aa r X i v : . [ m a t h - ph ] A ug ANALYTICITY FOR CLASSICAL GASSES VIA RECURSION
MARCUS MICHELEN AND WILL PERKINS
Abstract.
We give a new criterion for a classical gas with a repulsive pair potential toexhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure.Our improvement on the bound for analyticity is by a factor e over the classical clusterexpansion approach and a factor e over the known limit of cluster expansion convergence.The criterion is based on a contractive property of a recursive computation of the densityof a point process. The key ingredients in our proofs include an integral identity for thedensity of a Gibbs point process and an adaptation of the algorithmic correlation decaymethod from theoretical computer science. We also deduce from our results an improvedbound for analyticity of the pressure as a function of the density. Introduction
A central goal in classical statistical mechanics is to derive macroscopic properties of gasses,liquids, and solids given only details of microscopic interactions. The classical model involvesindistinguishable particles interacting in the continuum via a many-body potential. Even inthe case of a pair potential fundamental questions remain open, such as whether or not sucha model exhibits a phase transition from a gaseous to a solid state. One major success ofthe field is in the study of convergent expansions for thermodynamic quantities in powers ofthe activity or density; using these techniques, the model is proved to be in a gaseous statethroughout the region of convergence of these series. However, convergence can be limited bythe presence of non-physical singularities in the complex plane. Here, in the case of repulsivepair potentials, we extend the known range of the gaseous phase using analytic techniquesthat avoid such singularities and focus on small complex neighborhoods of the positive realaxis.1.1.
The model.
We consider classical particles interacting in a finite volume Λ ⊂ R d via asymmetric, translation invariant pair potential φ .The energy of a configuration { x , . . . , x n } ⊂ R d is given by U ( x , . . . , x n ) = X ≤ i The grand canonical partition function at activity λ ≥ ⊂ R d isgiven by:(1) Z Λ ( λ ) = 1 + X k ≥ λ k k ! Z Λ · · · Z Λ e − U ( x ,...x k ) dx · · · dx k . (Without losing any generality, we absorb the usual inverse temperature parameter β intothe potential φ ).If we let Λ V be the axis-parallel box of volume V centered at the origin in R d , then theinfinite volume pressure is p φ ( λ ) = lim V →∞ V log Z Λ V ( λ ) . See e.g. [34] for a proof of the existence of the limit. Non-analytic points of p φ ( λ ) on thepositive real axis mark phase transitions of the infinite volume system. An important topic inclassical statistical physics is proving the absence of phase transitions for certain parametervalues; that is, proving analyticity of the pressure. We remark that proving the existenceof a phase transition in the models considered here is a notoriously challenging problem;to the best of our knowledge it is not proved for any monatomic gas interacting through afinite-range or rapidly decaying pair potential (see [42, 21] for proofs of phase transitions incontinuous particle systems with different types of interactions). For more background onclassical gasses see [34].1.2. Main results. Our main result is a new criterion for analyticity of the pressure anduniqueness of the infinite volume Gibbs measure. Theorem 1. Let φ be a repulsive, tempered potential and let (2) C φ = Z R d − e − φ ( x ) dx . Then for λ ∈ [0 , e/C φ ) the infinite volume pressure p φ ( λ ) is analytic and there is a uniqueinfinite volume Gibbs measure. This improves by a factor e the classical bound of 1 / ( eC φ ) obtained by Groeneveld in 1962using the cluster expansion [14]. Extensions by Penrose [28] and Ruelle [33] a year later toa wider class of potentials via the Kirkwood–Salsberg equations match Groeneveld’s boundin the case of repulsive potentials. In a remarkable but little noticed paper, Meeron [26]proved uniqueness and analyticity for λ < /C φ using a novel interpolation scheme [25] andrecurrence relations for k -point densities related to the Kirkwood–Salsberg equations.New criteria for cluster expansion convergence have been given by Faris [9], Jansen [18],and Nguyen and Fern´andez [27] based on the work of Fern´andez and Procacci in discretesetting [10] and the extension by Fern´andez, Procacci, and Scoppola to hard spheres [11].Explicit improvements to the classical bounds from the new criteria have been worked out inthe case of hard spheres in dimension 2 (see Section 1.2.1 below).Moreover, 1 /C φ is an upper bound on the radius of convergence of the cluster expansionfor repulsive potentials [14, 28] (see remark 3.7 in [27]). Theorem 1 surpasses this limit (andMeeron’s bound) by a factor e . NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 3 Using Theorem 1 we can also deduce that the pressure is analytic with respect to theinfinite volume density defined by ρ φ ( λ ) := λ ddλ p φ ( λ ) . Theorem 2. For any repulsive, tempered potential φ , the infinite volume pressure p φ isanalytic as a function of the density ρ φ for ρ φ ∈ [0 , e e C φ ) . Previous results showed analyticity of the pressure as a function of the density by showingconvergence of the virial series in a disk around 0 in the complex plane [22, 30, 19, 27]. Thebest bound on convergence of the virial expansion for general repulsive potentials is thatproposed by Groeneveld [15] and proven by Ramawadh and Tate [31], showing convergencefor | ρ φ | ≤ . C φ .1.2.1. Example: the hard sphere model. One of the most studied classical gasses is the hardsphere model, with the potential φ ( x ) = + ∞ if k x k < r and 0 otherwise. A configuration( x , . . . , x k ) with U ( x , . . . , x k ) = 0 represents the centers of a packing of k spheres of radius r/ 2. This model provides a good testing ground for different criteria for uniqueness andanalyticity.For convenience, let us take r to be the radius of the ball of volume 1 in R d . Then theclassical Groeneveld–Penrose–Ruelle bound gives convergence of the cluster expansion anduniqueness for λ < /e ≈ . d = 2 their improved bound is λ < . λ < o d (1). In dimension 2, using the ‘high-confidence’ bound for Poisson-Booleanpercolation from [1], this approach gives uniqueness for λ < . r of the potential.Recently, Helmuth, Petti, and the second author [16] improved these bounds in all dimen-sions by showing uniqueness and exponential decay of correlations in the hard sphere modelfor λ < Corollary 3. Consider the hard sphere model with spheres of radius r/ where r is the radiusof the ball of volume in R d . Then for λ < e the pressure is analytic and there is a uniqueinfinite-volume Gibbs measure. Theorem 2 also gives an improvement on the bounds for analyticity of the pressure as afunction of the density of the hard sphere model. We write these bounds in terms of thepacking density of the model, which given our choice of normalization is 2 − d ρ φ ( λ ) since thevolume of the ball of radius r/ − d . Corollary 4. The infinite volume pressure of the hard sphere model in dimension d is analyticas a function of the density for packing densities at most e e − d . Moreover, for any ε > MARCUS MICHELEN AND WILL PERKINS and d large enough, the pressure of the hard sphere model in dimension d is analytic forpacking densities at most (1 − ε )2 − d . For instance, in dimension 2 this gives analyticity up to packing density . / . . − d is a natural barrier to analysis since below 2 − d free volume (space inwhich new centers can be placed) is guaranteed by a union bound. Improving the asymptoticbound in Corollary 4 by any constant factor would likely require significant new insight.1.3. Absence of zeros in the complex plane. To prove Theorem 1 we will work in thesetting of a multivariate, complex-valued partition function. Given any bounded, measurablefunction λ : Λ → C , we define(3) Z Λ ( λ ) = X k ≥ k ! Z Λ k k Y i =1 λ ( x i ) · e − U ( x ,...,x k ) dx · · · dx k . When λ is constant this definition coincides with (1).Following the Lee-Yang theory of phase transitions [43], absence of phase transitions andanalyticity of the pressure for activities in [0 , λ ] is closely related to the existence of a region R in the complex plane containing the segment [0 , λ ] so that for any λ ∈ R and any boundedregion Λ, Z Λ ( λ ) = 0. Theorem 1 thus follows, after a little complex analysis in Section 5,from the next theorem. Theorem 5. Let φ be a repulsive, tempered potential and suppose λ ∈ (0 , e/C φ ) . Then thereexist ε > and C > , so that the following holds. Let L be the ε -neighborhood of theinterval [0 , λ ] in the complex plane. Then for any measurable function λ : R d → C so that λ ( x ) ∈ L for all x , and any bounded, measurable Λ ⊂ R d , we have | log Z Λ ( λ ) | ≤ C | Λ | . In particular, Z Λ ( λ ) = 0 . Outline of techniques. The classical approach to proving absence of phase transi-tions, convergence of the cluster expansion, is limited by the possible presence of a complexzero of the partition function far from the positive real axis which determines the radius ofconvergence of the cluster expansion but does not affect the physical behavior of the system.In fact, for repulsive gasses, the closest singularity of the pressure to the origin lies on thenegative real axis [14]. Therefore to move beyond the limits of cluster expansion convergenceone must use properties of positive activities or utilize regions of the complex plane that arenot symmetric around 0. Two previous approaches in this direction are probabilistic: dis-agreement percolation [6, 17, 4] and Markov chain mixing [16]. In the case of the hard spheremodel these techniques surpass the bounds for uniqueness given by the cluster expansion.One drawback is that these arguments rely on a finite-range property of a potential (like hardspheres) and it is not clear how to extend these arguments to potentials satisfying the morenatural temperedness assumption.Our approach to proving Theorem 1 will instead be analytic, using essentially no prob-abilistic tools. One consequence of the convergence of the cluster expansion is that Z Λ ( λ ) NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 5 is not zero for λ in a disk in the complex plane, uniformly in the region Λ. To avoid thesingularity on the negative real axis, we instead prove that Z Λ ( λ ) is not zero in an asymmet-ric region in the complex plane that contains [0 , e/C φ ). We obtain this zero-free region byproving recursive bounds on the log partition function and on a generalization of the densityof a point process to complex parameters.Our approach is inspired by the ‘correlation decay method’, an algorithmic technique fromtheoretical computer science for approximating the partition function of a discrete spin model.The method was introduced by Weitz [40] in an influential paper on approximate countingand sampling from the hard core model on a graph G . At the core of Weitz’s argument isa recursion for the marginal probability that a given vertex v is chosen in the independentset in terms of the marginal probabilities of its neighbors in a modified model that removesthe dependence between neighbors of v . This ultimately allows Weitz to construct a ‘self-avoiding walk tree’ T with root r so that the marginal of v in G equals the marginal of r in T . A monotonicity argument then shows that if the infinite ∆-regular tree exhibitsstrong spatial mixing at activity λ then every graph of maximum degree ∆ does as well. Thethreshold for strong spatial mixing on the ∆-regular tree can be explicitly determined viafixed point equations as λ c (∆) = (∆ − ∆ − (∆ − ∆ . Consequences include an efficient algorithm forapproximating the partition function on any graph of maximum degree ∆ for λ < λ c (∆),and a lower bound on the uniqueness threshold of the hard-core model on Z d of λ c (2 d ). Thecorrelation decay method has since been refined and used to prove the best current lowerbounds on the uniqueness threshold for the hard-core lattice gas model on Z [39, 37].Most relevant for our approach is the paper of Peters and Regts [29] in which they use arecursion inspired by Weitz and ideas from complex dynamical systems to prove the existenceof a zero-free region for the independence polynomial of graphs of maximum degree ∆ in acomplex neighborhood containing [0 , λ c (∆)), thus solving a conjecture of Sokal [38]. Theirwork was in part motivated by another approach to approximate counting, the polynomialinterpolation method of Barvinok [3].Our goal in this paper is to develop a version of the correlation decay method for continuousparticle systems. This presents several challenges including determining the right analogueof self-avoiding walk trees and recurrence relations for point processes.We work with a generalization of the definition of the density of a point process to complex-valued activities (given in (4) below and used already in [33]). We develop three tools to workwith these densities. The first is Lemma 7, an integral expression for log Z Λ ( λ ) in terms ofthese generalized densities. This is what allows us to bound log Z Λ ( λ ) and prove the absenceof zeroes in a region in the complex plane. The next is Theorem 8 which gives a recursiveintegral identity for complex densities. This is somewhat analogous to one step of the self-avoiding walk tree construction of Godsil [13] and Weitz [40] in discrete settings. Finally ourconvergence criteria is determined by the contractive properties of the functional that definesthis recursive identity given in Lemma 12.At a very high level, our approach has some similarity to the approach of Penrose andRuelle via the Kirkwood-Salsberg equations and that of Meeron [26]: we write identitiesinvolving densities and show that for a certain range of complex parameters the operatordefining these identities is contractive in a suitable sense. In Ruelle’s argument, uniquenessfollows from invertibility of 1 − λK where K is an operator on a Banach space. When thenorm of λK is strictly less than 1 then invertibility, and thus uniqueness, follows. Withouta deeper understanding of the spectrum of this operator K , this approach inherently only MARCUS MICHELEN AND WILL PERKINS provides uniqueness for λ in a disk centered at 0. Conversely, we work entirely with values of λ near the positive interval [0 , e/C φ ) which allows us to avoid the non-physical obstructionsto analyticity on the negative real axis. Meeron’s interpolation between zero interaction andfull interaction also avoids the singularity on the negative real axis, but the operator defininghis recursion is contractive only for λ < /C φ ; it would be interesting to see if combininghis approach with a change of coordinates as in Section 3 below could match the bound ofTheorem 1. One could view successive applications of Theorem 8 below as a non-uniforminterpolation from activity 0 to activity λ .1.5. Future directions. While we work here only with repulsive potentials, we would bevery interested to see if the results could be generalized to the class of stable, temperedpotentials, the setting of the results of Penrose [28] and Ruelle [33] which include morephysically relevant examples such as the Lennard–Jones potential. For a discussion from thephysics perspective of the utility of purely repulsive potentials such as the hard sphere mode,see [41, 2].When specialized to real parameters, the functional contraction properties given in Sec-tion 3 can be used to show that a recursive computation of the density exhibits exponentiallysmall dependence on boundary conditions in the height of the recursion. This is turn canbe used to show strong spatial mixing for finite-range, repulsive potentials. In light of theseproperties and the algorithmic roots of our techniques, it would be interesting to use thesemethods to design efficient algorithms for estimating the pressure or density when λ < e/C φ .We believe that one could take advantage of the geometry of low dimensional Euclideanspace (e.g. d = 2 , 3) and an appropriate notion of the connective constant of R d to improvethe bounds of Theorem 1, as was done for the hard-core lattice gas by Sinclair, Srivastava,ˇStefankoviˇc, and Yin [37]. Doing this for the hard sphere model in dimension 2 would testthe limits of this method and the analogy with discrete spin systems.1.6. Notation. A region is a bounded, measurable subset of R d . The volume of Λ ⊂ R d isdenoted | Λ | . We denote by dist( x, y ) the Euclidean distance between x, y ∈ R d . An activityfunction on a region Λ is a bounded, measurable function from Λ → C . We use a boldsymbol, e.g. λ , for a non-constant activity function. Throughout, a complex neighborhood is a bounded open subset of C ; in each case here, the complex neighborhoods in questionare conformal images of the open unit disk, and thus are simply connected as well. Forpositive parameters λ and ε , we write N ( λ, ε ) to be the complex neighborhood { z ∈ C :dist( z, [0 , λ ]) < ε } . For a set A we write A to be the indicator function of the set A .2. Generalized densities To prove Theorem 5, we use a generalization of the density of a point process to complexactivity functions: the density of v ∈ Λ at activity λ is(4) ρ Λ , λ ( v ) = λ ( v ) Z Λ ( λ e − φ ( v −· ) ) Z Λ ( λ )if Z Λ ( λ ) = 0. If Z Λ ( λ ) = 0 then the density is undefined. Here λ e − φ ( v −· ) : Λ → C denotesthe function λ ( x ) e − φ ( v − x ) . NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 7 In Section 5 we will also work with multipoint densities: the k -point density (or k -pointcorrelation function) of ( v , . . . , v k ) ∈ Λ k at activity λ is(5) ρ Λ , λ ( v , . . . , v k ) = λ ( v ) · · · λ ( v k ) Z Λ ( λ e − P ki =1 φ ( v i −· ) ) Z Λ ( λ ) e − U ( v ,...,v k ) . Remark. There is a natural probabilistic interpretation of the densities when λ is non-negative. First define the Gibbs point process X to be the random point set in Λ withdensity e − U ( · ) against the Poisson process with intensity λ . Then the density ρ Λ , λ is thedensity of the measure that assigns to a set A the mass E [ | X ∩ A | ] with respect to Lebesguemeasure; a similar fact holds for the multipoint density with E | X ∩ A | replaced by a certainfactorial moment. When λ is non-negative, these may be taken as a definition of the density ρ Λ , λ and (4) and (5) become identities to check (see, e.g. [34, Chapter 4]). Since we areinterested in complex values of λ , we use (4) and (5) as our definition. This idea appears inRuelle’s classic text on statistical mechanics [34, Chapter 4] as well, in which this identityappears in a slightly different form to extend the definition of ρ Λ ,λ to complex λ . Whereour definition (4) differs from Ruelle, however, is the use of a “multivariate” λ rather thanconstant λ which allows us to write ρ Λ , λ as λ times a ratio of partition functions. This turnsout to be a crucial feature of (4) and essentially allow us to prove zero-freeness of Z Λ ( λ )inductively (see Theorem 19).In what follows in Sections 2, 3,and 4 the region Λ will be fixed, and so we will dropthe subscript Λ from the notation, writing Z ( λ ) for Z Λ ( λ ) and ρ λ ( x ) for ρ Λ , λ ( x ). We willinterpret an activity function λ on Λ as a function λ : R d → C that is 0 on R d \ Λ. In fact,Lemma 7 and Theorem 8 below hold for bounded, integrable functions λ : R d → C withoutrequiring bounded support. We call such functions activity functions .Given an activity function λ , let A λ = { λ ′ = α · λ : α : R d → [0 , 1] is measurable } . Wewill use the following hereditary notion of zero freeness. Definition 6. An activity function λ is totally zero-free if Z ( λ ′ ) = 0 for all λ ′ ∈ A λ . Our first lemma is an integral identity for the log partition function. Lemma 7. If an activity function λ is totally zero-free then log Z ( λ ) = Z R d ρ ˆ λ x ( x ) dx where ˆ λ x ( y ) = ( if y ∈ Λ x λ ( y ) if y / ∈ Λ x and Λ x = { y ∈ R d : dist(0 , y ) < dist(0 , x ) } .Proof. Define λ t ( y ) = dist( y, ≥ t λ ( y ) and note by assumption Z ( λ t ) = 0 for all t . We willapply the fundamental theorem of calculus and integrate ddt log Z ( λ t ). Since we want tocompute ddt log Z ( λ t ), we compute ddt Z ( λ t ); first change coordinates to write Z ( λ t ) − X j ≥ j ! Z ( R d ) j j Y i =1 λ t ( v i ) e − U ( v ,...,v j ) d v MARCUS MICHELEN AND WILL PERKINS = X j ≥ j ! Z ∞ Z ∂ ( B s (0) j ) j Y i =1 λ t ( v i ) e − U ( v ,...,v j ) d v ds = X j ≥ j ! Z ∞ j Z ∂B s (0) λ t ( w ) Z B s (0) j − j − Y i =1 λ t ( v i ) e − U ( v ,...,v j − ,w ) d v dw ds = X j ≥ j ! Z ∞ t j Z ∂B s (0) λ ( w ) Z B s (0) j − j − Y i =1 λ t ( v i ) e − U ( v ,...,v j − ,w ) d v dw ds and so we have ddt Z ( λ t ) = − X j ≥ j ! · j Z ∂B t (0) λ ( w ) Z ( R d ) j − j − Y i =1 λ t ( v i ) e − U ( v ,...,v j − ,w ) d v dw where the outermost integral is a d − t centered at 0. Rearranging the terms, we have ddt Z ( λ t ) = − Z ∂B t (0) λ ( w ) X j ≥ j ! Z ( R d ) j j Y i =1 (cid:16) λ t ( v i ) e − φ ( v i − w ) (cid:17) e − U ( v ,...,v j ) d v dw = − Z ∂B t (0) λ ( w ) Z ( λ t e − φ ( w −· ) ) dw . The fundamental theorem of calculus then giveslog Z ( λ ∞ ) − log Z ( λ ) = Z ∞ ddt log( Z ( λ t )) dt = − Z ∞ Z ∂B t (0) λ ( w ) Z ( λ t e − φ ( w −· ) ) Z ( λ t ) dw dt = − Z R d ρ ˆ λ w ( w ) dw . Noting Z ( λ ∞ ) = 1 and Z ( λ ) = Z ( λ ) completes the Lemma. (cid:3) Next we give a recursive identity for the densities. Fix an activity function λ . For v, w ∈ R d let λ v → w : R d → C be defined by λ v → w ( x ) = ( λ ( x ) e − φ ( v − x ) if dist( v, x ) < dist( v, w ) λ ( x ) if dist( v, x ) ≥ dist( v, w ) . In particular, λ v → w ∈ A λ . Theorem 8. Suppose an activity function λ is totally zero-free. Then for every v ∈ R d wehave (6) ρ λ ( v ) = λ ( v ) exp (cid:18) − Z R d ρ λ v → w ( w )(1 − e − φ ( v − w ) ) dw (cid:19) . Proof. For v fixed and each t ≥ λ t ( x ) = ( λ ( x ) e − φ ( v − x ) if dist( v, x ) < t λ ( x ) otherwise . NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 9 Note that λ t ∈ A λ , and so by assumption Z ( λ t ) = 0.It will be helpful to write λ t ( x ) = λ ( x )(1 − dist( v,x ) ≤ t (1 − e − φ ( v − x ) )). As in the proof ofLemma 7, we will apply the fundamental theorem of calculus to ddt log Z ( λ t ). We start bycomputing ddt Z ( λ t ). Write Z ( λ t ) = X k ≥ k ! Z ( R d ) k k Y j =1 (cid:16) λ ( x j )(1 − dist( v,x j ) ≤ t (1 − e − φ ( v − x j ) ) (cid:17) exp ( − U ( x , . . . , x k )) d x . Note that for each term in the product, we have ddt (1 − dist( v,x j ) ≤ t (1 − e − φ ( v − x j ) )) = − δ dist( v,x j ) ( t )(1 − e − φ ( v − x j ) )where δ dist( v,x j ) ( t ) is the Dirac delta function; equivalently, change coordinates to write Z ( λ t )as in the proof of Lemma 7 and use the fundamental theorem of calculus. By the productrule, this implies ddt Z ( λ t ) = − X k ≥ k ! Z ( R d ) k − Z ∂B t ( v ) k − Y j =1 λ t ( x j ) e − U ( x ) (1 − e − φ ( w − v ) ) · k λ ( w ) e − P k − j =1 φ ( x j − w ) d x dw = − Z ∂B t ( v ) Z ( λ t e − φ ( w −· ) )(1 − e − φ ( w − v ) ) dw . By the fundamental theorem of calculus, we then havelog( Z ( λ ∞ )) − log( Z ( λ )) = Z ∞ ddt Z ( λ t ) Z ( λ t ) dt = − Z ∞ Z ∂B t ( v ) λ ( w ) Z ( λ t e − φ ( w −· ) ) Z ( λ t ) (1 − e − φ ( w − v ) ) dw dt = − Z R d ρ λ v → w ( w )(1 − e − φ ( v − w ) ) dw . Noting λ ∞ = λ e − φ ( v −· ) and λ = λ and applying (4) completes the proof. (cid:3) Finally we need two continuity statements. Lemma 9. For any region Λ and any M > , the map λ Z Λ ( λ ) is uniformly continuousin the sup norm on the set of activity functions λ with | λ ( x ) | ≤ M for all x ∈ Λ . To prove this we need the following elementary lemma that appears in [7]; we reproducethe simple proof for completeness: Lemma 10. Let { z j } nj =1 and { w j } nj =1 be complex numbers with | z j | , | w j | ≤ θ for all j . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 z j − n Y j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ θ n − n X j =1 | w j − z j | . Proof. Calculate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 z j − n Y j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z n n − Y j =1 z j − z n n − Y j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z n n − Y j =1 w j − w n n − Y j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − Y j =1 z j − n − Y j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + θ n − | z n − w n | and induct on n . (cid:3) Now we prove Lemma 9. Proof of Lemma 9. Suppose λ , λ ′ are bounded by M ≥ | λ ( x ) − λ ′ ( x ) | ≤ δ for all x ∈ Λ. Then | Z Λ ( λ ) − Z Λ ( λ ′ ) | ≤ X k ≥ k ! Z Λ k e − U ( x ,...,x k ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 λ ( x i ) − k Y i =1 λ ′ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx k ≤ X k ≥ k ! Z Λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 λ ( x i ) − k Y i =1 λ ′ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx k ≤ X k ≥ k ! | Λ | k M k kδ ≤ δ | Λ | M e | Λ | M , and so Z Λ ( · ) is uniformly continuous. (cid:3) Lemma 11. For any region Λ , if an activity function λ on Λ is bounded and totally zero-freethen for every v ∈ Λ , ρ λ ′ ( v ) is uniformly continuous in λ ′ in a neighborhood of λ where themodulus of continuity is uniformly bounded as v varies.Proof. Since λ is totally zero-free, by the uniform continuity of Z , λ ′ is totally zero-free and ρ λ ′ ( v ) is well defined for λ ′ in a neighborhood of λ in the sup norm. Since Z Λ ( λ ) = 0, byLemma 9 there is some ε > | Z Λ ( λ ′ ) | is uniformly bounded away from 0 for λ ′ so that k λ − λ ′ k ∞ < ε . Another application of Lemma 9 shows that both Z Λ ( λ ′ ) and Z Λ ( λ ′ e − φ ( v −· ) )are uniformly continuous for k λ − λ ′ k ∞ < ε and so ρ λ ′ ( v ) = λ ′ ( v ) Z Λ ( λ ′ e − φ ( v −· ) ) Z Λ ( λ ′ ) is uniformlycontinuous in this neighborhood. (cid:3) Complex contraction For λ ∈ C , and ρ : R d → C bounded and integrable, define F ( λ, ρ ) = λ exp (cid:18) − Z R d ρ ( w )(1 − e − φ ( w ) ) dw (cid:19) . This function arises on the right hand side of the identity in (6). We will show that it iscontractive in an appropriate sense on a complex domain. This contractive property is similarto that shown for a different function in the case of discrete two-spins systems in [29, 35].For a positive reals s, ε we let N ( s, ε ) = { y ∈ C : ∃ x ∈ [0 , s ] , | y − x | < ε } and N ( s, ε ) = { y ∈ C : ∃ x ∈ [0 , s ] , | y − x | ≤ ε } . Lemma 12. For every λ ∈ (0 , e/C φ ) there exists ε > and complex neighborhoods U ⊂ U so that [0 , e/C φ ] ⊂ U with U ⊂ U so that the following holds. If λ ∈ N ( λ , ε ) and ρ ( x ) ∈ U for all x ∈ R d , then F ( λ, ρ ) ∈ U . NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 11 We will prove Lemma 12 in a sequence of steps. We will apply a change of coordinates inorder to make F a contraction mapping, as was done in the discrete setting in, e.g. [32, 23,29, 35]. With this in mind, set ψ ( x ) := log(1 + C φ x ). For a function z : R d → [0 , log(1 + e )],define g λ ( z ) := ψ ( F ( λ, ψ − ( z ))). First, we will consider only constant functions z ≡ z ; ourfirst main step is to show that | g ′ λ ( z ) | ≤ z and all λ ∈ [0 , e/C φ ]. Thus, by definition g λ ( z ) = log(1 + C φ λe · e − e z )and so g ′ λ ( z ) = − C φ λe · e z · e − e z C φ λe · e − e z . Lemma 13. For all z ≥ we have | g ′ e/C φ ( z ) | ≤ .Proof. Set r = e z ∈ [1 , ∞ ) and note that then we may write h ( r ) := | g ′ e/C φ ( r ) | = e re − r e e − r . Compute h ′ ( r ) = e e − r ( e − r − r + 1)(1 + e − r ) which has a unique zero at r = 2, showing that h ( r ) ≤ h (2) = 1. Noting h ( r ) ≥ (cid:3) Now we can show that for λ < e/C φ we have that | g ′ λ | is strictly less than 1, implying that g λ is a contraction. Lemma 14. For each λ ∈ [0 , e/C φ ) there exists δ > so that for all λ ∈ [0 , λ ] , we have max z ≥ | g ′ λ ( z ) | ≤ − δ . Proof. Set ε = e/C φ − λ > 0. Set β = C φ eλ so that β ∈ [0 , (1 − ε ) e ]. Assume β > 0, asotherwise g ′ λ = 0. Write max z ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ′ λ ( z ) g e/C φ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max z ≥ e − + e − e z β − + e − e z = max z ≥ (cid:18) − β − − e − β − + e − e z (cid:19) = 1 − β − − e − β − + e . Choosing δ := min β ∈ [0 , (1 − ε ) e ] 1 − βe − βe completes the proof. (cid:3) We now extend the definition of g λ ( z ) to take complex values of λ and complex-valuedfunctions z . Our next lemma is for the special case of constant, real z . Lemma 15. Fix λ ∈ [0 , e/C φ ) . Then there exists an M > so that for all ε , ε > sufficiently small we have max λ ∈N ( λ ,ε ) ,z ∈ [ − ε , log(1+ e )] (cid:12)(cid:12)(cid:12)(cid:12) ddλ g λ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M . Proof. This follows from computing ddλ g λ ( z ) = C φ e · e − e z C φ λe · e − e z . (cid:3) We now prove a version of Lemma 12 for the case of z ≡ z : Lemma 16. Fix λ ∈ [0 , e/C φ ) . There exists ε , ε > and ε ∈ (0 , ε ) so that for any λ ∈ N ( λ , ε ) and z ∈ N (log(1 + e ) , ε ) we have g λ ( z ) ∈ N (log(1 + e ) , ε ) .Proof. First note from Lemma 14, there is a δ > | g ′ λ ( z ) | ≤ − δ for all λ ∈ [0 , λ ]and z ∈ [0 , log(1 + e )]. Since g ′ λ is continuous in z and λ , we may take ε , ε sufficiently smallso that(7) | g ′ λ ( z ) | ≤ − δ/ , for all ( z, λ ) ∈ N (log(1 + e ) , ε ) × N ( λ , ε ) . To see this, note that for each λ ∈ [0 , λ ] and z ∈ [0 , log(1 + e )] we may find a neighborhoodof ( λ, z ) for which the above holds, thereby giving an open cover of [0 , λ ] × [0 , log(1 + e )]; bycompactness, we may reduce to a finite cover, thereby giving ε , ε .By Lemma 15, we may take ε small enough so that(8) max λ ∈N ( λ ,ε ) ,z ∈ [ − ε , log(1+ e )] (cid:12)(cid:12)(cid:12)(cid:12) ddλ g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ δε ε . For λ ∈ N ( λ , ε ) and z ∈ N (log(1 + e ) , ε ), find z ′ ∈ [0 , log(1 + e )] with | z − z ′ | ≤ ε and λ ′ ∈ [0 , λ ] with | λ − λ ′ | ≤ ε . Using the bounds (7) and (8), bound | g λ ( z ) − g λ ′ ( z ′ ) | ≤ | g λ ( z ) − g λ ( z ′ ) | + | g λ ( z ′ ) − g λ ′ ( z ′ ) | < (1 − δ/ ε + ε δ/ ε (1 − δ/ ε . Since g λ ([0 , log(1 + e )]) ⊂ [0 , log(1 + e )] we have g λ ( z ) ∈ [0 , log(1 + e )], thereby completingthe proof. (cid:3) To generalize Lemma 16 to non-constant z , we will use a convexity argument. We needthe following fact that appears in [29, Proof of Lemma 4 . Fact 17. For r > and ε sufficiently small, the image of N ( r, ε ) under the map z e z isconvex. This allows us to generalize Lemma 16 almost immediately. Lemma 18. Fix λ ∈ [0 , e/C φ ) . There exists ε , ε > and ε ∈ (0 , ε ) so that for any λ ∈ N ( λ , ε ) and any z so that z ( x ) ∈ N (log(1 + e ) , ε ) for all x , we have g λ ( z ) ∈ N (log(1 + e ) , ε ) .Proof. We will show that there is some z ∈ N (log(1 + e ) , ε ) so that g λ ( z ) = g λ ( z ). Thiswould follow if we find such a z so that C φ e z = Z R d e z ( w ) (1 − e − φ ( w ) ) dw . NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 13 Set α w := (1 − e − φ ( w ) ) /C φ , and note that α w ≥ R α w = 1; in particular, α w dw is aprobability measure. Fact 17 implies there is some such z so that e z = R e z ( w ) α w dw . Thenwe can apply Lemma 16 to complete the proof. (cid:3) Proof of Lemma 12. Find ε , ε and ε according to Lemma 18. Then we claim we may take ε = ε and U = ψ − ( N (log(1+ e ) , ε )) , U = ψ − ( N (log(1+ e ) , ε )). The only thing to checkis that U j are open and bounded, and the boundaries of N (log(1 + e ) , ε j ) are mapped to theboundaries of their inverse image under ψ ; since ψ − is conformal these properties hold. (cid:3) Zero freeness In this section we use the tools from the previous sections to prove Theorem 5. It iscontained in the following stronger theorem. Theorem 19. For every λ ∈ (0 , e/C φ ) there exists ε > and C > so that the followingholds for every region Λ ⊂ R d . Let λ be an activity function on Λ and suppose λ ( x ) ∈ N ( λ , ε ) for all x ∈ Λ . Then | log Z Λ ( λ ) | ≤ C | Λ | . In particular, Z Λ ( λ ) = 0 . Furthermore, for every v ∈ Λ , | ρ λ ( v ) | ≤ C .Proof. Fix Λ. Fix λ and let ε > U , U be as guaranteed in Lemma 12. Let C =max {| z | : z ∈ U } . Fix some λ so that λ ( x ) ∈ N ( λ , ε ) for all x . We will show that for all λ ′ ∈ A λ ,(9) ρ λ ′ ( v ) ∈ U for all v ∈ Λ . Towards this end let A ∗ ⊆ A λ be the subset of activity functions λ ′ for which (9) holdsfor all λ ′′ ∈ A λ ′ . In particular, if λ ′ ∈ A ∗ , then λ ′ is totally zero-free.First we observe that if λ ′ ∈ A ∗ , then since ˆ λ ′ x ∈ A λ ′ (where ˆ λ ′ x is as in Lemma 7), wehave | ρ ˆ λ t,x ( x ) | ≤ C . Applying Lemma 7 then gives that(10) | log Z ( λ ′ ) | ≤ Z Λ | ρ ˆ λ ′ x ( x ) | dx ≤ C | Λ | . Our goal is now to show that A ∗ = A λ , or equivalently that λ ∈ A ∗ . The identically 0activity function is in A ∗ since ρ ( v ) = 0. Now suppose for the sake of contradiction that A ∗ = A λ . Let λ be an arbitrary activity function in A λ \ A ∗ and for t ∈ [0 , 1] let λ t = t λ .We have 0 = λ ∈ A ∗ , λ / ∈ A ∗ , and λ t ∈ A ∗ ⇒ λ t ′ ∈ A ∗ if t ′ < t . We can then define t ∗ = sup { t : λ t ∈ A ∗ } and set λ ∗ = λ t ∗ . By the uniform continuity of Z ( · ) and ρ · ( x ) around λ ≡ t ∗ > t < t ∗ , λ t ∈ A ∗ and so by (10), | log Z ( λ t ) | ≤ C | Λ | . Since this is true for all t < t ∗ , by the uniform continuity of Z ( · ) we have that λ ∗ is totally zero-free and thus ρ · isuniformly continuous in a neighborhood of λ ∗ . This then shows λ ∗ ∈ A ∗ .Moreover, by uniform continuity again, we have that λ ′ is totally zero-free for all λ ′ in aneighborhood of λ ∗ in the sup norm and thus ρ λ ′ ( v ) is well defined for all x and all λ ′ in thisneighborhood.Now consider ρ λ ∗ ( x ) for an arbitrary x ∈ Λ. We know that for each w , ρ ( λ ∗ ) v → w ( w ) ∈ U since ρ ( λ ∗ ) v → w ∈ A λ ∗ and thus ρ ( λ ∗ ) v → w ∈ A ∗ . Then applying Theorem 8 and Lemma 12 we see that ρ λ ∗ ( v ) ∈ U . But then by the uniform continuity of ρ λ ∗ ( x ) in a neighborhood of λ ∗ and the fact that dist( U , U c ) > 0, we see that λ ′ ∈ A ∗ for all λ ′ in a neighborhood of λ ∗ .In particular, λ t ′ ∈ A ∗ for some t ′ > t ∗ , a contradiction. Thus we have that A ∗ = A λ asdesired.Now since λ ∈ A ∗ , applying (10) again gives | log Z ( λ ) | ≤ C | Λ | as desired. (cid:3) Proof of Theorem 1, Theorem 2, and Corollary 4 To prove analyticity of the pressure and uniqueness, we use Vitali’s Theorem (see, forinstance, [36, Theorem 6.2.8]): Theorem 20 (Vitali’s Convergence Theorem) . Let Ω ⊂ C be a complex neighborhood and f n a sequence of analytic functions on Ω so that | f n ( z ) | ≤ M for some M and for all n andall z ∈ Ω . If there is a sequence z m ∈ Ω with z m → z ∞ ∈ Ω so that lim n →∞ f n ( z m ) exists forall m , then f n converges uniformly on compact subsets of Ω to an analytic function f on Ω .Proof of Theorem 1. We first prove analyticity of the pressure. By Theorem 5, the finitevolume pressure p V ( λ ) := V log Z Λ V ( λ ) is an analytic function of λ in L and uniformlybounded. For λ ≥ p V ( λ ) converges as V → ∞ (see, e.g., [34]). Thus, by Vitali’s Theorem(Theorem 20), p V ( λ ) converges uniformly to an analytic function in L .As explained in [34, 18], to prove uniqueness of the infinite volume Gibbs measure, itis enough to show that for all k ≥ v , . . . , v k ) ∈ ( R d ) k , there exist functions ρ ∞ ,λ ( v , . . . , v k ) so that for all cofinal sequences of regions Λ n → R d ,(11) lim n →∞ ρ Λ n ,λ ( v , . . . , v k ) = ρ ∞ ,λ ( v , . . . , v k ) . This is true because when correlation functions satisfy the Ruelle bound (in the repulsivecase we have ρ Λ ,λ ( v , . . . , v k ) ≤ λ k ), the collection of k -point correlation functions determinesthe distribution of a point process. For λ ∈ C , | λ | < / ( eC φ ), (11) holds for all k ≥ v , . . . , v k ) ∈ ( R d ) k by [34, Theorem 4.2.3].Now write ρ Λ ,λ ( v , . . . , v k ) = λ k e − U ( v ,...,v k ) Z Λ ( λe − P ki =1 φ ( v i −· ) ) Z Λ ( λ )= λ k e − U ( v ,...,v k ) k Y j =1 Z λ ( λ j ) Z λ ( λ j − )= k Y j =1 ρ Λ , λ j ( v j )where λ j = λe − P ji =1 φ ( v i −· ) (in particular λ ≡ λ ). Then by (4) and Theorem 19 eachdensity ρ Λ , λ j ( v j ) is an analytic function of λ and uniformly bounded and so ρ Λ ,λ ( v , . . . , v k )is analytic and uniformly bounded in Λ. By Vitali’s Theorem, we then have that for every( v , . . . , v k ), ρ Λ n ,λ ( v , . . . , v k ) converges to an analytic function of λ in N ( e/C φ , ε ). By theidentity theorem for analytic functions, the limit must be unique since the limit is unique for | λ | sufficiently small (see [34]). (cid:3) NALYTICITY FOR CLASSICAL GASSES VIA RECURSION 15 To prove Theorem 2 we use a lower bound on the density as a function of λ for λ positive,closely related to an inequality of Lieb [24, Eq. (1.19)]. Fix the potential φ , and for a regionΛ, let ρ Λ ( λ ) = λ ddλ | Λ | log Z Λ ( λ ) be the finite volume density. Lemma 21. For any repulsive, tempered potential φ , any region Λ , and any λ ≥ , ρ Λ ( λ ) ≥ λ/ (1 + λC φ ) . The proof below is a generalization of that of [16, Lemma 18] for the case of hard spheres. Proof. We denote by X the random point set in Λ drawn from the Gibbs measure µ Λ ,λ , withdensity e − U ( · ) against the Poisson process of intensity λ on Λ. The finite volume density isthen | Λ | E | X | .We will use a simple consequence of inclusion–exclusion, that for each n and every sequenceof numbers x , . . . , x n , with x j ∈ [0 , 1] for all j , we have Q nj =1 (1 − x j ) ≥ − P nj =1 x j . Now since ρ Λ ( λ ) = λ Z ′ ( λ ) | Λ | Z ( λ ) we have ρ Λ ( λ ) = λ | Λ | Z Λ ( λ ) X k ≥ Z Λ k +1 λ k k ! Y ≤ i With this we can prove Theorem 2. Proof of Theorem 2. Fix λ ∈ [0 , e/C φ ). Then by Theorem 19, there is an ε > V log Z Λ V ( λ ) is analytic and bounded above for λ ∈ N ( λ , ε ); bythe proof of Theorem 1, the finite volume pressure converges uniformly on compact subsetsof N ( λ , ε ) and so its derivative with respect to λ (multiplied by λ ) converges uniformly oncompact subsets to the limit ρ φ ( λ ). In order to show that p φ may be taken as an analyticfunction of ρ φ , it is sufficient to show that λ may be taken as an analytic a function of ρ φ . To show this, we will use the inverse function theorem for analytic functions (e.g., [36,Theorem 2 . . ρ φ is an analytic function of λ , it is sufficient to show that ρ ′ φ ( λ ) = 0 for λ ∈ [0 , λ ]. This follows from, e.g., inequality (5) in [12] which gives a uniform lowerbound on the derivative of the finite volume density with respect to λ . This shows thatfor λ ∈ [0 , λ ] we have that ρ ′ φ ( λ ) = 0 and so ρ − φ is analytic in a complex neighborhood of[0 , ρ φ ( λ )]. By Lemma 21, this interval contains the interval h , λ λ C φ i . Sending λ → e/C φ proves analyticity of p φ in ρ φ in the interval h , e e C φ (cid:17) . (cid:3) To conclude we prove Corollary 4. Proof of Corollary 4. The first statement is an immediate consequence of Theorem 2. Thesecond statement follows by replacing the lower bound on the density from Lemma 21 withthe bound, specific to the hard sphere model, from [16, Theorem 19] (which in turn comesfrom [20]), which says that for λ fixed, the packing density in dimension d at fugacity λ is atleast (1 + o d (1)) W ( λ )2 − d , where W ( · ) is the W-Lambert function. In particular, W ( e ) = 1,which gives Corollary 4. 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