z ) developing to ols to control the b eha viour of the map F T ,n whic h will be crucial for our proof of Theorem 8 . 3. Non-autonomous affine dynamics Let λ > 0 and let ( λ n ) b e a sequence of num b ers in (0 , λ ]. Motiv ated b y Lemma 13 we study the non-autonomous dynamical system z 7→ F ( z ) = f λ n ◦ · · · ◦ f λ 1 ( z ) . (14) in this section. W e use throughout elemen tary properties of M¨ obius transformations, holo- morphic functions, and the P oincar ´ e distance that can b e found in any b o ok on geometric complex analysis, such as [ Zak21 ]. F or a starting v alue w 0 w e write ( w n ) for the orbit of w 0 under the asso ciated sequence of maps f λ i . In other w ords, w i = f λ n ( w i − 1 ) for i ≥ 1 . 3.1. A non-autonomous conjugation to affine maps. F or a sym bol ∗ ∈ { <, >, ≤ , ≥} and c ∈ R we define the half plane H ∗ c := { z ∈ b C | Re( z ) ∗ c } , where by con v en tion the p oin t ∞ is included if and only if the inequality is not strict. Lemma 15. L et λ > 0 . Given a se quenc e ( λ n ) in (0 , λ ] ther e exists a unique starting value w 0 for which the orbit ( w n ) is c ontaine d in the left half-plane H < 0 . Mor e over, the orbit ( w n ) is c ontaine d in ( − λ − 1 , − 1) . DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 11 Pr o of. Throughout w e write f n = f λ n . Observ e that the inv erse map f − 1 n is giv en by f − 1 n ( z ) = − 1 + λ n /z and maps the left half-plane into the shifted left half-plane H < − 1 . The Sch w arz-Pic k Lemma implies that each inv erse map f − 1 n defines a strict con traction with resp ect to the P oincar´ e metric on H < 0 . On compact subsets of H < 0 these con tractions are uniform ov er all λ n ∈ (0 , λ ], since each f − 1 n maps the left half-plane into H < − 1 . Consider the nested sequence of non-empt y , connected, compact subsets of the extended complex plane b C : = C ∪ {∞} : H ≤ 0 ⊃ f − 1 1 ( H ≤ 0 ) ⊃ ( f 2 ◦ f 1 ) − 1 ( H ≤ 0 ) ⊃ · · · Then the intersection \ n ∈ N ( f n ◦ · · · ◦ f 1 ) − 1 ( H ≤ 0 ) (15) is a non-empty , connected, compact subset of H ≤ 0 . Since all second preimages f − 1 n − 1 ◦ f − 1 n ( H ≤ 0 ) , n ∈ N are contained in a compact subset of H < 0 and the contraction of the maps f − 1 n is uniform o ver n , the diameter of the finite nested in tersections m ust conv erge to 0 and the in tersection consists of a single p oint w 0 , which is the only point whose entire forw ard orbit remains in H < 0 . Since the M¨ obius transformations f − 1 n ha ve real parameters, it follows that the in tersec- tion ( 15 ) is in v ariant under conjugation and therefore eac h w n is real (and hence negativ e). This implies that w n < − 1 for each n . Indeed, we ha v e w n = f − 1 n +1 ( w n +1 ) = − 1 + λ n +1 /w n +1 < − 1 using that w n +1 < 0 for all n . This in turn implies that w n ≥ − 1 − λ for each n . Indeed w n = f − 1 n +1 ( w n +1 ) = − 1 + λ n +1 /w n +1 > − 1 − λ n +1 ≥ − 1 − λ, using that 1 /w n +1 > − 1 for all n . This finishes the pro of. □ F or a sequence λ = ( λ n ) in (0 , λ ] w e write f n = f λ n for each n . Let ( w n ) = ( w n ( λ )) b e the associated orbit that sta ys in H < 0 W e note that the point w n = f n ◦ · · · ◦ f 1 ( w 0 ) is the unique p oin t whose orbit remains in H < 0 for the maps f m ◦ · · · ◦ f n +1 , with m > n . W e next study the effect of exp onentially smaller consecutiv e p erturbations of the para- meters λ n on the sequence ( w n ). Let us write ( b λ n ) (and correspondingly ( b w n )) for a second sequence of parameters. Lemma 16. Given λ > 0 , C > 0 and α ∈ (0 , 1) ther e exists C ′ > 0 such that the fol lowing holds. Given se quenc es ( λ n ) and ( b λ n ) in (0 , λ ] satisfying | λ n − b λ n | < C α n for al l n ∈ N , the asso ciate d orbits ( w n ) and ( b w n ) satisfy | w n − b w n | ≤ C ′ α n for al l n ∈ N . Pr o of. The sequence ( w n ) n w as chosen suc h that { w 0 } = \ m ∈ N ( f m ◦ · · · ◦ f 1 ) − 1 ( H < 0 ) . In particular, for the p oint − 1 ∈ H < 0 , we ha v e w 0 = lim m →∞ ( f m ◦ · · · ◦ f 1 ) − 1 ( − 1) and similarly w j = lim m →∞ ( f m ◦ · · · ◦ f j +1 ) − 1 ( − 1) = lim m →∞ w j,m , where w j,m := f − 1 j +1 ◦ · · · ◦ f − 1 m ( − 1) = f − 1 j +1 ( w j +1 ,m ) , 12 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS for m ≥ j . The analogous statemen ts hold for b w j . W e will show that there exists a constan t C 1 > 0 indep enden t of m such that d H < 0 ( w j,m , b w j,m ) < C 1 α j for all j ≤ m. (16) where d H < 0 denotes the Poincar ´ e distance on H < 0 . Before we pro ve ( 16 ) we first use it to pro ve the lemma. Note that all w j,m , b w j,m lie in the compact subset [ − λ − 1 , − 1] of H < 0 as follows by an easy induction, (cf. the pro of of Lemma 15 ). Since d H < 0 is equiv alent to the Euclidean distance on [ − λ − 1 , − 1], ( 16 ) implies that there is a constan t C ′ > 0 such that | w j − b w j | = lim m →∞ | w j,m − b w j,m | < C ′ α j for all j ∈ N pro ving the lemma. W e no w return to proving ( 16 ). Let C 1 = C 2 C α 1 − α , where C 2 > 0 is a constant suc h that d H < 0 ( ζ , b ζ ) ≤ C 2 | ζ − b ζ | for all ζ , b ζ in the compact subset [ − 1 − λ, − 1] of H < 0 . (This c hoice of C 1 will become clear later). W e will show ( 16 ) b y bac kw ards induction on j = m, m − 1 , . . . , 0, note that w m,m = b w m,m = − 1 and for j ≤ m we ha ve: d H < 0 ( w j − 1 ,m , b w j − 1 ,m ) ≤ d H < 0 ( f − 1 j ( w j,m ) , f − 1 j ( b w j,m )) + d H < 0 ( f − 1 j ( b w j,m ) , b f − 1 j ( b w j,m )) . By induction and con traction of the Poincar ´ e distance under the holomorphic map f − 1 j : H < 0 → H < 0 , we ha v e d H < 0 ( f − 1 j ( w j,m ) , f − 1 j ( b w j,m )) ≤ d H < 0 ( w j,m , b w j,m ) ≤ C 1 α j and by our c hoice of C 2 , we ha v e d H < 0 ( f − 1 j ( b w j,m ) , b f − 1 j ( b w j,m )) = d H < 0  − 1 + λ j b w j,m , − 1 + b λ j b w j,m  ≤ C 2    λ j b w j,m − b λ j b w j,m    ≤ C 2 | λ j − b λ j | < C 2 C α j , where the first inequality uses that b w j,m ≤ − 1 and the second inequality uses the assump- tion in the lemma. Combining these estimates, we obtain: d H < 0 ( w j − 1 ,m , b w j − 1 ,m ) ≤ ( C 1 α + C 2 C α ) α j − 1 = C 1 α j − 1 b y our c hoice of C 1 . This completes the pro of of ( 16 ) and hence the lemma. □ W e use the orbit ( w n ) to change co ordinates non-autonomously via: ϕ n ( z ) = ϕ w n ( z ) = 1 z − w n , and obtain a sequence of affine maps: g n = ϕ n ◦ f n ◦ ϕ − 1 n − 1 , (17) see Figure 2 for the asso ciated commutativ e diagram. Since f n maps w n to w n +1 and b C b C b C b C · · · b C b C b C b C · · · f 1 ϕ 0 f 2 ϕ 1 f 3 ϕ 2 ϕ 3 g 1 g 2 g 2 Figure 2. A commutativ e diagram displaying the relation b et w een the f n and the g n . DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 13 ϕ n ( w n ) = ∞ , the M¨ obius transformation g n maps ∞ to ∞ , and is therefore affine . Let us write down the precise formula for g n : g n ( ζ ) = ϕ n ◦ f n ◦ ϕ − 1 n − 1 ( ζ ) = ϕ n ◦ f n ( 1 ζ + w n − 1 ) = ϕ n  λ n ζ 1 + ζ w n − 1 + ζ  = 1 + (1 + w n − 1 ) ζ λ n ζ − w n (1 + (1 + w n − 1 ) ζ ) = − 1 + ( w n − 1 + 1) ζ w n , where we used the fact that g n is affine to obtain the last equality . Let G n = g n ◦ · · · ◦ g 1 . Then G ′ n ( ζ ) = ( g n ◦ · · · ◦ g 1 ) ′ ( ζ ) = ( − 1) n n Y j =1 w j − 1 + 1 w j = ( − 1) n w 0 + 1 w n · n − 1 Y j =1 w j + 1 w j . (18) F ollowing the notation from Lemma 16 we write b g n and b G n for the maps induced by the second sequence ( b λ n ). Corollary 17. Given λ + > λ − > 0 , C > 0 and α ∈ (0 , 1) ther e exists a c onstant A > 1 such that the fol lowing holds: if ( λ n ) and ( b λ n ) in [ λ − , λ + ] satisfy | λ n − b λ n | < C α n for al l n , then 1 / A ≤     G ′ n b G ′ n     ≤ A. (19) Pr o of. By ( 18 ), we kno w for all n that     G ′ n b G ′ n     = n Y j =1 w j − 1 + 1 b w j − 1 + 1 b w j w j . (20) On the compact set [ − λ + − 1 , − λ − ] 2 b oth b w w and w +1 b w +1 ha ve b ounded gradient and are equal to 1 on the diagonal, so there is a constan t C 3 > 0 such that w e hav e b w w , w + 1 b w + 1 < 1 + C 3 | w − b w | for all w , b w ∈ [ − λ + − 1 , − λ − ]. Since all w j , b w j lie in this in terv al, ( 20 ) then implies     G ′ n b G ′ n     ≤ n Y j =1 (1 + C 3 | w j − b w j | ) 2 < n Y j =1 (1 + C 3 C ′ α j ) 2 . with C ′ from Lemma 16 . This pro duct is b ounded by A :=  e C 3 C ′ α 1 − α  2 . This shows the second inequality in ( 19 ) and the first inequalit y follo ws by symmetry . □ Next, we relate the supremum for a compact set K , ∥ F ′ n ∥ K : = sup z ∈ K | F ′ n ( z ) | of the absolute v alue of the deriv ative of F n = f n ◦ · · · ◦ f 1 to that of G n = g n ◦ · · · ◦ g 1 . 14 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS Lemma 18. L et K, L ⊆ C b e disjoint c omp act subsets. Then ther e exists a c onstant C > 0 (dep ending only on K and L ) such that for any differ entiable map F : K → K and G = ϕ w ′ ◦ F ◦ ϕ − 1 w : ϕ w ( K ) → ϕ w ′ ( K ) wher e ϕ w ( z ) = 1 z − w with w , w ′ ∈ L , we have 1 C ∥ F ′ ∥ K < ∥ G ′ ∥ ϕ w ( K ) < C ∥ F ′ ∥ K . Pr o of. As the deriv ativ es ϕ ′ w ( z ) = − 1 ( z − w ) 2 are uniformly b ounded aw a y from 0 and ∞ for z ∈ K and w ∈ L , there is a uniform b ound √ C > 0 on the deriv atives of ϕ w | K and ϕ − 1 w | ϕ w ( K ) for all w ∈ L . Hence, we ha v e ∥ G ′ ∥ ϕ w ( K ) ≤ ∥ ϕ ′ w ′ ∥ K ∥ F ′ ∥ K ∥ ( ϕ − 1 w ) ′ ∥ ϕ w ( K ) ≤ C ∥ F ′ ∥ K and vice versa. □ Corollary 19. L et λ > 0 , let K ⊂ H > − 1 / 2 b e a c omp act set. Then ther e exists a c onstant C > 0 such that the fol lowing holds. F or any se quenc e λ = ( λ n ) in (0 , λ ] , let f n := f λ n , F n ( z ) = f n ◦ · · · ◦ f 1 and let ( w n ) b e the unique orbit that stays in H < 0 . L et ( g n ) b e the asso ciate d se quenc e of affine tr ansformations define d by ( 17 ) and let G n ( z ) := g n ◦ · · · ◦ g 1 . Then, if ∥ G ′ n ∥ ϕ 0 ( K ) ≤ η , then ∥ F ′ n ∥ K ≤ C η and c onversely, if ∥ F ′ n ∥ K ≤ η , then ∥ G ′ n ∥ ϕ 0 ( K ) ≤ C η . Pr o of. F or any λ ′ ∈ [0 , λ ], the M¨ obius transformation f λ ′ maps lines and circles to lines and circles and preserv es angles. Since f λ ′ maps the real line to itself and is decreasing there, f λ ′ ( H > − 1 / 2 ) is the disc B λ ′ in the interior of the circle centred on the real line through f λ ′ ( ∞ ) = 0 and f λ ′ ( − 1 / 2) = 2 λ ′ . Let K ′ ⊆ H > − 1 / 2 b e a compact set (e.g. a large closed ball) con taining b oth K and the closed ball B λ . Then for an y λ ′ ∈ [0 , λ ], w e hav e f λ ′ ( H > − 1 / 2 ) = B λ ′ ⊆ B λ ⊆ K ′ and hence F n ( K ′ ) ⊆ K ′ indep enden tly of the sequence ( λ n ). As G n = ϕ n ◦ F n ◦ ϕ − 1 0 , taking L = [ − λ − 1 , − 1], whic h is clearly disjoint from K ′ , the statement follo ws from Lemma 18 □ The next lemma giv es us information ab out the deriv ative of F n = f n ◦ · · · ◦ f 1 . Lemma 20. L et D b e a disc c entr e d on the r e al line and K ⊆ D a c omp act subset. L et c 1 and c 2 b e the two p oints wher e the b oundary of D interse cts the r e al line R . Then ther e exists a c onstant C > 0 such that for any r e al M¨ obius tr ansformation F with F ( D ) ⊆ C , we have | F ( z ) − F ( w ) | < C | F ( c 2 ) − F ( c 1 ) | · | z − w | for al l z , w ∈ K . Pr o of. On the compact subset K ⊆ D , the P oincar ´ e metric d D on D is comparable to the Euclidean metric. So there exists a constant C > 0 such that for z , w ∈ K , w e hav e | z − w | > 1 C d D ( z , w ) . As before, the image F ( D ) under the real M¨ obius transformation F is b ounded by the circle cen tred on the real line through F ( c 1 ) and F ( c 2 ). Since it is con tained in C b y assumption, F ( D ) m ust b e the interior of that circle. The P oincar´ e distance on a disc of radius r is bounded from b elo w b y 1 /r times the Euclidean distance. Hence, b y non- expansiv enes of the Poincar ´ e distance under the holomorphic map F , we conclude: C | z − w | > d D ( z , w ) ≥ d F ( D ) ( F ( z ) , F ( w )) ≥ 2 | F ( c 2 ) − F ( c 1 ) | | F ( z ) − F ( w ) | , where d F ( D ) denotes the Poincar ´ e distance on F ( D ). □ DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 15 The following corollary will b e used in the pro of of our main theorem. Corollary 21. L et ∆ ∈ N and let λ > 0 . L et K ⊂ B λ/ 2 ( λ/ 2) b e a c omp act set c ontaining a neighb ourho o d of [ ℓ ∆ ( λ ) , r ∆ ( λ )] . Ther e exists a c onstant C > 0 such that for any se quenc e ( λ n ) in [ ℓ ∆ ( λ ) , λ ] , for al l n ∈ N and the maps F n = f λ n ◦ · · · ◦ f λ 1 , we have ∥ F ′ n ∥ K ≤ C | F n (0) − F n ( λ ) | . (21) Pr o of. Possibly enlarging K , we ma y assume K to b e connected. Fix n ∈ N . Clearly F n ( B λ/ 2 ( λ/ 2)) ⊂ C . So Lemma 20 applied to K and D = B λ/ 2 ( λ/ 2) provides a constant C > 0 such that for all x, y ∈ K w e hav e | F n ( x ) − F n ( y ) | ≤ C | F n (0) − F n ( λ ) || x − y | . Since K is connected, this implies ( 21 ). □ 4. Proof of Theorem 8 In this section w e prov e Theorem 8 . W e use the framew ork dev elop ed in Section 3 abov e to provide a pro of that VSSM implies absence of zeros. W e start with a simple but crucial monotonicity prop erty of the map F T ,n as defined in ( 12 ). Lemma 22. L et ( T , v ) b e a r o ote d tr e e. If n is even the function F T ,n is monotonic al ly in- cr e asing in e ach c o or dinate, while if n is o dd the function F T ,n is monotonic al ly de cr e asing in e ach c o or dinate. Pr o of. It suffices to prov e this for F T ,n ; u for any u ∈ S ∗ T ( v , n ) and any fixed inputs r u ′ for the remaining u ′ ∈ S ∗ T ( v , n ). By Lemma 13 w e hav e F T ,n ; u = f λ n ◦ · · · f λ 1 for certain λ i > 0. Now since f ′ λ i ( z ) = − λ i (1+ z ) 2 < 0 for all z ≥ 0, and since f λ i maps R ≥ 0 in to R ≥ 0 , it follo ws by the chain rule that ( − 1) n d dz f λ n ◦ · · · ◦ f λ 1 ( z ) > 0 . This finishes the pro of. □ Let ( T , v ) b e a ro oted tree. Let n ∈ N and recall that S ∗ T ( v , n ) denotes the collection of those vertices at distance n from v that are not lea ves. The following is a direct consequence of the previous lemma. Corollary 23. L et λ > 0 . F or any r o ote d tr e e ( T , v ) and any r = ( r u ) ∈ [0 , λ ] S ∗ T ( v ,n ) we have F T ,n (  0) ≤ F T ,n ( r u ) ≤ F T ,n (  λ ) if n is even, and (22) F T ,n (  λ ) ≤ F T ,n ( r u ) ≤ F T ,n (  0) if n is o dd. (23) Fix a tree T with v ertex v and let n b e an p ositive integer. Iden tify S ∗ T ( v , n ) with [ ℓ ] := { 1 , . . . , ℓ } . Let for i = 0 , . . . , ℓ and λ > 0 the v ector δ i ∈ { 0 , λ } ℓ b e defined by δ i j = 0 if j ≤ i and δ i j = λ if j > i . Let for i = 1 , . . . , ℓ ϵ i ( T , n ) := F T ,n ( δ i ) − F T ,n ( δ i − 1 ) (24) and note that by Lemma 22 ϵ i ( T , n ) is p ositive for all i = 1 , . . . , ℓ if n is ev en and negative for all i = 1 , . . . , ℓ if n is o dd. Observe that b y construction we ha ve R ( T ,σ n +1 , 0 ) ,v ( λ ) − R ( T ,σ n +1 , 1 ) ,v , ( λ ) = F T ,n (  λ ) − F T ,n (  0) = ℓ X i =1 ϵ i ( T , n ) , (25) where σ n +1 , 0 (resp. σ n +1 , 1 ) denotes the b oundary condition on the v ertices at distance n + 1 from v in T assigning all vertices to 0 (resp. 1). 16 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS The follo wing prop osition tells us that the functions F T ,n ; i for i ∈ S ∗ T ( v , n ) defined before Lemma 13 con tract with rate a constant times | ϵ i ( T , n ) | , pro vided T is the self-av oiding w alk tree of a graph G contained in a family satisfying VSSM at λ . Prop osition 24. L et λ > 0 and let ∆ ∈ N . L et G b e a family of gr aphs of maximum de gr e e at most ∆ that is close d under taking induc e d sub gr aphs and that satisfies VSSM with r ate α ∈ [0 , 1) and c onstant C ′ > 0 for the har d-c or e mo del at fugacity λ . L et T denote the family of self avoiding walk tr e es of gr aphs in G . L et K ⊂ B λ/ 2 ( λ/ 2) b e a c omp act set c ontaining a neighb ourho o d of [ ℓ ∆ ( λ ) , r ∆ ( λ )] . Then ther e exists a c onstant C > 0 such that for any ( T , v ) ∈ T , n ∈ N , u ∈ S ∗ T ( v , n ) , and any ve ctor ( r u ′ ) ∈ [ ℓ ∆ ( λ ) , λ ] S ∗ T ( v ,n ) \{ u } we have ∥ F ′ T ,n ; u ∥ K ≤ C | ϵ u ( T , n ) | . Pr o of. Let us fix ( T , v ) = ( T SA W( G,v ) , v ) ∈ T , an integer n , any u ∈ S ∗ T ( v , n ) and a v ector r ∈ [ ℓ ∆ ( λ ) , λ ] S ∗ T ( v ,n ) \{ u } . After identifying S ∗ T ( v , n ) with [ ℓ ] we may assume that u ∈ [ ℓ ]. W e will write ϵ = | ϵ u ( T , n ) | . Let λ j , for j = 1 , . . . , n b e obtained from Lemma 13 from the input v ector δ u as defined righ t after Corollary 23 . Let ˜ λ j for j = 1 , . . . , n b e obtained from the input vector r . By setting λ n + i = ˜ λ n + i = λ for i ≥ 1 we can of course turn these into infinite sequences. W e write f j = f λ j and ˜ f j = f ˜ λ j . Let ( w j ) (resp.) ( ˜ w j ) b e the asso ciated sequences obtained from Lemma 15 . Finally , let ( g j ) (resp. ( ˜ g j ) b e the corresp onding sequences of affine maps. W e first claim that for all j ≥ 1 we ha v e | λ j − ˜ λ j | ≤ C α j . (26) It suffices to pro ve ( 26 ) for j ≤ n . By Lemma 13 we kno w that λ j = F T n,j ,w j ( δ u ) , where w j is the vertex on the unique path P from v to u in T at distance n − j from v and where T n,j is the comp onen t of T \ ( P − w j ) con taining the vertex w j (see Figure 1 ). By Lemma 22 it th us follows that λ j is sandwiched b etw een the v alues F T n,j ,w j (  0) and F T n,j ,w j (  λ ). Similarly we ha v e ˜ λ j = F T n,j ,w j ( r ) , and since r i ∈ [0 , λ ] for all i ∈ [ ℓ ], it follows from Lemma 22 that also ˜ λ j is sandwic hed b et w een the v alues F T n,j ,w j (  0) and F T n,j ,w j (  λ ). Fix j and denote the neighbours of w j that are at distance n − j + 1 from v , but are not on the path from u to v by u 1 , . . . , u d ′ and denote b y S 1 , . . . , S d ′ the ro oted trees con taining u i and all vertices at distance larger than n − j + 1 whose unique path to v go es through u i . Lemma 12 tells us that ( S i , u i ) is the self av oiding w alk tree of ( H i , u i ) where H i is an induced subgraph of G . Since the family G satisfies VSSM at λ , there exists C ′ > 0 and α ∈ (0 , 1) such that | R ( S i ,σ j, 0 ) ,u i ( λ ) − R ( S i ,σ j, 1 ) ,u i ( λ ) | ≤ C ′ α j . No w by the construction of F T ,n w e hav e F T n,j ,w j (  0) = λ Q d ′ i =1 (1 + R ( S i ,σ j, 1 ) ,u i ( λ )) , and F T n,j ,w j (  λ ) = λ Q d ′ i =1 (1 + R ( S i ,σ j, 0 ) ,u i ( λ )) , DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 17 and thus there exists a constant C > 0 only dep ending on λ, C ′ and d such that | F T n,j ,w j (  0) − F T n,j ,w j (  λ ) | ≤ C α j . This prov es ( 26 ). Let us next write G n = g n ◦ · · · ◦ g 1 , ˜ G n = ˜ g n ◦ · · · ˜ g 1 and F n = f n ◦ · · · ◦ f 1 and ˜ F n = ˜ f n ◦ · · · ˜ f 1 . Observ e that with this notation we hav e ˜ F n ( · ) = F T ,n ( · , r ) and F n ( · ) = F T ,n ( · , δ u ). W e will transfer kno wledge ab out ∥ F ′ n ∥ K to ∥ ˜ F ′ n ∥ K using G n and ˜ G n . Let A > 1 b e the constan t from Corollary 17 on input of λ + = λ , λ − = ℓ ∆ ( λ ), C and α . Then we kno w that 1 / A ≤     G ′ n ˜ G ′ n     ≤ A. (27) Let C (1) b e the constant from Corollary 21 . Then ∥ F ′ n ∥ K ≤ C (1) ϵ. By Corollary 19 w e thus ha ve that | G ′ n | = ∥ G ′ n ∥ ϕ 0 ( K ) ≤ C (2) ϵ. Consequen tly by ( 27 ) and another application of Corollary 19 we obtain ∥ ˜ F ′ n ∥ K ≤ C (3) ϵ, where C (3) is equal to the constant from Corollary 19 multiplied by AC (2) . Since the constan t C (3) only dep ends on λ, C ′ , α and ∆, this finishes the pro of. □ W e next use the previous prop osition to show that small p erturbations of the functions F T ,n ha ve no big effect on the v alue of the functions. T o this end let us denote for a rooted tree ( T , v ) and  λ = ( λ u ) u ∈ V ( T ) the map F T ,n,  λ for the map obtained from F T ,n b y replacing in each step of the recursion ( 11 ) at some ro ot vertex u the v alue of λ by λ u . Prop osition 25. L et λ > 0 and ∆ ∈ N . L et G b e a family of gr aphs of maximum de gr e e at most ∆ that is close d under taking induc e d sub gr aphs and satisfies VSSM with r ate α ∈ [0 , 1) and c onstant C ′ > 0 for the har d-c or e mo del at fugacity λ . L et T denote the family of self avoiding walk tr e es of gr aphs in G . Then ther e exists n ∈ N and η 0 > 0 such that for e ach 0 < η 1 < η 0 ther e is an η 2 > 0 such that for any ( T , v ) ∈ T , r ∈ [ ℓ ∆ ( λ ) , r ∆ ( λ )] S ∗ T ( v ,n ) , r ′ ∈ B ℓ ∞ ( r , η 1 ) and  λ = ( λ u ) u ∈ V ( T ) with λ u ∈ B ( λ, η 2 ) for e ach u we have | F T ,n,  λ ( r ′ ) − F T ,n ( r ) | ≤ η 1 . (28) Pr o of. Cho ose n large enough so that, with the constant C from Prop osition 24 ab ov e, up on input of λ and a compact set K containing a neigh b ourho o d of [ ℓ ∆ ( λ ) , r ∆ ( λ )] and α, C ′ w e hav e C C ′ α n ≤ 1 / 4. Since there are only finitely many ro oted trees of maximum degree at most ∆ and depth n + 1, it follo ws that there exists only finitely man y associated functions F T ,n . Fix any ro oted tree ( T , v ) ∈ T . Prop osition 24 implies that for each u ∈ S ∗ T ( v , n ) and r ∈ [ ℓ ∆ ( λ ) , λ ] S ∗ T ( v ,n ) \{ u } the absolute v alue of the deriv ativ e of F T ,n ; u is b ounded b y C | ϵ u ( T , n ) | on a neigh b ourho o d of [ ℓ ∆ ( λ ) , r ∆ ( λ )]. This implies that the ℓ 1 -norm of the gradien t, ∥∇ F T ,n ( r ) ∥ 1 , is b ounded b y ∥∇ F T ,n ( r ) ∥ 1 = X u ∈ S ∗ T ( v ,n )     ∂ ∂ r u F T ,n ( r )     = X u ∈ S ∗ T ( v ,n ) | F ′ T ,n ; u ( r u ) | ≤ X u ∈ S ∗ T ( v ,n ) C | ϵ u ( T , n ) | 18 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS for all r ∈ [ ℓ ∆ ( λ ) , r ∆ ( λ )] S ∗ T ( v ,n ) . Thus from ( 25 ), the fact that all ϵ u ( T , n ) ha ve the same sign, the assumption of VSSM, and our choice of n , it follows that ∥∇ F T ,n ( r ) ∥ 1 ≤ C C ′ α n ≤ 1 / 4 , for any r ∈ [ ℓ ∆ ( λ ) , r ∆ ( λ )] S ∗ T ( v ,n ) and all ( T , v ) ∈ T . By con tin uit y of the map r 7→ ∇ F T ,n ( r ) and the fact there exists only finitely many maps F T ,n , it follows that that there exists η 1 > 0 such that for all r ∈ B ([ ℓ ∆ ( λ ) , r ∆ ( λ )] , η 1 ) S ∗ T ( v ,n ) and all ( T , v ) ∈ T we ha v e ∥∇ F T ,n ( r ) ∥ 1 ≤ 1 / 2 . This in turn implies that for eac h tree ( T , v ) and eac h r, r ′ ∈ B ([ ℓ ∆ ( λ ) , r ∆ ( λ )] , η 1 ) S ∗ T ( v ,n ) w e hav e | F T ,n ( r ) − F T ,n ( r ′ ) | ≤ 1 / 2 ∥ r − r ′ ∥ ∞ . (29) Next w e lo ok at the p erturb ed maps F T ,n,  λ . By contin uit y of (  λ, r ′ ) 7→ F T ,n,  λ ( r ′ ) and the fact that there only finitely man y maps F T ,n , there exists η 2 > 0 such that for all trees ( T , v ) ∈ T , r ′ ∈ B ([ ℓ ∆ ( λ ) , r ∆ ( λ )] , η 1 ) S ∗ T ( v ,n ) and any  λ = ( λ u ) u ∈ V ( T ) ∈ B ( λ, η 2 ) V ( T ) , w e ha ve | F T ,n,  λ ( r ′ ) − F T ,n ( r ′ ) | < η 1 / 2 . (30) No w combining ( 30 ) and ( 29 ) with the triangle inequalit y , w e obtain that for any tree ( T , v ) ∈ T , an y r ∈ [ ℓ ∆ ( λ ) , r ∆ ( λ )] S ∗ T ( v ,n ) , r ′ ∈ B ℓ ∞ ( r , η 1 ) and any  λ = ( λ u ) u ∈ V ( T ) ∈ B ( λ, η 2 ) V ( T ) , | F T ,n,  λ ( r ′ ) − F T ,n ( r ) | ≤ | F T ,n,  λ ( r ′ ) − F T ,n ( r ′ ) | + | F T ,n ( r ′ ) − F T ,n ( r ) | ≤ η 1 / 2 + 1 / 2 ∥ r − r ′ ∥ ∞ ≤ η 1 , as desired. □ W e can now pro v e Theorem 8 . Pr o of of The or em 8 . T o pro ve the implication we assume that G is family of graphs of maxim um degree at most ∆, is closed under taking induced subgraphs, and satisfies VSSM at fugacity λ > 0 with rate α ∈ [0 , 1) and constan t C > 0. Let n ∈ N , η 1 ∈ (0 , 1), and η 2 > 0 b e the guaran teed constants from Prop osition 25 . Consider the collection T of all ro oted trees of maximum degree at most ∆ and depth at most n . Since T consists of only finitely man y trees, there exists ε ∈ (0 , η 2 ) suc h that for each ( T , v ) ∈ T and any  λ ∈ B ( λ, ε ) V ( T ) w e hav e R T ,v (  λ ) ∈ B ( R T ,v ( λ ) , η 1 ) . W e will sho w that for all G = ( V , E ) ∈ G and ( λ u ) u ∈ V suc h that λ u ∈ B ( λ, ε ) for all u ∈ V w e hav e (i) Z G ( λ u )  = 0, and (ii) R G,v ( λ u ) ∈ B ( R G,v ( λ ) , η 1 ) for any v ertex v ∈ V ( G ) . In case V = ∅ this is trivial and th us we ma y assume that | V | ≥ 1. W e claim that it suffices to sho w (ii). Indeed, by induction we kno w that Z G − v ( λ u )  = 0, and th us b y ( 8 ) it suffices to show that R G,v ( λ u )  = − 1 for an y vertex v ∈ V . Since R G,v ( λ ) ≥ 0 and η 1 < 1 it follows that (ii) implies (i). T o sho w (ii) let T = T SA W( G,v ) . W e will show that R T ,v ( ˜ λ w ) ∈ B ( R T ,v ( λ ) , η 1 ), whic h by Lemma 12 implies that R G,v ( λ u ) ∈ B ( R G,v ( λ ) , η 1 ). In case the depth of T is at most n , w e hav e that R T ,v ( ˜ λ w ) ∈ B ( R T ,v ( λ ) , η 1 ) b y our c hoice of ε . So we ma y assume that the depth of T is at least n . W e know that R T ,v ( ˜ λ w ) = F T ,n,  λ ( R T u ,u ( ˜ λ w )) , where for u ∈ S ∗ T ( v , n ), T u denotes the subtree of T ro oted at u consisting of all vertices whose path to v go es trough u and where  λ denotes the v ector of v ariables obtained from DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 19 ( ˜ λ w ). Note that T u consists of at least one edge since u ∈ S ∗ T ( v , n ). By Lemma 12 , w e kno w that each such subtree T u is the tree of self a v oiding walks of a strict induced subgraph H of G and hence by induction and another application of Lemma 12 we obtain that R T u ,u ( ˜ λ w ) = R H,u ( λ u ′ ) ∈ B ( R H,u ( λ ) , η 1 ) = B ( R T u ,u ( λ ) , η 1 ) for each u ∈ S ∗ T ( v , n ). Note that by Lemma 13 and Lemma 14 we hav e R T u ,u ( λ ) ∈ [ ℓ ∆ ( λ ) , r ∆ ( λ )] and thus b y Prop osition 25 we ha v e | F T ,n,  λ ( R u (( ˜ λ w ))) − F T ,n ( R T u ,u ( λ )) | ≤ η 1 . Since R G,v ( λ u ) = R T ,v ( ˜ λ w ) = F T ,n,  λ ( R u (( ˜ λ w ))) and R G,v ( λ ) = R T ,v ( λ ) = F T ,n ( R u ( λ )) by y et another application of Lemma 12 this sho ws (ii) for ( G, v ) and finishes the pro of of Theorem 8 . □ 5. VSSM a t gro wing dist ance allows zer os In this section w e prov e Theorem 10 . The family of graphs G that w e consider consists of trees of the following form: Let d = ∆ − 1 ≥ 2. F or k ∈ N let T d k b e the ro oted tree in which the ro ot v ertex has degree d and is connected to d iden tical copies of T d k − 1 , where T d 0 is just a single vertex. F or k , m ∈ N let T d k , 1 m b e obtained from T d k b y attac hing a path (tree of do wn-degree 1) of depth m to each leaf. All these trees ha ve maxim um degree ∆. The zeros of the indep endence polynomials of trees T d k , k ∈ N are known to accum ulate at λ c (∆), as shown via dynamical instabilit y in [ PR19 ]. W e will give a v arian t of this pro of and sho w that the dynamical instabilit y p ersists for the trees T d k , 1 m , while the con tracting dynamics of long paths ensures φ -VSSM. F or the ro oted tree T = T d k , 1 m , equation ( 11 ) simplifies to R T ,v ( λ ) = λ (1 + R T ′ ,v ′ ) d , where v ′ is a neigh b our of v in T and T ′ is the comp onen t of T − v with ro ot v ′ . In other w ords, R T ,v is the image of R T ′ ,v ′ under the rational map f d,λ : C → C , defined b y R 7→ λ (1 + R ) d . By recursion, this giv es us a represen tation of the ratio of T d k , 1 m at the ro ot v 0 as the comp osition: R T d k , 1 m ,v 0 ( λ ) = f ◦ k d,λ ◦ f ◦ m 1 ,λ (0) . (31) W e can thus study these ratios via the dynamical b eha vior of f 1 ,λ and f d,λ . The next subsection contains a summary of useful prop erties of these maps. 5.1. Dynamics of f d,λ . Let us first fix some language and notation: A rational map f with real or complex co efficien ts defines a holomorphic map from the extended complex plane b C to itself. A fixed p oin t x = f ( x ) of f is attracting if | f ′ ( x ) | < 1, rep elling if | f ′ ( x ) | > 1, and parab olic if f ′ ( x ) is a ro ot of unity . The basin of a fixed p oin t x of f is B f ( x ) := { y ∈ b C | lim n →∞ f n ( y ) = x } . If x is rep elling, there can b e no orbit conv erging to x non-trivially , i.e. that is not equal to x after finitely many iterates. The dynamics of f d,λ ha ve been studied in [ PR19 ], where the first tw o authors sho w in particular: Lemma 26. L et d ∈ N . Then: 20 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS (1) F or λ > 0 , f d,λ has a fixe d p oint x d ( λ ) ∈ (0 , λ ) and x d ( λ ) is attr acting for λ < λ c ( d + 1) , p ar ab olic for λ = λ c ( d + 1) , and r ep el ling for λ > λ c ( d + 1) . (2) The fixe d p oint x d ( λ ) p ersists for λ in a c omplex neighb ourho o d of [0 , ∞ ) and dep ends holomorphic al ly on λ . Note 27 . F or d = 1, λ c ( d + 1) = + ∞ , so x 1 ( λ ) is attracting for all λ > 0. W e can further understand the limit b eha viour of orbits on the entire in terv al [ − 1 , ∞ ]: Lemma 28. L et d ∈ N > 0 and λ > 0 . Then x d ( λ ) is the unique fixe d p oint of f d,λ in the interval [ − 1 , ∞ ] and: (1) If λ ≤ λ c ( d + 1) , then the entir e interval [ − 1 , ∞ ] lies in the b asin of the attr acting or p ar ab olic fixe d p oint x d ( λ ) . (2) If λ > λ c ( d + 1) , then f d,λ has an attr acting 2 -cycle ( x 1 , x 2 ) = ( x d, 1 ( λ ) , x d, 2 ( λ )) ∈ (0 , λ ) 2 (i.e. x 1 and x 2 ar e attr acting fixe d p oints of f 2 d,λ with f d,λ ( x 1 ) = x 2 ) and the entir e interval [ − 1 , ∞ ] exc ept the fixe d p oint x d ( λ ) lies inside the b asin of the attr acting cycle. R emark 29 . The con vergence in both cases is uniform on compact subsets of the basin. If x d ( λ ) is attracting, con v ergence to x d ( λ ) is locally uniform in b oth x and λ . More precisely , for each λ < λ c ( d + 1) and every x in the basin B f d,λ ( x d ( λ )) there exist neighbourho o ds Λ ⊆ C of λ and U ⊆ b C of x suc h that lim n →∞ f n d,λ ′ ( x ′ ) = x d ( λ ′ ) uniformly in ( λ ′ , x ′ ) ∈ Λ × U (see [ Mil06 ] and [ MSS83 ]). Pr o of. F or any λ > 0, f d,λ is strictly decreasing on [ − 1 , + ∞ ], so g : = f 2 d,λ is strictly increasing, and hence injectiv e. W e know g ([ − 1 , ∞ ]) = [0 , λ ], so \ n ∈ N g n ([ − 1 , ∞ ]) = [ x 1 , x 2 ] , (32) for some 0 < x 1 ≤ x 2 < λ . By monotonicit y , x 1 , x 2 are fixed p oin ts of g that are either attracting or parab olic and f d,λ ( x 1 ) = x 2 . W e recall that ev ery non-rep elling fixed p oint of a rational map is related to the orbit of a critical p oin t (see [ Mil06 ]). The critical p oin ts of g are ∞ and − 1 and are contained in the basins of x 1 and x 2 resp ectiv ely , so all further fixed p oin ts of g must b e rep elling. Moreo ver, any fixed p oints of g in [ − 1 , ∞ ] m ust b e con tained in [ x 1 , x 2 ], including the fixed p oin t x 0 = x d ( λ ) of f d,λ . (1) Let now λ ≤ λ c ( d + 1). Then the fixed p oin t x 0 is attracting or parabolic for f d,λ and hence for g , so x 0 ∈ { x 1 , x 2 } , but since f d,λ maps x 1 and x 2 to each other, g has the unique fixed p oin t x 0 = x 1 = x 2 on [ − 1 , ∞ ] and, by ( 32 ), the entire in terv al [ − 1 , ∞ ] is in the basin of x 0 . (2) Let λ > λ c ( d + 1). Then x 0 is rep elling, so x 1 < x 0 < x 2 and if x > x 0 is small enough, we ha v e g ( x ) > x . As b efore, the monotonicity of g then implies that \ n ∈ N g n ([ x, x 2 ]) = [ x 3 , x 2 ] with x 3 = lim n →∞ g n ( x ) ∈ ( x 0 , x 2 ], which is an attracting or parabolic fixed p oin t of g , so x 3 m ust b e x 2 . Th us we hav e shown that lim n →∞ g n ( x ) = x 2 lo cally uniformly for all x ∈ ( x 0 , ∞ ]. An analogous argument sho ws that lim n →∞ g n ( x ) = x 1 lo cally uniformly for all x ∈ [ − 1 , x 0 ). W e ha ve thus shown that all orbits in [ − 1 ., ∞ ] except the rep elling fixed point x 0 con verge to either the fixed point x 1 or x 2 and these are either b oth attracting or b oth parab olic. W e recall that, for ev ery orbit conv erging to a parab olic fixed p oin t, there m ust DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 21 also b e an orbit of a critical p oin t con verging to it from the same direction (see [ Mil06 ]). x 1 and x 2 eac h hav e orbits conv erging to them from ab o v e and from below, but g only has t wo critical p oin ts, so x 1 and x 2 m ust b e attracting fixed p oints. □ T o demonstrate the relation of zero-freeness and dynamical stabilit y , w e use Lemma 28 to give a short pro of of a weak er v ersion of a result in [ PR19 ]: Lemma 30. The indep endenc e p olynomials of the family T = { T d k } k have no zer os on a neighb ourho o d of [0 , λ c ( d + 1)) , but have zer os ac cumulating at λ c ( d + 1) . The pro of uses the notion of normal families and Montel’s Theorem, whic h are funda- men tal notions in the study of complex dynamical systems, see for example [ Bea91 ],[ CG93 ], [ Mil06 ]. Normality is a notion of dynamical stability defined as follows: Definition 31. A family of holomorphic functions F is normal at λ ∈ C , if there exists a neigh b ourho od U of λ such that for each sequence ( g n ) n in F there exists a subsequence ( g n k ) k that conv erges uniformly on U to a holomorphic map g : U → b C . Mon tel’s theorem relates normality of a family to omitted v alues: Theorem 32 (Montel) . L et F b e a family of holomorphic functions fr om an op en set U ⊆ C to the Riemann spher e C ∪ {∞} that al l omit thr e e values in C ∪ {∞} . Then F is normal on U . W e can now pro v e the lemma. Pr o of of L emma 30 . By Remark 29 , conv ergence of the sequence of ratios ( R T d n ( λ ) = f n d,λ (0)) n to x d ( λ ) p ersists on a small complex neighbourho o d of [0 , λ c ( d + 1)). In particular none of these ratios are − 1, so there are no zeros of the indep endence p olynomial in that neigh b ourho od. Assume that R T d n ( λ ) = f n d,λ (0) is not equal to − 1 for all n ∈ N and all λ in a neigh- b ourhoo d Λ of λ c ( d + 1). Then, since x = − 1 is the only solution to f d,λ ( x ) = ∞ and ∞ is the only solution to f d,λ ( x ) = 0, the family { R T d n ( λ ) = f n d,λ (0) } n a voids − 1 , ∞ , and 0. Hence, by Mon tel’s theorem 32 , the family is normal on Λ. Ho wev er, the c hange of dynamics in Lemma 28 shows that the family F = { R T d n ( λ ) = f n d,λ (0) } n cannot be normal near λ c ( d + 1): Assume there is a neigh b ourhoo d Λ of λ c ( d + 1) on whic h F is normal. W e ma y assume Λ to b e connected. Then, by definition, for the sequences ( f 2 n d,λ ) n and ( f 2 n +1 d,λ ) n there exist subsequences ( n k ) k and ( n ′ k ) k with holomorphic limit functions: g 1 ( λ ) = lim k →∞ f 2 n k d,λ (0) , g 2 ( λ ) = lim k →∞ f 2 n ′ k +1 d,λ (0) . By Lemma 28 , w e hav e g 1 ( λ ) = x d ( λ ) = g 2 ( λ ) for λ ≤ λ d , but g 1 ( λ ) = x d, 1 ( λ )  = x d, 2 ( λ ) = g 2 ( λ ) for λ > λ d . This is a contradiction to the iden tit y theorem for the holomorphic functions g 1 and g 2 on Λ. Hence, for any neigh b ourho od Λ of λ c ( d + 1) there exist λ ∈ Λ and T d n ∈ T , such that R T d n ( λ ) = f n d,λ (0) = − 1, that is, Z T d n ( λ ) = 0. □ W e will show that this dynamical deriv ation of accum ulating zeros can still b e applied when we attac h long paths to the leav es. 22 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS 5.2. Zeros accumulating at λ c (∆) . F or a sequence of positive in tegers ( m k ) k , w e denote T k = T d k , 1 m k . Then we ha v e R T k ,v 0 = f d,λ ( R T ′ k ,v ′ ) = f 2 d,λ ( R T ′′ k ,v ′′ ) (33) with T ′ k = T d k − 1 1 m k and T ′′ k = T d k − 2 1 m k . The following proposition giv es us one part of Theorem 10 . Prop osition 33. L et ( m k ) k b e a se quenc e of non-ne gative inte gers. Then at le ast one of the families T = { T k } k , T 1 = { T ′ k } k , and T 2 = { T ′′ k } k has zer os of its indep endenc e p olynomial ac cumulating at λ c (∆) . Pr o of. Our strategy is the same as for the homogeneous case ab o v e: Assume for con tra- diction, Z T k ( λ ), Z T ′ k ( λ ), and Z T ′′ k ( λ ) are non-zero for ev ery λ in a neighbourho o d Λ of λ d and all k ∈ N . Then R T k ,v , R T ′ k ,v ′ , and R T ′′ k ,v ′′ all omit the v alue − 1 on Λ, and, by ( 33 ), R T k ,v omits − 1, ∞ = f d,λ ( − 1) and 0 = f 2 d,λ ( − 1). This would imply that R T k ,v 0 ( λ ) is normal on Λ by Theorem 32 (Montel’s theorem). T o reach a con tradiction, we therefore need to show that the family { R T k ,v 0 } k cannot b e normal near λ c (∆). Recall that R T k ,v 0 ( λ ) = f ◦ k d,λ ◦ f ◦ m k 1 ,λ (0) and, since 0 is con tained in the attr acting basin of the fixed point x 1 ( λ ) of f 1 ,λ for all λ , w e kno w that f m k 1 ,λ (0) conv erges to x 1 ( λ ) uniformly on a neigh b ourho o d of λ c (∆). Moreov er, for λ ∈ (0 , λ c (∆)], the limit x 1 ( λ ) is real and hence con tained in the basin of x d ( λ ). T aking z ( λ ) = x 1 ( λ ) and z k ( λ ) = f m k 1 ,λ (0) in Lemma 35 b elow, shows that { R T k ,v 0 } k cannot b e normal, completing our pro of by con tradiction. □ R emark 34 . Remo ving the ro ot (and its neigh b ours) to create the families T ′ (and T ′′ ) enabled our argument similar to [ Buy21 , Lem. 13]. In fact, a more technical pro of using the Theorem of Lav aurs [ Lav89 ] (see also [ Dou94 ]) sho ws that it is enough to only consider the family T (see the app endix of [ PR19 ]). Since the ab o ve result is sufficient for our purp oses and its pro of is muc h more elementary , we leav e the pro of of the more elegant statemen t to the interested reader. Lemma 35. L et ∆ ∈ N and let z ( λ ) b e a holomorphic function in λ on a neighb ourho o d of λ c (∆) . Assume that z ( λ )  = x d ( λ ) for al l λ an d that, mor e over, for non-ne gative r e al λ ≤ λ c (∆) , z ( λ ) is r e al and c ontaine d in the b asin of the attr acting or p ar ab olic fixe d p oint x d ( λ ) . Mor e over, let ( z k ( λ )) k b e a se quenc e of holomorphic functions in λ such that z k ( λ ) − − − → k →∞ z ( λ ) uniformly in λ ne ar λ c (∆) . Then the family { f n d,λ ( z n ( λ )) } n is not normal in λ ne ar λ c (∆) . Pr o of. F or fixed λ ≤ λ c (∆) and n large enough, z n ( λ ) is contained in a compact subset K λ of the basin of x d ( λ ), and by uniform conv ergence w e hav e lim n →∞ f n d,λ ( z n ( λ )) = x d ( λ ) . F or λ > λ c (∆), z ( λ ) remains real by the identit y theorem for holomorphic functions and if λ is small enough, since z ( λ d )  = x d ( λ d ), we still hav e z ( λ ) ∈ [ − 1 , ∞ ] \ { x d ( λ ) } , so z ( λ ) is in the basin of the attracting p erio dic cycle ( x d, 1 ( λ ) , x d, 2 ( λ )) of f d,λ and we ha v e f 2 n d,λ → x d, 1 ( λ ) , f 2 n +1 d,λ → x d, 2 ( λ ) uniformly on a compact neighbourho o d K λ of z ( λ ). Assume { f n d,λ ( z n ( λ )) } n is normal on a connected neigh b ourho o d Λ of λ c (∆). Then there exists holomorphic limit functions g 1 ( λ ) = lim k →∞ f 2 n k d,λ ( z n ( λ )) , g 2 ( λ ) = lim k →∞ f 2 n k +1 d,λ ( z n ( λ )) DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 23 with g 1 ( λ ) = x d ( λ ) = g 2 ( λ ) for λ < λ d , but g 1 ( λ ) = x d, 1 ( λ )  = x d, 2 ( λ ) = g 2 ( λ ) for λ > λ d . This is a contradiction to the iden tit y theorem for the holomorphic functions g 1 and g 2 on Λ. □ 5.3. φ -VSSM for { T k } k . Giv en an unbounded increasing function φ , we no w show that if we chose the sequence ( m k ) k gro wing fast enough, the family of trees T = { T k = T d k , 1 m k } k ∈ N satisfies φ -VSSM for all parameters λ ∈ [0 , λ ∗ ], allo wing us to prov e The- orem 10 . Prop osition 36. L et λ ⋆ > 0 . F or e ach k ∈ N , ther e exists a c onstant M k > 0 such that for al l λ ∈ [0 , λ ⋆ ] , al l induc e d sub gr aphs H of T k , and al l ℓ > 2 k we have | R ( H,v ) ,τ 1 ( λ ) − R ( H,v ) ,τ 2 ( λ ) | < M k c ℓ (34) for al l v ∈ V ( H ) and al l b oundary c onditions τ 1 , τ 2 : S H ( v , ℓ ) → { 0 , 1 } . Before the pro of, let us show how Prop osition 36 together with Prop osition 33 implies Theorem 10 . Pr o of of The or em 10 . Fix a function φ . Let G = { T d k , 1 m k } k ∪ { T d k − 1 , 1 m k } k ∪ { T d k − 2 , 1 m k } k for some increasing un b ounded sequence ( m k ) to b e determined b elo w. By Prop osition 33 we kno w that zeros of the independence p olynomial of graphs in G accum ulate at λ c (∆) for an y such sequence ( m k ). So it suffices to sho w that there exists a sequence ( m k ) for which G satisfies φ -VSSM at any λ ∈ [0 , λ ∗ ]. Let c < c 1 < 1 and M k , k ∈ N from Prop osition 36 . Then for eac h k ∈ N , there exists an ℓ k > 2 k such that for all ℓ ≥ ℓ k , we hav e M k c ℓ < c ℓ 1 for all ℓ ≥ ℓ k . Now take m k large enough that φ ( | V ( T d k − 2 1 m k ) | ) ≥ ℓ k . Then for every T = ( V , E ) ∈ { T d k , 1 m k } k , every induced subgraph H of T , w e hav e | R ( H,v ) ,τ 1 ( λ ) − R ( H,v ) ,τ 2 ( λ ) | < c ℓ 1 for all v ∈ V ( H ), all b oundary conditions τ 1 , τ 2 : S H ( v , ℓ ) → { 0 , 1 } with ℓ ≥ φ ( | V ( T d k 1 m k ) | ) and λ ∈ [0 , λ ∗ ]. In other words, the hard-core mo del on T = { T d k 1 m k } k satisfies φ -VSSM at an y λ ∈ [0 , λ ∗ ]. Note that the same is true for the families T ′ = { T d k − 1 , 1 m k } k and T ′′ = { T d k − 2 , 1 m k } k . This finishes the pro of. □ W e now pro ve the prop osition. W e note that an alternative pro of can b e deriv ed from b ounds on the connectiv e constant follo wing the arguments in [ SS ˇ SY17 ]. Pr o of of Pr op osition 36 . Fix k ∈ N . Recall that for a b oundary condition τ : S T ( v , ℓ ) → { 0 , 1 } on a tree T , w e still hav e a recursion form ula: R ( T ,v ) ,τ = λ Q n j =1 (1 + R ( T j ,v j ) ,τ ) , where v 1 , . . . , v n are the neigh b ours of v in T and T j is the comp onen t of T − v containing v j for j = 1 , . . . , n . Let v b e a v ertex in a connected induced subgraph H of T k and T H = H ∩ T d k . W e ensure the estimate ( 34 ) by the following heuristic in each case: (1) v ∈ T H . Then for ℓ ≫ 2 k , B ℓ ( v ) con tains T H and a long part of each attached path. The difference of the b oundary conditions is reduced by the contraction under each f 1 ,λ along each long path. (See Figure ( 3 ), left) 24 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS T H v v 1 P 1 u 1 u n − 1 v n P n S H ( v , ℓ ) τ ( u j ) λ f m j 1 ,λ f m ′ 1 ,λ R ( P j ,v j ) ,τ R ( H,v ) ,τ F v,T H T H v v 1 v ′ u 1 P 1 v ′′ v 2 P 2 u 2 v n P n u n S H ( v , ℓ ) τ ( u j ) R ( P j ,v j ) ,τ R ( H \ ( P 1 − v 1 ) ,v 1 ) ,τ R ( T ′ ,v ′ ) ,τ R ( H,v ) ,τ τ ( u 1 ) R ( T ′′ ,v ′′ ) ,τ f m j 1 ,λ F v,T H f m 1 1 ,λ f ℓ 1 ,λ Φ Figure 3. How ratios propagate up w ards in the tree to wards a root v ertex v in T H (left) and in P 1 (righ t). Shown is B ℓ ( v ) ⊆ H . τ is a b oundary condition on S H ( v , ℓ ). In the left picture, P n examplifies a path that do es not in tersect the b oundary S H ( v , ℓ ), so τ has no influence along this path. (2a) v / ∈ T H , but close to T H . F or ℓ large enough, B ℓ ( v ) contains T H and a long part of eac h attached path. As b efore, the difference is reduced along the paths not con taining v un til reac hing T H , where a bounded expansion may happ en and then not expanded by the path from T H to v . (See Figure ( 3 ), righ t, m j large) (2b) v / ∈ T H and far a w a y from T H . If T H is far enough from v , then the path from T H to v leads to enough iterations of f 1 ,λ that any difference is reduced sufficien tly . (See Figure ( 3 ), righ t, m 1 large) Let us quan tify the maximal gain of correlation passing through T H for giv en λ ∈ [0 , λ ∗ ]. Let L T H = { v 1 , . . . v n } b e the set of leav es of T H . Let P 1 , . . . , P n b e the paths attac hed to the resp ectiv e leav es v 1 , . . . v n of T H to complete H . (1) Let v in T H . Then there is a rational map F v ,T H ,λ : [0 , ∞ ] L T H \{ v } → [0 , ∞ ] , dep ending analytically on λ , that maps ( R ( P j ,v j ) ,τ ) v j  = v , to R ( H,v ) ,τ for any b ound- ary condition τ on S H ( v , ℓ ) with ℓ > 2 k . (Note that v may b e equal to a leaf v j for some j .) (2) Let v in P α , v  = v α . Then there exists a rational map F v ,T H ,λ = F P α ,T H ,λ : [0 , ∞ ] L T H \{ v α } → [0 , ∞ ] , dep ending analytically on λ , that maps ( R ( P j ,v j ) ,τ ) v j  = v α to R ( H \ ( P α − v α ) ,v α ) ,τ for an y boundary condition τ on S H ( v , ℓ ) suc h that S H ( v , ℓ ) does not contain vertices of T H . Let c ∈ (0 , 1) b e a constant such that | f ′ 1 ,λ ( x 0 ( λ )) | < c for all λ ∈ [0 , λ ∗ ], where we recall that x 1 ( λ ) denotes the attracting fixed p oint of f 1 ,λ in R + . Since [0 , λ ∗ ] is compact, there exist uniform neighbourho o ds U λ = B r ( x 1 ( λ )) of x 1 ( λ ) for some fixed r > 0 for all λ ∈ [0 , λ ∗ ], such that for all x, y ∈ U λ , we ha v e f 1 ,λ ( x ) , f 1 ,λ ( y ) ∈ U λ and | f 1 ,λ ( x ) − f 1 ,λ ( y ) | < c | x − y | . (35) The compact in terv al I = [0 , λ ∗ ] is con tained in the basin B λ of the attracting fixed p oint x 1 ( λ ) of f 1 ,λ for every λ ∈ [0 , λ ∗ ]. By Remark 29 , there is a uniform num b er m 0 ∈ N of iterations of f 1 ,λ that maps the in terv al I into U λ , that is f m 0 1 ,λ ( I ) ⊆ U for all λ ∈ [0 , λ ∗ ]. DECA Y OF CORRELA TIONS AND ZEROS FOR THE HARD-CORE MODEL 25 Iterating the estimate ( 35 ), it follo ws: | f m 1 ,λ ( x ) − f m 1 ,λ ( y ) | < | x − y | c m    ∥ f ′ 1 ,λ ∥ m [0 ,λ ] c m , if m ≤ m 0 ∥ f ′ 1 ,λ ∥ m 0 [0 ,λ ] c m 0 , if m > m 0 for all x, y ∈ I , λ ∈ [0 , λ ∗ ], and m ≥ 0. Therefore, there exists a constant M 1 > 1 such that | f m 1 ,λ ( x ) − f m 1 ,λ ( y ) | < M 1 c m | x − y | (36) for all x, y ∈ I , λ ∈ [0 , λ ∗ ], and m ≥ 0. Since f 1 ,λ ([0 , ∞ ]) = [0 , λ ] ⊆ I , there moreo ver exists a constant M 2 > 1 such that | f m 1 ,λ ( x ) − f m 1 ,λ ( y ) | < M 2 c m (37) for all x, y ∈ [0 , ∞ ], λ ∈ [0 , λ ∗ ] and m ≥ 1. Note that if v j / ∈ S H ( v , ℓ ), then R ( P j ,v j ) ,τ is in the image f 1 ,λ ([0 , ∞ ]) = [0 , λ ] ⊆ I . So the input v alues of each F v ,T ,λ are actually from the compact I L T H . Since F v ,T ,λ ( x ) is con tinuously differen tiable in ( x, λ ) on a neigh b ourhoo d of I L T H × [0 , λ ∗ ] and the family { F v ,T ,λ } v ,T is finite (finitely man y subtrees T of T d k with finitely man y v ertices v of T ), there is a uniform b ound L > 1 on the sup-norm of the deriv ative F ′ v ,T ,λ for all v , T and λ ∈ [0 , λ ∗ ]. No w we can com bine this with the deca y of correlation along long paths: Let v b e a v ertex of H , ℓ > 2 k , and τ 1 , τ 2 b e tw o boundary conditions on S H ( v , ℓ ), and λ ∈ [0 , λ ∗ ]. W e again consider the three separate cases: (1) v ∈ T H . Then T H ⊆ B ℓ ( v ), since ℓ ≥ 2 k . F or eac h j , if S H ( v , ℓ ) ∩ P j = ∅ , then R ( P j ,v j ) ,τ 1 = R ( P j ,v j ) ,τ 2 . Otherwise, let { u j } = S H ( v , ℓ ) ∩ P j . Then by ( 37 ), w e ha ve | R ( P j ,v j ) ,τ 1 − R ( P j ,v j ) ,τ 2 | = | f m j 1 ,λ ( τ 1 ( u j )) − f m j 1 ,λ ( τ 2 ( u j )) | < M 2 c m j , where m j = d ( T H , u j ) > ℓ − 2 k . By the deriv ativ e b ound, we ha ve | R ( H,v ) ,τ 1 − R ( H,v ) ,τ 2 | < LM 2 c ℓ − 2 k . (2a) v ∈ P 1 and T H ⊆ B ℓ ( v ). Let m 1 = d ( v , T H ) and m 2 the minimal n umber of vertices of one of the paths P j , j ≥ 2 in B ℓ ( v ). Again, for each j , if S H ( v , ℓ ) ∩ P j = ∅ , then R ( P j ,v j ) ,τ 1 = R ( P j ,v j ) ,τ 2 . Otherwise, let { u j } = S H ( v , ℓ ) ∩ P j . Then by ( 37 ), w e hav e | R ( P j ,v j ) ,τ 1 − R ( P j ,v j ) ,τ 2 | = | f m j 1 ,λ ( τ 1 ( u j )) − f m j 1 ,λ ( τ 2 ( u j )) | < M 2 c m 2 , where m j = d ( T H , u j ) > m 2 . By the deriv ative b ound, w e then ha ve | R ( H \ ( P 1 − v 1 ) ,v 1 ) ,τ 1 − R ( H \ ( P 1 − v 1 ) ,v 1 ) ,τ 2 | < M 2 Lc m 2 . Let v ′ , v ′′ b e the neighbours of v , where v ′ is the neighbour of v on the path from v to T H and let T ′ , T ′′ b e the corresp onding comp onen ts of H − v . Then, by ( 36 ), w e hav e | R ( T ′ ,v ′ ) ,τ 1 − R ( T ′ ,v ′ ) ,τ 2 | < M 1 c m 1 · M 2 Lc m 2 < 2 M 1 M 2 Lc ℓ − 2 k and by the same arguments as in the previous case: | R ( T ′′ ,v ′′ ) ,τ 1 − R ( T ′′ ,v ′′ ) ,τ 2 | < 2 M 2 c ℓ < M 1 M 2 Lc ℓ − 2 k . The final ratio R ( H,v ) ,τ j is obtained from the t wo ratios ab o v e through the rational map Φ λ ( x, y ) = λ (1+ x )(1+ y ) , so | R ( H,v ) ,τ 1 − R ( H,v ) ,τ 2 | < L ′ M 1 M 2 Lc ℓ − 2 k where L ′ > 1 is a b ound on the deriv ative of Φ λ on I 2 that is uniform in λ ∈ [0 , λ ∗ ]. 26 HAN PETERS, JOSIAS REPPEKUS, AND GUUS REGTS (2b) v ∈ P 1 and T H ⊆ B ℓ ( v ). Then there is a path of length at least ℓ − 2 k from v to T H and by ( 37 ), we ha v e | R ( T ′ ,v ′ ) ,τ 1 − R ( T ′ ,v ′ ) ,τ 2 | < M 2 c ℓ − 2 k . No w we can contin ue as in the previous case. In conclusion, there exists a constan t M k > 0 such that for ℓ > 2 k , we ha v e | R ( H,v ) ,τ 1 − R ( H,v ) ,τ 2 | < M k c ℓ for all H , v , τ 1 , τ 2 , and λ as required. □ R emark 37 . The results in this section can b e generalized b y replacing the paths (trees of down-degree 1) b y trees of arbitrary down-degree d 2 with 1 < d 2 < ∆ − 1. In this case the same argumen ts imply that zeros accumulate at λ c (∆), while φ -VSSM holds for λ < λ c (∆ 2 ) > λ c (∆), where ∆ 2 = d 2 + 1. The failure of φ -VSSM for λ > λ c (∆ 2 ) follows from the emergence of the attracting 2-cycle of f d 2 ,λ (see Section 5.1 ), which do es not o ccur for d 2 = 1. 6. Concluding remarks and open questions Here w e state tw o questions that arise naturally from our w ork, and discuss p ossible approac hes for these questions. The first question concerns a p ossible con verse to Theorem 8 . Question 1. L et G b e a family of b ounde d de gr e e gr aphs that is close d under taking induc e d sub gr aphs, and let λ ⋆ > 0 . Supp ose that ther e exists an op en set U ⊂ C that c ontains [0 , λ ⋆ ] and such that for al l λ ∈ U and G ∈ G one has Z G ( λ )  = 0 . Do es it fol low that the har d-c or e mo del on G satisfies VSSM at λ ⋆ ? F ollowing [ Reg23 ] a natural w a y to approach this question would b e to try to sho w that the ratios R T SA W( G,v ) ,σ,v ( λ ) with σ : S G ( v , ℓ ) → { 0 , 1 } form a normal family on an op en set con taining [0 , λ ⋆ ] for G ∈ G and v ∈ V ( G ) and ℓ > 0. The zero-freeness assumption implies that the ratios R T SA W( G,v ) ,v ( λ ) form a normal family . It is ho w ever not ob vious ho w to deduce normality of the ratios R T SA W( G,v ) ,σ,v ( λ ) from this. In fact if we w ould argue by con tradiction and supp ose that R T SA W( G,v ) ,σ,v ( λ ) do not form a normal family near some giv en λ ∈ [0 , λ ⋆ ], then we would like to derive that R T SA W( G,v ) ,v ( λ ) is not normal at some λ ′ ∈ [0 , λ ⋆ ]. This is exactly what w e sho wed in our proof of Prop osition for the sp ecial case of the trees T d k , 1 m . Our pro of of this actually relies on sp ecific prop erties of the function f λ,d and do es not seem to generalize easily to the general setting. Our next question concerns the relation b etw een VSSM and SSM, and can b e seen as a v ariation of the first question. Question 2. L et G b e a family of b ounde d de gr e e gr aphs that is close d under taking induc e d sub gr aphs and let λ ⋆ > 0 . If the har d-c or e mo del on G satisfies SSM at λ ⋆ , do es it also satisfy VSSM at λ ⋆ ? If the answ er to this question is ‘y es’, then this w ould also imply a p ositiv e answer to the previous question b y [ Reg23 ]. 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