Central limit theorems, Lee-Yang zeros, and graph-counting polynomials
aa r X i v : . [ m a t h . C O ] A ug Central limit theorems, Lee-Yang zeros, andgraph-counting polynomials
J. L. Lebowitz , , B. Pittel , D. Ruelle , , and E. R. Speer Department of Mathematics, Rutgers University,Piscataway NJ 08854-8019 USA Department of Physics, Rutgers University,Piscataway NJ 08854-8019 USA Department of Mathematics, The Ohio State University,231 W. 18 th Avenue, Columbus, OH 43210 USA IHES, 91440 Bures sur Yvette, France
July 30, 2018
MSC: 05C30, 05C31, 05C80, 05A16, 60C05, 60F05, 82B05, 82B20Keywords: graph polynomial, grand canonical partition function, Lee-Yang,combinatorial, asymptotic enumeration, limit theorems
Abstract
We consider the asymptotic normalcy of families of random vari-ables X which count the number of occupied sites in some large set.We write Prob( X = m ) = p m z m /P ( z ), where P ( z ) is the generatingfunction P ( z ) = P Nj =0 p j z j and z >
0. We give sufficient criteria,involving the location of the zeros of P ( z ), for these families to satisfya central limit theorem (CLT) and even a local CLT (LCLT); the the-orems hold in the sense of estimates valid for large N (we assume thatVar( X ) is large when N is). For example, if all the zeros lie in theclosed left half plane then X is asymptotically normal, and when thezeros satisfy some additional conditions then X satisfies an LCLT. Weapply these results to cases in which X counts the number of edgesin the (random) set of “occupied” edges in a graph, with constraintson the number of occupied edges attached to a given vertex. Our re-sults also apply to systems of interacting particles, with X counting he number of particles in a box Λ whose size | Λ | approaches infinity; P ( z ) is then the grand canonical partition function and its zeros arethe Lee-Yang zeros. In this note we investigate the asymptotic normalcy of the number X ofelements in a random set M when the expected size of M is very large.We shall be concerned in particular with the case in which M is a randomset of edges, called occupied edges , in some large graph G , under certainrules which constrain the admissible configurations of occupied edges. Ouranalysis is however not restricted to such examples; in particular, it includesmany cases of interest in statistical mechanics, for which X is the number ofoccupied sites in some region Λ ⊂ Z d (or the number of particles in Λ ⊂ R d ).The probability that X = m is written asProb { X = m } := p m z m P ( z ) , (1.1)where P ( z ) := N X m =0 p m z m (1.2)is a polynomial of degree N and z is a strictly positive parameter; we willoften take z = 1. The coefficient p m will be, in the graph counting case,the number of admissible configurations of occupied edges of size m . Byconvention we take p m = 0 if m > N or m <
0. In some cases we willconsider P as the fundamental object of study and will then write X P and N P for X and N .A simple example is that in which a configuration is admissible if thenumber of occupied edges attached to each vertex v , d M ( v ), is zero or one.In this case the polynomial P ( z ) coincides with one of several definitions ofthe matching polynomial of the graph, properties of which have been studiedextensively in the graph theory literature. In particular, a local central limittheorem (see below) for X has been proved in the case z = 1 [12]. Ourprimary examples in this paper will be graph-counting polynomials , whicharise when the restriction d M ( v ) ∈ { , } discussed above is generalized to d M ( v ) ∈ C ( v ) for some set C ( v ); we will obtain a local central limit theoremfor X when C ( v ) = { , , } for all v .2he above examples are also natural objects of study in equilibrium sta-tistical mechanics; there one refers to the case with d M ( v ) ∈ { , } as asystem of monomers and dimers , and to that with d M ( v ) ∈ { , , } as asystem of monomers and unbranched polymers . In this setting one thinksof the edges belonging to M as occupied by particles, and the parameter z is then the fugacity of these particles. The restriction d M ( v ) ∈ C ( v )with C ( v ) = { , , . . . , c v } corresponds to hard core interactions betweenthe particles, and is a special or limiting case of a more general model forwhich a configuration M is assigned a Gibbs weight w M := e − βU ( M ) , with U ( M ) the interaction energy of M and β the inverse of the temperature, and p m := P { M || M | = m } w M . p m is then called the canonical partition function for m particles and P ( z ) the grand canonical partition function of the system.In this statistical mechanics setting the graph G is usually a subset of aregular lattice. For example, the vertices may be the sites of the lattice Z d which belong to some cubical box B = { , . . . , L } d ⊂ Z d , with edges, usuallycalled bonds, joining nearest-neighbor sites; one also considers such a boxwith periodic boundary conditions, in which an additional bond joins anypair of sites whose coordinate vectors differ in only one component, in whichthe values for the two sites are are 1 and L . Such a box contains | B | verticesand ∼ d | B | edges. The particles are most often thought of as occupying thesites of the lattice, that is, the vertices of the graph, but for our examplesthey occupy the bonds, as noted above. For the monomer-dimer problemon such a box B one would have N ∼ | B | /
2. Considering potentials U forthe periodic box which are translation invariant and sufficiently regular weare then in the usual situation for equilibrium statistical mechanics, see e.g.[27, 10].In the statistical mechanics setting there are many cases in which one canprove that E [ X ] ∼ c N and Var( X ) ∼ c N for some c , c > X satisfies a central limit theorem (CLT), that is, thatProb (cid:8) X ≤ E [ X ] + x p Var( X ) (cid:9) ∼ G ( x ) (1.3)when N → ∞ , where G ( x ) is the cumulative distribution function of thestandard normal random variable. A discussion of different proofs is givenin [10, p. 469]; most of these make use of the approximate independence ofdistant regions of Z d to write X as a sum of many approximately independentvariables, and do not extend directly to general graphs without any spatialstructure. See also [5] for a broad review of proof methods in the context3f combinatorial enumeration. Here, inspired by a proof due to Iagolnitzerand Souillard [16] in a statistical mechanics context, we prove a CLT thatrequires only that for large N there be no zeros of P ( z ) in some disc ofuniform size around z , and that Var( X ) grow faster than N / as N → ∞ .We describe the method in Section 2 and in Section 6 verify the variancecondition, and thus obtain a CLT, for the random variables associated witha class of graph-counting polynomials and for the particle number in somestatistical mechanical systems. We note here and will show later that whenthe zeros of P ( z ) lie in the left half plane it is sufficient for the CLT thatVar( X ) → ∞ as N → ∞ .Once one has a CLT for X , in the usual sense (1.3) of convergence ofdistributions, one would like also a local CLT (LCLT), that is, one would liketo show that for large N ,Prob { X = m } ∼ p π Var( X ) e − ( m − E [ X ]) / X ) . (1.4)If (1.4) holds for m belonging to some set S of integers then one speaks ofan LCLT on S , but in the cases we will consider we will prove an LCLTon all of Z . In the statistical mechanics setting such a result was estab-lished for certain systems in [7]; see also [10]. An LCLT for dimers on gen-eral graphs was given by Godsil [12], with a very different proof. EarlierHeilmann and Lieb [15] proved that all the zeros of the attendant match-ing polynomial P ( z ), whose coefficients p m enumerate incomplete matchings(monomer-dimer configurations) by the number m of edges (dimers), lie onthe negative real axis. Harper [14] was the first to recognize—in a particularcase of Stirling numbers—that such a property of a generating function P ( z )meant that the distribution of the attendant random variable is one of asum of independent, (0 , { p m } , undera constraint on the ground graph guaranteeing that the variance tends to in-finity. Significantly, since Heilmann-Lieb’s result and Menon’s theorem [23]implied log-concavity of { p m } , Godsil was able to prove the stronger LCLTby using the quantified version of Bender’s LCLT for log-concave distribu-tions [3] due to Canfield [4]. We refer the reader to Kahn [17] for severalnecessary and sufficient conditions under which the variance of the randommatching size tends to infinity, and to Pitman [24] for a broad range surveyof the probabilistic bounds when the generating function has real roots only.4ears later Ruelle [28] found that the polynomial P ( z ) whose coefficientsenumerate the unbranched subgraphs (2-matchings) of a general graph G hasroots in the left half of the z -plane, but not necessarily on the negative realline. Our key observation is that here again the related random variable X is,in distribution, a sum of independent random variables, this time each havinga 3-element range { , , } . Since the range remains bounded, a CLT forunbranched polymers follows whenever Var X goes to infinity with the degreeof P . However, only when the roots are within a certain wedge enclosing thenegative real axis can we prove log-concavity of the distribution of X . Stillwe are able to prove an LCLT, with an explicit error term, under certainmild conditions on G .We now summarize briefly some consequences of our results (not neces-sarily the optimal ones). Assuming that the mean E [ X ] and variance Var( X )go to infinity as N → ∞ , then:1. The random variable X satisfies a CLT for all z > ζ of P satisfy Re ζ ≤ X satisfies an LCLT for all z > ζ are in a wedge of angle 2 π/ ζ ≤ − δ , δ >
0, and Var( X ) grows faster than N / .3. The random variable X satisfies a CLT if there are no zeros of P in a discof radius δ > z and Var( X ) grows faster than N / (see [16]).4. Finally, we show that certain of the above conditions are satisfied bymany graph-counting polynomials and statistical mechanical systems—forexample, unbranched polymers—and hence obtain a CLT or LCLT in thesecases. The result mentioned in 2(a) above has also been used [9] to establishan LCLT for determinantal point processes.The outline of the rest of the paper is as follows. In Section 2 we apply themethod of [16] to derive a CLT for the random variable X from rather weakhypotheses on the location of the zeros of P ( z ), and in Section 3 we obtain anLCLT under the stronger hypothesis that the zeros lie in the left half plane. InSection 4 we describe more precisely the class of graph-counting polynomialsand what can be said about the location of their zeros. In Section 5 weobtain central limit theorems and, in some cases, local central limit theoremsfor graph-counting polynomials from the results of Section 3, and in Section 6obtain, from the results of Section 2, central limit theorems for further graph-counting examples and for some statistical mechanical systems. Throughoutour discussions we will, rather than considering sequences of polynomials,5ay that a family P of polynomials, of unbounded degrees, satisfies a CLT oran LCLT when one can give estimates for the errors in the approximations(1.3) and (1.4), respectively, which are valid for all polynomials in P andwhich vanish as the degree N of the polynomial goes to infinity. In this section we first consider a fixed polynomial P ( z ) = P Nm =0 p m z m , asin (1.2), and assume throughout that p m ≥ p N >
0, i.e., that P isin fact of degree N . We fix also a number z > X to be a random variable with probabilitydistribution given by (1.1). We will let ζ j , j = 1 , . . . , N , denote the roots of P . Our first result is an estimate corresponding to an (integrated) centrallimit theorem. To state it we define, for x ∈ R , F ( x ) := 1 P ( z ) X m ≤ E [ X ]+ x √ Var( X ) p m z m = Prob ( X − E [ X ] p Var( X ) ≤ x ) , (2.1) G ( x ) := (2 π ) − / Z x −∞ e − u / du. (2.2) Theorem 2.1.
Suppose that there exists a δ > such that z ≥ δ and | z − ζ j | ≥ δ for all j , j = 1 , . . . , N . Then there exist constants N , B , B > ,depending only on δ and z , such that for N ≥ N , sup x ∈ R | F ( x ) − G ( x ) | ≤ B N Var( X ) / + B N / Var( X ) / . (2.3) Remark 2.2.
We record here some standard results, adopting the notationof Theorem 2.1. For z in the disk D := { z ∈ C | | z − z | < δ } we will definelog P ( z ) by log P ( z ) := log p N + N X j =1 log (cid:0) z − ζ j (cid:1) , (2.4)with log p N real andlog( z − ζ j ) := log( z − ζ j ) + log z − ζ j z − ζ j , (2.5)6hereIm log( z − ζ j ) ∈ ( − π, π ) and Im log z − ζ j z − ζ j ∈ ( π/ , π/ . (2.6)In (2.6) the first specification is possible since ζ j cannot be a positive realnumber and the second since (cid:12)(cid:12) ( z − ζ j ) / ( z − ζ j ) − (cid:12)(cid:12) < z ∈ D ; inparticular, log( z − ζ j ) / ( z − ζ j ) is analytic for z ∈ D . Moreover, log P ( z ) isreal for real z , because non-real roots occur in complex conjugate pairs, andfurthermore log P ( z ) − log P ( z ) = N X j =1 log z − ζ j z − ζ j , z ∈ D. (2.7)Then for all z in D , z ddz log P ( z ) = P m mp m z m P ( z ) , (cid:18) z ddz (cid:19) log P ( z ) = P m m p m z m P ( z ) − (cid:18) P m mp m z m P ( z ) (cid:19) , (2.8)and so z ddz log P ( z ) (cid:12)(cid:12)(cid:12) z = z = E [ X ] , (cid:18) z ddz (cid:19) log P ( z ) (cid:12)(cid:12)(cid:12) z = z = Var( X ) . (2.9)From (2.9) we also have ddu log P ( e u z ) (cid:12)(cid:12)(cid:12) u =0 = E [ X ] , d du log P ( e u z ) (cid:12)(cid:12)(cid:12) u =0 = Var( X ) . (2.10)To state the next lemma we observe that there exists an ǫ >
0, dependingonly on δ and z , such that if | u | ≤ ǫ then | e u z − z | ≤ min { δ/ , | z |} , sothat for | u | ≤ ǫ we may define, as in Remark 2.2, f ( u ) := log E [ e uX ] = log P ( e u z ) − log P ( z )= N X j =1 log e u z − ζ j z − ζ j . (2.11)7 emma 2.3. Let δ be as in Theorem 2.1 and let ǫ = ǫ ( z , δ ) be as above.Then for K = 2 log 2 /ǫ , f ( u ) = uE [ X ] + u X ) + u R ( u ) , with | R ( u ) | ≤ N K. (2.12)
Proof.
Suppose that | u | ≤ ǫ/
2. Then we have, by Cauchy’s integral formulaand (2.10), f ( u ) = f (0) + uf ′ (0) + u f ′′ (0) + u R ( u )= uE [ X ] + u X ) + u R ( u ) , (2.13)where R ( u ) := 12 πi I | v | = ǫ f ( v ) v ( v − u ) dv. (2.14)Then from (2.11), | R ( u ) | ≤ N X j =1 (cid:12)(cid:12)(cid:12)(cid:12) πi I | v | = ǫ log (cid:18) e v z − ζ j z − ζ j (cid:19) dvv ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ N X j =1 sup | v | = ǫ (cid:12)(cid:12)(cid:12)(cid:12) log e v z − ζ j z − ζ j (cid:12)(cid:12)(cid:12)(cid:12) < ǫ N log 2 . (2.15)Here we have used | ( e v z − ζ j ) / ( z − ζ j ) | < ( δ/ /δ = 1 / | v | = ǫ and | log(1 − t ) | ≤ − log(1 − | t | ) for | t | <
1; the latter is easily verified for examplefrom the expansion log(1 − t ) = − P k ≥ t k /k . Proof of Theorem 2.1.
The proof follows closely the proof of the Esseen-Berry Theorem given in Feller [8, Section XVI.5] and in particular is basedon the “smoothing inequality” [8, Section XVI.4, Lemma 2]. If we specializeto the particular application we need then the latter implies that for any
T >
0, sup x ∈ R | F ( x ) − G ( x ) | ≤ π Z T − T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( t ) − e − t / t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt + 24 π √ πT , (2.16)where ψ ( t ) = E [ e itY ] is the characteristic function of Y = ( X − E [ X ]) /σ ,with σ = p Var( X ). We will apply this inequality with T = σ/N / . For8 t | ≤ T , then, | t/σ | ≤ N − / , so that for N ≥ N := 8 /ǫ we have t/σ ≤ ǫ/ ψ ( t ) = e − itE [ X ] /σ e f ( it/σ ) = e − t / − it R ( it/σ ) /σ , (2.17)with | R ( it/σ ) | ≤ N K and hence | it R ( it/σ ) /σ | ≤ K . Now let K ∗ =max | u |≤ K | ( e iu − /u | , so that | e − it R ( it/σ ) /σ − | ≤ | t/σ | N KK ∗ for N ≥ /ǫ and t ≤ T . (2.18)Then Z T − T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( t ) − e − t / t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ N KK ∗ σ Z T − T t e − t / dt ≤ N KK ∗ σ Z ∞−∞ t e − t / dt = N KK ∗ √ πσ . (2.19)Inserting this estimate into (2.16) we obtain (2.3) with B := r π KK ∗ , B := 24 π √ π . (2.20)In Section 6 we will apply Theorem 2.1 to obtain central limit theoremsfor families of graph-counting polynomials and for families of polynomialsarising from statistical mechanics. To do so we must establish that, for P inthe family under consideration, Var( X P ) grows faster than N / P . Our toolfor this will be a result due to Ginibre [11], which we recall as Theorem 6.1below; our next result, which is similar to Theorem 2.1, will be needed in theapplication of Ginibre’s result to graph-counting polynomials. Proposition 2.4.
Suppose that p and p are nonzero and that c and δ arepositive constants such that (i) p ≥ c p N and (ii) | ζ j | ≥ δ , j = 1 , . . . , N .Then there exists a constant M > , depending only on c , δ , and z , suchthat E [ X ] ≥ M N .Proof.
For z real and nonnegative, log P ( z ) is well defined by the requirementthat it be real; further, E [ X ] = z ddz log P ( z ) (cid:12)(cid:12) z = z (2.21)9nd z ddz E [ X ] = Var( X ) > , (2.22)so that E [ X ] is an increasing function of z . Thus it suffices to verify theconclusion for sufficiently small z . Now we allow z to be complex, and for | z | < δ define as in (2.7) g ( z ) := log P ( z ) − log P (0) = N X j =1 log ζ j − zζ j , (2.23)where again Im log(( ζ j − z ) /ζ j ) ∈ ( − π/ , π/ | z | < δ / zg ′ ( z ) = z ddz (cid:18) g (0) + zg ′ (0) + z πi I | y | = δ / g ( y ) y ( y − z ) dy (cid:19) = z p p + z R ( z ) , (2.24)with R ( z ) := 12 πi I | y | = δ / (2 y − z ) g ( y ) y ( y − z ) dy. (2.25)Since for | y | = δ / | z | ≤ δ / / | y | = 4 /δ , 1 / | y − z | ≤ /δ , | y − z | < δ /
4, and | g ( y ) | ≤ log 2 (see (2.15)), we find that | R ( z ) | ≤ δ N log 2 . (2.26)Let z ∗ = min { δ / , c δ / (80 log 2) } ; then for 0 < z ≤ z ∗ , E [ X ] = zg ′ ( z ) (cid:12)(cid:12)(cid:12) z = z ≥ z p p − z δ N log 2 ≥ z c N . (2.27)Thus E [ X ] ≥ M N holds with M = z c / z ≤ z ∗ and with M = z ∗ c / Remark 2.5.
Theorem 2.1 strengthens and gives a complete proof of theresult in [16] that F ( x ) → G ( x ) as N → ∞ . [16] considered specifically theIsing model, for which it is known that Var( X ) ≥ cE ( X ) ≥ kN , c, k >
0; seeSection 6. We also note here that Dobrushin and Shlosman [6] proved a local“large and moderate deviation” result for X which implies a LCLT under a10urther locality condition, which rules out situations in which all the zerosare close to the imaginary axis. The locality condition is in turn implied bya certain bound on the characteristic function E [ e itX ], which they showed tohold for the Ising model at zero magnetic field and high temperature. Thebound in question is somewhat stronger than the bound (3.19) which weobtain from the condition that the roots all lie in the negative half of thecomplex plane. In this section we again consider a polynomial P ( z ) as in (1.2), and continueto assume that P is of degree N and that all the coefficients p m are nonnega-tive. Moreover, we assume that all roots of P lie in the closed left-half plane,and no root is zero, i. e. p >
0. For convenience we now write these rootsas − η j , so thatRe( η j ) ≥ , ( j = 1 , . . . , N ) , and P ( z ) = p N N Y j =1 ( z + η j ) . (3.1)We will take the fugacity z to be 1, but our results extend easily to any z > Under the assumption (3.1) the derivation of a CLT given in Section 2 canbe simplified; moreover, the result is strengthened since we require only thatVar( X P ) → ∞ as N P → ∞ , in contrast to the power growth condition neededto apply Theorem 2.1. The key idea is to write X P as a sum of independentrandom variables; the central limit theorem then follows, for example fromthe Berry-Esseen theorem. In the case in which all the η j are nonnegativethe method goes back to Harper [14]. [4],To decompose X P as such a sum, we partition { , . . . , N } as J ∪ J ∪ J ′ ,where j ∈ J iff η j is real and j ∈ J (respectively j ∈ J ′ ) iff Im( η j ) > η j ) < P ( z ) is P ( z ) = p N Y j ∈ J ( z + η j ) Y j ∈ J ( z + 2 Re( η j ) z + | η j | ) . (3.2)11e then introduce independent random variables X j , j ∈ J ∪ J , where if j ∈ J (respectively j ∈ J ) then X j takes values 0 and 1 (respectively 0, 1,and 2). With P j ( z ) = z + η j for j ∈ J and P j ( z ) = z + 2 z Re( η j ) + | η j | for j ∈ J , the individual distribution of these random variables isPr { X j = 0 } = 1 P j (1) , Pr { X j = 1 } = η j P j (1) , ( j ∈ J );Pr { X j = 0 } = | η j | P j (1) , Pr { X j = 1 } = 2 Re( η j ) P j (1) , Pr { X j = 2 } = 1 P j (1) , ( j ∈ J ) . Then E [ z X j ] = P j ( z ) /P j (1) and so E [ z P j ∈ J ∪ J X j ] = Y j ∈ J ∪ J P j ( z ) P j (1) = P ( z ) P (1) = E [ z X P ] (3.3)for all z . Thus X P and P j ∈ J ∪ J X j have the same distribution, and we mayidentify these two random variables. Theorem 3.1.
Let P be a family of polynomials as in (1.2) , of unboundeddegrees, all of which satisfy (3.1) . Then for each P ∈ P , sup x ∈ R | F P ( x ) − G ( x ) | ≤ p Var( X P ) . (3.4) Consequently, if Var ( X P ) → ∞ as N P → ∞ in P then P satisfies a CLT inthe sense described in Section 1.Proof. From [8, Section XVI.5, Theorem 2] and | X j | ≤ X ) / X j ∈ J ∪ J E (cid:0)(cid:12)(cid:12) X j − E ( X j ) (cid:12)(cid:12) (cid:1) ≤ X ) / X j ∈ J ∪ J Var( X j ) . (3.5)This theorem calls for explicit bounds for Var( X P ). From Remark 2.2,Var( X P ) = (cid:18) z ddz (cid:19) p N N Y j =1 ( z + η j } !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =1 = N X j =1 η j (1 + η j ) = N X j =1 Re( η j )(1 + | η j | ) + 2 | η j | | η j | (3.6)12hen since | η j | = 1+2 Re( η j )+ | η j | ≥ | η j | and | η j | / (1+ | η j | ) ≤ / X P ) ≤ N X j =1 (cid:18) Re( η j )1 + | η j | + 12 (cid:19) ≤ N. (3.7)On the other hand, (3.6) also yieldsVar( X P ) ≥ W ( X P ) := 14 N X j =1 Re( η j )1 + | η j | . (3.8)In our proof of the general case of the LCLT we will need Var( X P ) (respec-tively W ( X P )) to bound (cid:12)(cid:12) E [ e itX P ] (cid:12)(cid:12) for “small” | t | (respectively for “large” | t | ).Here is a useful upper bound for Var( X P ). Introduce α P = max j | arg( η j ) | ,( α P ∈ [0 , π/ α < π/
2, thenVar( X P ) ≤ α P ) W ( X P ) . (3.9)Indeed, denoting r j = Re( η j ), α j = | arg( η j ) | , we bound the j -th term in(3.6) by r j r j sec α j + 2 r j sec α j (1 + r j sec α j ) ≤ r j r j sec α j + 2 r j sec α j / (2 r j sec α j )1 + r j sec α j ≤ Re( η j )1 + | η j | · (1 + sec α j ) , and (3.9) follows. Thus, as N P → ∞ , Var( X P ) and W ( X P ) are of the sameorder of magnitude if α P is bounded away from π/ W ( X P ) that can make it easier to provethat W ( X P ) → ∞ . To this end we define, for P ∈ P ,∆ (= ∆ P ) := min ≤ j ≤ N | η j | Re( η j ) , f (= f P ) := p p . (3.10)Notice that θ j := 1 /η j , j = 1 , . . . , N , satisfy p N Q j θ j · N Y k =1 ( z + θ k ) = N X m =0 z m p N − m .
13o equating the coefficients by z N and z N − we have p N Q j θ j = p , p N Q j θ j X k θ k = p = ⇒ X k θ k = p p . Consequently f = p p = N X j =1 θ j = N X j =1 Re( θ j ) . (3.11)In addition, Re( θ j ) = Re( η j ) | η j | = | θ j | · | η j | Re( η j ) ≥ ∆ | θ j | . (3.12)Then Jensen’s inequality for the convex function 1 / (1 + x ), with (3.11) and(3.12), yields X j Re( η j )1 + | η j | = X j Re( θ j )1 + | θ j | = f X j Re( θ j ) f
11 + | θ j | ≥ f f − P j Re( θ j ) | θ j | ≥ f f − P j | θ j | ≥ f / ∆ ≥ f { , ∆ } . (3.13)Thus we have proved Lemma 3.2.
Var( X P ) ≥ W ( X P ) := 14 N X j =1 Re( η j )1 + | η j | ,W ( X P ) ≥ f { , ∆ } , with ∆ = ∆ P and f = f P as defined in (3.10) . Let us show that the CLT proved in Section 3.1 implies an LCLT when thelocations of the roots ζ j of the polynomials P (see (3.1)) are further confinedto a sharp wedge enclosing the negative axis in the complex plane.14 efinition 3.3. A sequence a n , n ≥
0, of nonnegative real numbers is log-concave if for all n ≥ a n ≥ a n − a n +1 .In the factorization (3.2) of P the coefficients η j and 1 of each linearfactor, augmented from the right with an infinite tail of zeros, obviouslyform a log-concave sequence, and so do the coefficients | η j | , 2 Re( η j ), and 1of each quadratic factor, provided that4(Re( η j )) ≥ | η j | ⇔ | arg( η j ) | ≤ π/ . (3.14)In terms of the roots ζ j = − η j , the last condition is equivalent to | arg( ζ j ) | ∈ [2 π/ , π ] , (3.15)for all non-zero roots ζ j . Since the convolution of log-concave sequencesis log-concave (Menon [23]), we see that, under the condition (3.14), thecoefficients of P are also log-concave. This result appears as a special casein Karlin [18] (Theorem 7.1, p. 415). (See Stanley [31] for a more recent,comprehensive, survey of log-concave sequences.)We say that a random variable X taking nonnegative integer values is log-concave distributed if the sequence { Pr { X = n }} is log-concave. Bender [3]discovered that an LCLT holds for a sequence { X n } of log-concave distributedrandom variables if lim n →∞ sup x ∈ R | F X n ( x ) − G ( x ) | = 0; remarkably, X n doesnot have to be a sum of independent random variables. Later Canfield [4]quantified Bender’s theorem. For this he needed a stronger notion of log-concavity. Definition 3.4.
A sequence a n , n ≥
0, of nonnegative real numbers is prop-erly log-concave if(i) there exist integers L and U such that a n = 0 iff n < L or n > U (in theterminology of [31], { a n } has no internal zeros );(ii) for all n ≥ a n ≥ a n − a n +1 , with equality iff a n = 0.Canfield showed that the convolution of properly log-concave sequencesis also properly log-concave. Observe that the linear and quadratic factorsof our polynomial P ( z ) are properly log-concave iff | arg( ζ j ) | ∈ (2 π/ , π ].Subject to this stronger condition, the coefficients of P ( z ) form therefore aproperly log-concave sequence.Here is a slightly simplified formulation of Canfield’s result.15 heorem 3.5. (Canfield) Suppose that X has a properly log-concave distri-bution and that sup x ∈ R | F X ( x ) − G ( x ) | ≤ K p Var( X ) . If K > , K/ Var( X ) / < − , K/ Var( X ) / < − , then sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X = m ) − p π Var( X ) exp (cid:18) − ( m − E [ X ]) X ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c Var( X ) / , with c := 14 . K + 4 . . This theorem and Theorem 3.1 imply an LCLT for X P with the roots ζ j satisfying the condition | arg( ζ j ) | ∈ (2 π/ , π ]. Corollary 3.6.
If the roots ζ j of P ( z ) satisfy | arg( ζ j ) | ∈ (2 π/ , π ] , and Var( X P ) > × , then sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X P = m ) − p π Var( X P ) exp (cid:18) − ( m − E [ X p ]) X P ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X P ) / . While we proved the LCLT for the roots ζ j in the wedge | arg( ζ j ) | > π/ X P ) → ∞ , we cannot expect this condition besufficient in general. A trivial example is P ( z ) with purely imaginary, non-zero roots, in which case the distribution of X P is supported by the positiveeven integers only. We will see shortly, however, that a stronger condition, f P min { , ∆ P } → ∞ fast enough, does the job perfectly.We first state the fundamental estimate, in terms of the variance Var( X P )and its lower bound W ( X P ) defined in (3.8). Theorem 3.7.
Suppose
Var( X P ) ≥ . Then setting X := X P , sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X = m ) − p π Var( X ) exp (cid:18) − ( m − E [ X ]) X ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π / Var( X ) / W ( X ) exp (cid:18) − / π W ( X )Var( X ) / (cid:19) + 24 π Var( X ) . (3.16)16 orollary 3.8. If W ( X P ) ≥ π · / Var( X P ) / log(Var( X P )) , (3.17) then for X := X P , sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X = m ) − p π Var( X ) exp (cid:18) − ( m − E [ X ]) X ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π Var( X ) . Remark 3.9. (a) For | arg( ζ j ) | ∈ (2 π/ , π ], the estimate (3.9) leads to W ( X P ) ≥ (1 / X P ). Therefore the condition (3.17) is satisfied forVar( X P ) > . × , the last number being a close upper bound for thelarger root of v = 12 π · / v / log v. The resulting error estimate, 25 / ( π Var( X P )), is noticeably better than thebound 180 / Var( X P ) / in Corollary 3.6.(b) In general, by (3.7) and Lemma 3.2,Var( X P ) ≤ N P , W ( X P ) ≥ p p min { , ∆ P } , (∆ P := min j | η j | Re( η j )) . So the condition (3.17) is certainly met if p p min { , ∆ P } ≥ π · / N / P log N P . (3.18)For the proof of Theorem 3.7 we introduce the characteristic functions φ ( t ) of X and φ ∗ ( t ) of X ∗ = X − E [ X ]: φ ( t ) := E [ e itX ] and φ ∗ ( t ) := E [ e itX ∗ ] = e − itE [ X ] φ ( t ). The next two lemmas give estimates for these func-tions. In Lemma 3.10 we use crucially the fact that all roots of P ( z ) lie inthe left hand plane; this is also used in the proof of Lemma 3.11, althoughsome version of this result could be obtained as in Section 2, using only thefact that a neighborhood of z = 1 is free from zeros of P ( z ). Lemma 3.10.
For all t ∈ [ − π, π ] , | φ ( t ) | ≤ exp (cid:18) − t π W ( X P ) (cid:19) . (3.19)17 roof. First of all, φ ( t ) = P ( e it ) P (1) = Y j η j + e it η j + 1 . (3.20)So, using 1 + u ≤ e u for u real, 1 − cos t = 2 sin ( t/ ≥ t /π for t ∈ [ − π, π ],and | η j | ≤ | η j | ), | φ ( t ) | = Y j | η j + e it | | η j + 1 | = Y j (cid:18) η j (cos t −
1) + 2 Im η j sin t | η j + 1 | (cid:19) ≤ exp X j Re( η j )(cos t − | η j | ! ≤ exp − t π X j Re( η j )1 + | η j | ! . Invoking the definition of W ( X P ) in (3.8) then yields the bound (3.17) im-mediately.Unlike Lemma 3.10, the next claim and its proof are more or less standard;we give the argument to make presentation more self-contained. Lemma 3.11. If | t | ≤ then φ ∗ ( t ) = exp (cid:18) − t X ) + D ( t ) (cid:19) with | D ( t ) | ≤ | t | Var( X ) . (3.21) Proof.
We write X = P j ∈ J ∪ J X j as in Section 3.1. It is easy to checkthat Var( X j ) ≤
1, and Var( X j ) = 1 iff Pr( X j = 0) = Pr( X j = 2) = 1 / X ∗ j = X j − E [ X j ], j ∈ J ∪ J , we write φ ∗ ( t ) = Y j ∈ J ∪ J φ ∗ j ( t ) , φ ∗ j ( t ) := E [ e itX ∗ j ]; (3.22)here, see Feller [8, Section XVI.5], φ ∗ j ( t ) = 1 − t X j ) + R j ( t ) , | R j ( t ) | ≤ | t | E (cid:2) | X ∗ j | (cid:3) ≤ | t | X j ) , | X ∗ j | ≤
2. Denoting u j := t Var( X j ) − R j ( t ), and using Var( X j ) ≤
1, wesee that, for | t | ≤ | u j | ≤ t X j ) + | t | X j ) ≤ t Var( X j ) ≤ . So, using log(1 − u ) = − P j> u j /j , we obtain φ ∗ j ( t ) = exp (cid:2) log(1 − u j ) (cid:3) = exp (cid:2) − u j + S j ( t ) (cid:3) , where | S j ( t ) | ≤ X ℓ ≥ | u j | ℓ ℓ ≤ u j − | u j | ) ≤ u j ≤ t Var( X j ) . Therefore φ ∗ j ( t ) = exp (cid:20) − t X j ) + D j ( t ) (cid:21) , where | D j ( t ) | = | R j ( t ) + S j ( t ) |≤ | t | X j ) + 25 t
12 Var( X j ) ≤ | t | Var( X j ) . Consequently, for | t | ≤ φ ∗ ( t ) = Y j φ ∗ j ( t ) = exp − t X j Var( X j ) + D ( t ) ! = exp (cid:18) − t X ) + D ( t ) (cid:19) , (3.23)with D ( t ) := P j D j ( t ), and | D ( t ) | ≤ X j | D j ( t ) | ≤ | t | Var( X ) . (3.24)19 roof of Theorem 3.7. For any T ∈ [0 , π ] we write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X = m ) − p π Var( X ) exp ( m − E [ X ]) X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) π Z π − π φ ( t ) e − itm dt − π Z ∞−∞ e − t Var( X ) / e − it ( m − E [ X ]) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:12)(cid:12)(cid:12)(cid:12)Z T ≤| t |≤ π φ ( t ) e − itm dt (cid:12)(cid:12)(cid:12)(cid:12) + 12 π (cid:12)(cid:12)(cid:12)(cid:12)Z | t |≥ T e − t Var( X ) / e − it ( m − E [ X ]) dt (cid:12)(cid:12)(cid:12)(cid:12) + 12 π Z | t |≤ T (cid:12)(cid:12)(cid:12) φ ∗ ( t ) − e − t Var( X ) / (cid:12)(cid:12)(cid:12) dt. (3.25)Let us denote the three terms in the final expression in (3.25) by I , I , and I , respectively. Then from Lemma 3.10 and the inequality Z | y |≥ x e − ay / dy ≤ ax e − ax / (3.26)we have, for any T ∈ (0 , π ], I ≤ π W ( X ) T exp (cid:18) − T π W ( X ) (cid:19) ; I ≤ π Var( X ) T exp (cid:18) − T X ) (cid:19) . (3.27)We now turn to I . Let us pick T = (4 Var( X )) − / ; then T < X ) ≥
1. Also, for | t | ≤ T , | D ( t ) | in Lemma 3.11 is at most 3 / <
1. Sousing that lemma and the inequality | e x − | ≤ | x | − | x | , ( | x | < , we have that for | t | ≤ T , (cid:12)(cid:12) φ ∗ ( t ) − e − t Var( X ) / (cid:12)(cid:12) ≤ e − t Var( X ) / D ( t )1 − | D ( t ) |≤ X ) | t | e − t Var( X ) / . I ≤
12 Var( X )2 π Z ∞−∞ | t | e − t Var( X ) / dt = 24 π Var( X ) . (3.28)For this choice of T , the bounds (3.27) become I ≤ π / X ) / W ( X ) exp (cid:18) − / π W ( X )Var( X ) / (cid:19) ; (3.29) I ≤ / π Var( X ) − / exp (cid:18) − Var( X ) / · / (cid:19) . We notice that the top bound exceeds the bottom bound since Var( X ) ≥ W ( X ) and π >
8. Adding the bound (3.28) and the double bound (3.29),we get the bound claimed in Theorem 3.7.
Let G be a finite graph with vertex set V and edge set E ; an edge e ∈ E connects distinct vertices v ( e ) and v ( e ), and different edges may connectthe same two vertices. We identify the subgraphs of G with the subsets M ⊂ E . For v ∈ V we let d v be the degree of v in G and d M ( v ) be the degreeof v in the subgraph M ; to avoid trivialities we assume that d v > v .Now suppose that for each v ∈ V we choose a finite nonempty subset C ( v )of nonnegative integers and define a set ( C ) of subgraphs of G , associatedwith the family ( C ( v )) v ∈ V , by M ∈ ( C ) ⇔ d M ( v ) ∈ C ( v ) for all v ∈ V. (4.1)We assume throughout that ( C ) = ∅ . Then the graph-counting polynomial associated with ( C ) is P ( C ) ( z ) = X M ∈ ( C ) z | M | . (4.2)For example, as discussed in Section 1, if C ( v ) = { , } for each v ∈ V then( C ) corresponds to the set of matchings in G or, in the language of statisticalmechanics, to the set of monomer-dimer configurations on G , while if C ( v ) = { , , } for all v then ( C ) is the set of unbranched polymer configurations.21f C ( v ) = { , } for all v then the subgraphs in ( C ) are unions of disjointcircuits.The proofs of the CLT and LCLT given in later sections depend on infor-mation about the locations of the zeros of the polynomials P ( C ) , and this canbe obtained from corresponding information for certain subsidiary polynomi-als associated with the vertices. Given a nonempty finite set C of nonnegativeintegers and a positive integer d we define p C,d ( z ) = X k ∈ C (cid:18) dk (cid:19) z k ; (4.3)we will often write p v = p C ( v ) ,d v . The next two results control respectivelythe magnitudes and arguments of the roots of P ( C ) in terms of correspondinginformation for the roots of the p v . Theorem 4.1.
Suppose that, for each v ∈ V , there is a constant r v > suchthat | ζ | ≥ r v for each root ζ of p v . Then every root ξ of P ( C ) satisfies | ξ | ≥ R ,where R = min e ∈ E r v ( e ) r v ( e ) . (4.4)Notice that p C,d (0) = 0 if and only if 0 / ∈ C , so that the hypotheses ofTheorem 4.1 imply that 0 ∈ C ( v ) for each v ∈ V . Proof of Theorem 4.1.
The proof uses Grace’s Theorem, the notion ofAsano contraction, and the Asano-Ruelle Lemma; these topics are reviewedin Appendix A. Let E v ⊂ E be the set of edges of G incident on the vertex v . To each polynomial p v there corresponds a unique symmetric multi-affinepolynomial q v in the d v variables ( z v,e ) e ∈ E v such that q v ( z, . . . , z ) = p v ( z ).Since p v ( z ) = 0 for | z | < r v , Grace’s Theorem implies that q v = 0 if | z v,e | < r v , ∀ e ∈ E v . Now we define a multi-affine polynomial Q (0) (cid:0) ( z v,e ) v ∈ V,e ∈ E v (cid:1) = Y v ∈ V q v (cid:0) ( z v,e ) e ∈ E v (cid:1) (4.5)and generate, by repeated Asano contractions ( z v ( e ) ,e , z v ( e ) ,e ) → z e , a se-quence of polynomials Q (0) , Q (1) , . . . , Q ( | E | ) , where Q ( k ) depends on k vari-ables z e and ( | E | − k ) pairs of uncontracted variables z e,v ( e ) , z e,v ( e ) . Fromthe Asano-Ruelle Lemma and an inductive argument, Q ( k ) (( z e ) , ( z v,e )) =0 when the variables satisfy | z e | < r v ( e ) r v ( e ) , | z e,v | < r v . In particu-lar, Q ( | E | ) (( z e ) e ∈ E ) = 0 when | z e | < R for all e ∈ E . But P ( C ) ( z ) = Q ( | E | ) ( z, z, . . . , z ), completing the proof.22 heorem 4.2. Suppose that for each v ∈ V there is an angle φ v ∈ [0 , π/ such that each nonzero root ζ of p v satisfies | arg( ζ ) | ∈ [ π − φ v , π ] . Let S = { θ ∈ [ − π, π ] | ∃ ( θ v ) v ∈ V , | θ v | ≤ π/ − φ v , θ v ( e ) + θ v ( e ) = θ, e ∈ E } . (4.6) Then every nonzero root ξ of P ( C ) satisfies | arg( ξ ) | ∈ [max S, π ] . Our applications of this theorem will always be those of the next corollary.
Corollary 4.3. (a) Suppose that there is an angle φ ∈ [0 , π/ such that, foreach v ∈ V , each nonzero root ζ of p v satisfies | arg( ζ ) | ∈ [ π − φ, π ] . Thenevery nonzero root ξ of P ( C ) satisfies | arg( ξ ) | ∈ [ π − φ, π ] .(b) Suppose that the graph G is bipartite, so that V may be partitioned as V = V ∪ V with each e ∈ E satisfying v ( e ) ∈ V , v ( e ) ∈ V . Supposefurther that there are angles φ , φ ∈ [0 , π/ such that, for each v ∈ V i , eachnonzero root ζ of p v satisfies | arg( ζ ) | ∈ [ π − φ i , π ] for i = 1 , . Then everynonzero root ξ of P ( C ) satisfies | arg( ξ ) | ∈ [ π − φ − φ , π ] .Proof. For (a) we see that π − φ ∈ S by taking θ v = π/ − φ for all v ∈ V ;for (b) we have similarly π − φ − φ ∈ S from θ v = π/ − φ i if v ∈ V i . Proof of Theorem 4.2.
It suffices to consider the case max
S >
0. Weadopt the notations q v and Q ( k ) from the proof of Theorem 4.1, and for ε > p v,ε ( z ) = p v ( z + ε ) and q v,ε (( z v,e ) e ∈ E v ) = q v (( z v,e + ε ) e ∈ E v ); q v,ε is theunique symmetric multi-affine polynomial such that q v,ε ( z, . . . , z ) = p v,ε ( z ).We also define Q (0) ε (cid:0) ( z v,e ) v ∈ V,e ∈ E v (cid:1) = Y v ∈ V q v,ε (cid:0) ( z v,e ) e ∈ E v (cid:1) , (4.7)and let Q (0) ε , Q (1) ε , . . . , Q ( | E | ) ε be obtained by Asano-Ruelle contractions, as inthe proof of Theorem 4.1. Finally, we define P ε by P ε ( z ) = Q ( | E | ) ε ( z, z, . . . , z ).Fix θ with | θ | < max S . We claim that if, for each e ∈ E , z e belongs tothe ray ρ θ = { e iθ x | x > } , then Q ( | E | ) ε (cid:0) ( z e ) e ∈ E (cid:1) = 0. It follows then that P ε ( z ) = 0 for z ∈ ρ θ , so that P ε does not vanish on the open set G := { z ∈ C | z = 0 , | arg( z ) | < max S } . (4.8)But lim ε → P ǫ = P ( C ) uniformly on compacts, and P ( C ) does not vanish iden-tically since ( C ) = ∅ . So, by an application on G of the theorem of Hurwitz, P ( C ) ( z ) = 0 if z ∈ G . This is the desired conclusion.23e now prove the claim. Clearly max S ≤ π and S = [ − max S, max S ].Consider θ ∈ ( − max S, max S ), and let ( θ v ) v ∈ V be as in the definition of S .If θ = 0 then we may take θ v = 0 for all v . If θ = 0 then necessarilymin j =1 , | θ v j ( e ) | < π/ e ∈ E , (4.9)since otherwise θ v ( e ) + θ v ( e ) = θ ∈ ( − π, π ) is inconsistent with | θ v j ( e ) | ≤ π/ θ , we may assume that (4.9) holds.Now let H and H denote respectively the open and closed right halfplanes, and for ǫ > K ǫ ( v ) = − ( ǫ + e iθ v H ) . (4.10)No root ζ of p v ( z ) can belong to e iθ v H ; for ζ = 0 this is trivial and for ζ = 0follows from | arg( ζ ) | ∈ [ π − φ v , π ] and | θ v | ≤ π/ − φ v . Thus p v,ε ( z ) = 0 if z + ε ∈ e iθ v H , that is, if z + ε / ∈ − e iθ v H or equivalently if z / ∈ K ǫ ( v ). Grace’sTheorem then implies that q v,ǫ (( z ve ) e ∈ E v ) = 0 if z v,e / ∈ K ǫ ( v ) for all e ∈ E v .Repeatedly using the Asano-Ruelle Lemma, as in the proof of Theorem 4.1,we then conclude that Q ( | E | ) (cid:0) ( z e ) e ∈ E (cid:1) = 0 if z e / ∈ − K ǫ ( v ( e )) × K ǫ ( v ( e )) forall e ∈ E .Now, the set − K ε ( v ( e )) × K ε ( v ( e )) and the ray ρ θ v e ) + θ v e ) = ρ θ do notintersect. Otherwise there would exist ( s ≥ , t ), ( s ≥ , t ) and x > − ( ε + e iθ v e ) ( s + it ))( ε + e iθ v e ) ( s + it )) = xe i ( θ v e ) + θ v e ) ) , or equivalently y y = ρe iπ , y j = e − iθ vj ( e ) ε + ( s j + it j ) , j = 1 , . (4.11)From the second equation in (4.11) we have | arg( y j ) | ≤ π/
2, since Re( y j ) ≥ j ; this isinconsistent with the first equation in (4.11). This completes the proof ofthe claim. In this section we consider various infinite families of graphs, each with anassociated assignment ( C ( v )) v ∈ V of finite sets to vertices; we let G denote such24 family and P = P ( G ) denote the class of associated graph polynomials,which we now denote by P G . We will measure the size of a graph G by thesize of its edge set E = E ( G ) and let d max = d max ( G ) denote the maximumdegree of any vertex of G ; for convenience we assume that d max ≥ d max = 1 is trivial to analyze).For simplicity we restrict our attention to the two cases implicit in Corol-lary 4.3, and thus assume that either (a) there is a fixed angle φ ∈ [0 , π/ G ∈ G and each v ∈ V ( G ), every nonzero root ζ of p v satisfies | arg( ζ ) | ∈ [ π − φ, π ], or (b) each graph in G is bipartite, with V ( G ) partitioned as V ( G ) ∪ V ( G ), and there are fixed angles φ , φ ∈ [0 , π/ G and each v ∈ V i ( G ), i = 1 ,
2, every nonzero root ζ of p v satisfies | arg( ζ ) | ∈ [ π − φ i , π ]. We will give examples in which the results ofSection 4 imply that the roots of each P ∈ P lie in the left half plane, andthen apply the results of Section 3 to obtain a CLT or LCLT for P .Note that the proofs of CLT and LCLT in Section 3 require two sorts ofhypotheses: on the one hand, the roots of the polynomials must lie in theleft hand plane, or in some more restricted region; on the other, the varianceof the random variable X P , or more precisely the related quantity W ( X P ),must grow sufficiently fast with N P (see, for example, Remark 3.9). Whenthe graphs in the family under consideration have bounded vertex degree thelatter condition is, in our examples, automatically satisfied. For the moregeneral situation with unbounded degrees one must impose conditions ontheir growth to obtain the result; we will work this out in detail only forsome of our examples. Example 5.1.
When C ( v ) = C = { , } for each vertex v the admissibleedge configurations are matchings or monomer-dimer configurations, as dis-cussed in the introduction. It is well known [15] that in this case all rootsof P ( z ) lie on the negative real axis. This follows also from Corollary 4.3(a);one may take φ = 0 there, using the fact that for any vertex v the vertexpolynomial p v ( z ) = 1 + d v z has negative real root − /d v . To obtain an LCLTfrom Corollary 3.6 we need to find the quantities ∆ and f defined in (3.10).Corollary 4.3 implies that the roots − η j of P G are negative real numberssatisfying η j > /d max , so that ∆ = min ≤ j ≤ N | η j | Re( η j ) ≥ /d . Further, p = 1 and p = | E | , since any subgraph with exactly one edge is admissible,so that f = p /p = | E | . Then from Lemma 3.2,Var( X ) ≥ | E | d , (5.1)25nd an LCLT follows immediately from Corollary 3.8 and Remark 3.9(a),whenever d max ( G ) grows more slowly than | E ( G ) | / in the class of graphs G : Theorem 5.1.
If for each G ∈ G , C ( v ) = { , } for each vertex v , and | E ( G ) | ≥ . × d ( G ) , then sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X P = m ) − e − ( m − E [ X P ]) X P ) p π Var( X P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d ( G ) π | E ( G ) | . We note that Godsil [13] used the work of Heilmann and Lieb [15] to ob-tain the estimate Var( X P ) ≥ | E ( G ) | / (4 d max − (see Lemma 3.5 in [13]), andapplied Canfield’s theorem for log-concave distributions to get his LCLT for X P with the error bound O (cid:0) d / / | E | / (cid:1) . Godsil’s bound is better (respec-tively worse) than ours for d max ≫ | E | / (respectively for d max ≪ | E | / ). Example 5.2.
When C ( v ) = { , , } for each vertex v the admissible edgeconfigurations are unbranched subgraphs, as discussed in the introduction.In this case the vertex polynomial is p v ( z ) = 1 + d v z + d v ( d v − z . (5.2)If d v = 1 then p v has root ζ v = −
1, while if d v ≥ ζ ± v := − d v ± i p d v − d v d v ( d v − . (5.3)From | ζ ± v | = 2 / (cid:0) d v ( d v − (cid:1) we see that each root ζ of p v satisfies | ζ | ≥ d max ( d max −
1) ; (5.4)note that when d v = 1 this follows from our convention d max ≥
2. Thus fromTheorem 4.1 each root − η j of P G satisfies | η j | ≥ d max ( d max − . (5.5)26imilarly, each root ζ of p v satisfies | arg( ζ ) | = π − φ v with φ v ≤ φ max := sin − s d max − d max −
1) ; (5.6)when d v = 1 this is trivial and for d v ≥ | arg( − η j )) | ≥ π − φ max . Sincecos(2 φ max ) = 1 − φ max = 1 d max − > , (5.7)all the roots η j lie in the left half plane; moreover, from (3.10),∆ = min j | η j | Re( η j ) ≥ min j | η j | cos(2 φ max ) ≥ d ( d max − . (5.8)As in Example 5.1, f = p /p = | E | , so that from Lemma 3.2,Var( X ) ≥ | E | d ( d max − . (5.9)An LCLT then follows from Corollary 3.8 and Remark 3.9 when d max ( G )grows logarithmically slower than | E ( G ) | / in the class of graphs G (theprecise condition is (5.10)). Theorem 5.2.
Suppose that for each G ∈ G and vertex v of G , C ( v ) is { , } or { , , } . If | E ( G ) | is large enough so that | E ( G ) | ≥ / π d ( G ) λ ( G ) / log λ ( G ) , (cid:0) λ ( G ) := min {| E ( G ) | , | V ( G ) |} (cid:1) , (5.10) (for instance, if | E ( G ) | ≥ d max ( G ) log | V ( G ) | ), then sup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X P = m ) − e − ( m − E [ X p ]) X P ) p π Var( X P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d ( G ) π | E ( G ) | . (5.11) Proof.
By Remark 3.9(b), condition (3.17) of Corollary 3.8 is met if (3.18)holds, and by p /p = | E ( G ) | and (5.8), the latter is true if4 | E ( G ) | d ( G ) ≥ π · / N / P G log N P G . (5.12)27ow N P G ≤ λ ( G ) = min {| E ( G ) | , | V ( G ) |} , since 2 N P G ≤ P v c v ≤ | V ( G ) | .Therefore (5.12) follows from the condition (5.10). Thus when (5.10) issatisfied the condition of Corollary 3.8 holds, and with (5.9) this implies(5.11). Remark 5.3. If d max ( G ) ≤ G ∈ G , for example if the graphs in G are all finite subgraphs of the planar hexagonal lattice, then φ max = π/ − η j of P G satisfy the condition (3.15) that | arg( − η j ) | ∈ [2 π/ , π ]. Then from Corollary 3.8, Remark 3.9(a), and (5.1)we obtain an LCLT with the error bound · π | E | , provided that | E | > . · .In the next four examples we consider families of bipartite graphs, as-suming, as discussed above, that the vertex set V ( G ) of each graph G ispartitioned as V ( G ) = V ( G ) ∪ V ( G ). We assume that there is a uniformbound on the vertex degrees; specifically, d v ≤ d i for v ∈ V i ( G ), i = 1 , G ∈ G . In some cases this assumption is made for simplicity and one could,in principle, dispense partially or completely with it, but in others it is strictlynecessary, at least for our methods. Example 5.3.
Here we take C v = { , } for v ∈ V ( G ) and, for v ∈ V ( G ), C v = { , , . . . , k } with k either 2, 3, or 4. For v ∈ V , p v ( z ) = 1 + d v z as in Example 5.1, with a single negative real root. Moreover, for v ∈ V ,each root ζ of p v ( z ) satisfies | arg( ζ ) | ∈ [ π − φ v , π ], where φ v ≤ φ max < π/ φ max which depends on k and d ; for k = 2 this was shownin Example 5.2 above (with φ max = π/
4) and for k = 3 or 4 was shownin [20] (see Theorem 5.1 there). Thus taking φ = 0 and φ = φ max inCorollary 4.3(b) we see that the roots − η j of P G satisfy | arg( − η j ) | ∈ [ π − φ max , π ]. On the other hand, each root ζ of any p v will satisfy | ζ | ≥ r forsome r >
0, so that ∆ = min ≤ j ≤ N | η j | Re( η j ) ≥ ∆ > G ; for notational simplicity we may assume that ∆ ≤
1. We stillhave f = p /p = | E ( G ) | , so that Var( X P G ) ≥ ∆ | E | / X P G ) ≤ (1 + sec φ max ) W ( X P G ) , and therefore the condition (3.17) of Corollary 3.8 is satisfied if Var( X P G ) ≥ v ∗ where v ∗ is the larger root of v / = π (1 + sec φ max )3 · / ln v.
28o for Var( X P G ) ≥ v ∗ from Corollary 3.8 we obtain an LCLT in the formsup m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr( X P = m ) − e − ( m − E [ X P ]) X P ) p π Var( X P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | E ( G ) | , (5.13)with C = 200 /π ∆ .With more precise information on the location of the roots of p v for v ∈ V ( G ) one could extend this result to families in which the vertex degreesare not bounded, in the style of Theorem 5.2. For k = 2 the necessaryinformation was obtained in the discussion of Example 5.2; for k = 3 , Example 5.4.
Here C v = { , , } for v ∈ V ( G ) and C v = { , , , } for v ∈ V ( G ), with d arbitrary and d ≤ C = { , . . . , k } with k = 1 or 2 are covered by earlier examples). For v ∈ V ( G ) a root ζ of p v ( z ) satisfies | arg( ζ ) | < π/
4; for v ∈ V ( G ) all roots of p v ( z ) are ζ = − d v ≤
3, while when d v = 4 the roots of p v ( z ) = 1 + 4 z + 6 z + 4 z are − / − ± i ) /
2, so that all roots ζ satisfy | arg( ζ ) | ≤ π/
4. Thusfrom Corollary 4.3(b) the roots − η j of P G satisfy | arg( − η j ) | ∈ [ π − φ max , π ]for some φ max < π/
2. As in Example 5.3 we find again ∆ > ∆ for some d -dependent ∆ , leading to an LCLT of the form (5.13). Again, one mayalso find as in Example 5.2 an LCLT for a family of graphs in which d ( G )can increase with | E ( G ) | . Example 5.5.
This example relies on numerical computations, althoughone could probably justify these by obtaining rigorous bounds. We take C v = { , , } for for v ∈ V ( G ) and, for v ∈ V ( G ), C v = { , , . . . , k } with k either 3 or 4. The possible values of d and d are shown in Table 1; forexample, one may take d = 3, k = 3, and d = 5, 6, or 7. There are atotal of five possible examples. Also shown are angles φ , φ , obtained bycomputation with Maple, such that for v ∈ V i ( i = 1 , ζ of p v ( z )lies in [ π − φ i , π ]. Since in each case φ + φ < π/ Example 5.6.
In the examples considered above, each C v has been of theform { , , . . . , k } for some k . Now we take C v = { , } for v ∈ V ( G ), butfor v ∈ V ( G ) take C v to be either { , } or { , , } . To avoid vertices29 = 3 k = 4 d φ d φ d φ . · · · π . · · · π . π . · · · π . · · · π Table 1: Possible values of d and d with corresponding values of φ and φ .which are effectively disconnected from the rest of the graph we assume that d v ≥ v ∈ V ( G ), and again assume that d v ≤ d i for v ∈ V i ( G ), i = 1 , d and d fixed. Again p v ( z ), v ∈ V , has a single negative real root,while for v ∈ V ( G ), p v ( z ) = ˜ p v ( z ), and one finds easily that ˜ p ( w ), which iseither linear or quadratic, has only negative real roots, so that p v has purelyimaginary roots. Thus taking φ = 0 and φ = π/ − η j of P G satisfy Re( − η j ) ≤
0, so that a CLT will followfrom Theorem 3.1 once we verify that Var( X P ) → ∞ as N P → ∞ in thefamily P under consideration.Since in this case the roots − η j of P may lie on the imaginary axis, theestimates that we have been using for the variance, which begin with (3.8),are no longer effective. On the other hand, from (3.6) we haveVar( X P ) ≥ N P X j =1 | η j | (1 + | η j | ) ≥ N P j (cid:8) | η j | , | η j | − (cid:9) . (5.14)Since d and d are fixed we have upper and lower bounds 0 < r ≤ | ζ | ≤ R on the magnitudes of the roots ζ of the p v ( z ), and Theorem 4.1, togetherwith a corresponding result, with a similar proof, for upper bounds, impliesthat r ≤ | η j | ≤ R . N P is the size of the largest admissible configurationof occupied edges in G ; let M ⊂ E be an admissible configuration with | M | = N P . Each edge of M is incident on a unique vertex of V , and everyvertex of V must be joined by an edge of E to one of these vertices, sinceif v ∈ V were not so joined then two edges incident on v could be added to M . Thus | V | ≤ d N P , and since | E | ≤ d | V | , N P ≥ | E | /d d . From (5.14)we thus have Var( X P ) ≥ | E | d d min (cid:8) r , R − (cid:9) . (5.15)30 Further central limit theorems
In this section we give applications of Theorem 2.1, obtaining central limittheorems (but not local central limit theorems) in cases in which the zeros of P avoid a neighborhood of the point z on the positive real axis. Section 6.1presents examples for families of graph-counting polynomials and Section 6.2for families of polynomials arising from statistical mechanics. To apply thetheorem we will establish that, for the family of polynomials in question,Var( x P ) grows as N P . For this we will use the following result, due to Ginibre[11]: Theorem 6.1.
Let X be a random variable taking nonnegative integer valuesand let T m := m ! Pr { X = m } . If for some A > − and all m , ≤ m ≤ N − , T m +2 T m +1 ≥ T m +1 T m − A, (6.1) then Var( X ) ≥ E [ X ]1 + A . (6.2)
Proof.
The proof is elementary. Write E [ X ] (1 + A ) = (cid:18) E (cid:20) T X +1 T X + XA (cid:21)(cid:19) ≤ E "(cid:18) T X +1 T X + XA (cid:19) (6.3)and expand the right hand side, using (6.1) . In order to apply Theorem 6.1 to graph-counting polynomials, we show that(6.1) holds for these under a mild condition on the sets C ( v ) defining admis-sibility of subgraphs. Proposition 6.2.
Suppose that G is a graph with graph-counting polynomial P ( z ) and that for each vertex v of G , C ( v ) = { , , . . . , k v } (6.4) for some k v ≥ Then for all z > the quantities T m = m ! p m z m /P ( z ) satisfy (6.1) with A = (2 α + 1) z , where α := max v ∈ V [ d v − k v ] + .
31o prove Proposition 6.2 we first establish a lemma relating p m +1 and p m +2 to p m . Let M m be the set of admissible subgraphs with m edges, sothat p m = |M m | , and for each M ∈ M m let K ( M ) and K ( M ) be thenumber of subgraphs in M m +1 and M m +2 , respectively, which contain M ;equivalently, we may introduce E ( M ) = { e | e ∈ E \ M, { e } ∪ M ∈ M m +1 } , (6.5) E ( M ) = {{ e , e } | e , e ∈ E \ M, { e , e } ∪ M ∈ M m +2 } , (6.6)and define K ( M ) = | E ( M ) | , K ( M ) = | E ( M ) | . We will regard K and K as random variables, furnishing M m with the uniform probability measureProb( M ) = 1 /p m . Lemma 6.3. p m +1 = 1 m + 1 X M ⊂M m K ( M ) = E [ K ] m + 1 p m , (6.7) p m +2 = 2( m + 2)( m + 1) X M ⊂M m K ( M ) = 2 E [ K ]( m + 2)( m + 1) p m . (6.8) Proof.
Let S = { ( M, e ) | M ∈ M m , e ∈ E ( M ) } and notice that | S | = P M ∈M m K ( M ). S may be put in bijective correspondence with S ′ = { ( M ′ , e ) | M ′ ∈ M m +1 , e ∈ M ′ } , via the correspondence ( M, e ) ↔ ( M ′ , e )with M ′ = M ∪ { e } ; here we use the fact that each C ( v ) has the form(6.4), which implies that the subgraph obtained by deleting an edge froman admissible subgraph is admissible. Clearly | S ′ | = ( m + 1) p m +1 , and(6.7) follows from | S | = | S ′ | . Similarly, (6.8) is obtained from the cor-respondence of S = { ( M, { e , e } ) | M ∈ M m , { e , e } ∈ E ( M ) } with S ′ = { ( M ′ , { e , e } ) | M ′ ∈ M m +2 , e , e ∈ M ′ , e = e } . Proof of Proposition 6.2.
With A = (2 α + 1) z , (6.1) becomes, fromLemma 6.3, 2 E [ K ] − E [ K ] ≥ − (2 α + 1) E [ K ] . (6.9)Now notice that we may obtain E ( M ) by choosing a pair { e , e } of edgesfrom E ( M ) and then rejecting this pair if { e , e } ∪ M is not admissible,which can happen only if e and e share a vertex v with d M ( v ) ≥ k v −
1. Thusif we first choose e with vertices v, v ′ we will reject at most d v − k v + d v ′ − k v ′ e , e ); this counts unordered edge pairs twice, and wethus find that K ( M ) ≥ (cid:18) K ( M )2 (cid:19) − αK ( M ) . Thus 2 E [ K ] − E [ K ] ≥ E [ K ] − E [ K ] − (2 α + 1) E [ K ] , (6.10)verifying (6.9). Example 6.1.
Consider a family G of graphs such that for each vertex v of any G ∈ G , C v = { , , . . . , k v } with 1 ≤ k v ≤
4, and assume thatthe maximum degrees of the graphs are bounded by some fixed d max . Asdiscussed in Example 5.3, there is then an angle φ max (which may dependon d max ), with 0 ≤ φ max < π/
2, such that, for any v , each root ζ of p v ( z )satisfies | arg( ζ ) | ∈ [ π − φ max , π ]. Thus taking φ = φ max in Corollary 4.3(a)we see that the roots ζ j of P G satisfy | arg( ζ j ) | ∈ [ π − φ max , π ], and so forany z > z , which can be chosen uniformlyin G , which is free from zeros of P G .A CLT for the family P ( G ) will now follow from Theorem 3.1 once we showthat Var( X P ) grows faster than N / P in P ( G ), and with Proposition 6.2 thiswill follow from Ginibre’s result, Theorem 6.1, if we can show that E [ X P ]grows faster than N / P . But in fact it follows from Proposition 2.4 that E [ X P ] ≥ M N P , once we verify the hypotheses of that result. But since forany P G , p = 1 and p = | E ( G ) | ≥ N P G , condition (i) of the proposition,that p ≥ c p N P , is satisfied with c = 1. Moreover, since for v a vertex ofany G ∈ G the degree d v is uniformly bounded by d max , the possible roots of p v ( z ) are uniformly bounded away from zero, and by Corollary 4.3(b) so arethe roots of P G . This verifies condition (ii) and completes the proof of theCLT for P ( G ).We remark that, although the methods of Section 4 do not show that theroots of the graph-counting polynomials for the graphs considered here lie inthe left half plane, we do not have an example in which we know that someof these roots in fact lie in the right half plane. We consider an
Ising spin system in a finite subset Λ of the lattice Z d , thatis, a collection σ of spin variables σ ( x ), x ∈ Λ, taking values σ ( x ) = ±
1. Let33 ( σ ) be the number of sites for which σ = 1 (the number of “up spins”).The partition function of the system is P ( β, z ; Λ) = X σ z m ( σ ) e − βU ( σ ) = | Λ | X m =0 p m ( β ; Λ) z m , (6.11)where p m ( β ; Λ) = X { σ | m ( σ )= m } e − βU ( σ ) (6.12)Here U ( σ ) is the interaction energy for the spin configuration σ and β is theinverse temperature. The parameter z is the the magnetic fugacity , relatedto the (uniform) magnetic field h by z = e βh .In this section we will adopt the spin language above because it is thetraditional one for the discussion of the location of the zeros (in the variable z ) of P . Alternatively, however, one may make contact with the discussionin Section 1 by viewing this model as a system of particles, with site x ∈ Λoccupied by a particle if σ ( x ) = 1 and empty if σ ( x ) = − m ( σ ) is then thetotal number of particles in the system.For finite Λ there can be no zeros of P ( β, z ; Λ) for the physically relevantvalues of the fugacity—those on the positive real axis. This means that thethermodynamic pressure, Π( β, z ; Λ) = | Λ | − log P ( β, z ; Λ), is real analytic forall physically relevant fugacities and there can be no phase transitions , thatis, no non-analyticity in the pressure as a function of z .The situation is different in the thermodynamic limit Λ ր Z d . This limit,with translation invariant interactions U ( σ ) = − X x ∈ Z d X A J A + x Y y ∈ A σ ( x + y ) , (6.13)where P A runs over subsets A ⊂ Z d with 0 ∈ A and | A | ≥
2, and the J A are real coupling constants , which we always assume for simplicity satisfy P A | J A | < ∞ , is the right model for a macroscopic system containing, say,10 atoms, when we are not considering surface effects. In this limit thethermodynamic pressure is given byΠ( β, z ) = lim Λ ր Z d log P ( β, z ; Λ) | Λ | ; (6.14)the existence of this limit can be proved for very general J A . In the limit,however, the zeros of P ( β, z ; Λ) can approach the positive z -axis and thus34ause singularities in the pressure Π( β, z ). This is a standard mechanism forthe occurrence of phase transitions in statistical mechanical systems [32, 27].Suppose, on the other hand, that z is a point of analyticity of Π( β, z ),so that some neighborhood | z − z | < δ is free of zeros for | Λ | large. Let X := X β,z ;Λ be the random variable defined by (1.1) with p m = p m ( β, Λ) asin (6.11); X is the total number of up spins (or particles) in the system in Λat fugacity z and inverse temperature β . If we assume for the moment thatlim Λ ր Z d Var( X ) / | Λ | / = ∞ , (6.15)then Theorem 2.1 shows that the family of these random variables, as Λincreases, satisfies a CLT. Various cases in which such a fugacity z existsare known. We briefly describe some of these below.In a seminal paper [21], Lee and Yang proved that for ferromagnetic pairinteractions, U ( σ ) = − X x,y ∈ Λ J ( x, y ) σ ( x ) σ ( y ) , J ( x, y ) ≥ , (6.16)all the zeros of P ( z, β ; Λ) lie on the unit circle, | z | = 1. Translation invarianceis not needed here. In the translation invariant situation described above,however, the Lee-Yang result implies that Π( β, z ) is analytic in z for | z | 6 = 1,so that the number of up spins satisfies a CLT for z = 1. We remark thatRuelle [29] gave a general characterization of polynomials satisfying the Lee-Yang property, that all roots satisfy | z | = 1. He showed in particular that forIsing systems the only interactions U ( σ ) for which this property holds for all β are ferromagnetic pair interactions, the systems covered by the Lee-Yangtheorem. More recent references about Lee-Yang zeros can be found in [30].In the translation invariant case, which we shall consider from now on,more is known about the analyticity in z , at fixed β , of Π( β, z ). One canshow in particular [27] that (i) Π( β, z ) is analytic on the positive real z -axis, if β is sufficiently small (no phase transitions at high temperature),and (ii) P ( β, z ; Λ) is nonzero, and hence Π( β, z ; Λ) is analytic, in a disc | z | ≤ R ( β ; Λ), with R ( β ) := inf Λ R ( β ; Λ) >
0, for all β >
0, so that Π( β, z )is analytic for | z | < R ( β ). Each of these results yields a CLT for the corre-sponding real fugacities z .The behavior of the zeros for other interactions has been investigatedextensively, both analytically and numerically (see [20, 19] and references35herein). One can show [20], for certain classes of interactions U ( σ ), that forsome δ > P ( β, z ; Λ) satisfies Re ζ < − δ ; for these systems, X β, Λ satisfies the conditions of Corollary 3.8 and thus an LCLT. In othercases one can prove [20, 19] that for β large the zeros stay away from thepositive z -axis and X β, Λ thus satisfies a CLT by Theorem 3.1. Such CLThave been obtained by other methods; see for example [7] and the discussionin [10].In some cases in which the zeros do approach the real z -axis at some z in the Λ ր Z d limit it is known that the fluctuations in X β,z ;Λ are in factnot Gaussian in the Λ ր Z d limit [22, 1].We finally want to justify the assumption (6.15) made above. FromProposition 2.4 we can conclude that E [ X ] ≥ M | Λ | for some M >
0, oncewe verify the hypotheses of that result. Condition (i), that p ≥ c p | Λ | , fol-lows from (6.12): the sum defining p ( β ; Λ) contains only one term and thatdefining p ( β ; Λ) contains | Λ | terms, each nonzero, and the ratio e − βU ( σ ) /p ,for m ( σ ) = 1, is independent of σ by translation invariance, at least upto “boundary effects,” and these can be ignored for | Λ | large. Condition(ii) follows from the fact, mentioned above, that no zeros of P ( β, z ; Λ) liein the disc | z | < R ( β ). With this, Ginibre’s result Theorem 6.1 givesVar( X β, Λ ) ≥ M | Λ | / (1 + A ). We need to know, of course, that (6.1) holds forthe spin systems under consideration here. In fact this is true more generally,as we show in Appendix B. Acknowledgments.
The work of J.L.L. was supported in part by NSFGrant DMR 1104500. The research of B.P. was supported by the NSF underGrant No. DMS 1101237. We thank S. Goldstein for a helpful discussionand Dima Iofee for bringing [6] to our attention.
A Grace’s Theorem and Asano contractions
Theorem A.1 ( Grace’s theorem).
Let P ( z ) be a complex polynomial inone variable of degree at most n , and let Q ( z , . . . , z n ) be the unique multi-affine symmetric polynomial in n variables such that Q ( z, . . . , z ) = P ( z ) .If the n roots of P are contained in a closed circular region K and z / ∈ K, . . . , z n / ∈ K , then Q ( z , . . . , z n ) = 0 . Here a closed circular region is a closed subset K of C bounded by a circleor a straight line. If P is in fact of degree k with k < n then we say that36 − k roots of P lie at ∞ and take K noncompact. For a proof of the resultsee Polya and Szeg¨o [25, V, Exercise 145]. Lemma A.2 ( Asano-Ruelle Lemma [2, 26]).
Let K , K be closed subsetsof C , with K , K . If Φ is separately affine in z and z , and if Φ( z , z ) ≡ A + Bz + Cz + Dz z = 0 whenever z / ∈ K and z / ∈ K , then ˜Φ( z ) ≡ A + Dz = 0 whenever z / ∈ − K · K . Here we have written − K · K = {− uv | u ∈ K , v ∈ K } . The mapΦ ˜Φ is called Asano contraction ; we denote it by ( z , z ) → z . B Ginibre’s theorem for particle systems
We consider a set Λ of N sites and populate these with a random configura-tion of distinguishable particles, at most one particle per site, in such a waythat the probability of having exactly m sites occupied is given as in (1.1)by p m z m /P ( z ), where P ( z ) = P Nm =0 p m z m and p m = 1 m ! X Y m e − U ( Y m ) . (B.1)In (B.1) the sum is over ordered m -tuples Y m = ( y , . . . , y m ) with y i = y j for i = j , and U ( Y m ) = U ( y , . . . , y m ) is the potential energy of the system whensite y i is occupied by particle i , i = 1 , . . . , m , and the remaining N − m sitesare empty. The energy U is invariant under permutation of its arguments.It will be convenient to allow sums such as that of (B.1) to run over all Y m ∈ Λ m , so we define U ( y , . . . , y m ) = + ∞ whenever y i = y j for any i, j .Thus the quantity T m appearing in (6.1) is T m = z m P ( z ) X Y m ∈ Λ m e − U ( Y m ) . (B.2)37et us define functions V ( Y m | x m +1 ) and W ( Y m | x m +1 , x m +2 ) by the re-quirement that they be + ∞ when any two arguments coincide, and otherwisesatisfy U ( Y m +1 ) = U ( Y m ) + V ( Y m | y m +1 ) , (B.3) U ( Y m +2 ) = U ( Y m ) + V ( Y m | y m +1 ) + V ( Y m | y m +2 )+ W ( Y m | y m +1 , y m +2 ) . (B.4)Note that V ( Y m +1 | y m +2 ) = V ( Y m | y m +2 ) + W ( Y m | y m +1 , y m +2 ) . (B.5)For any function F ( Y m ) we define F + = max { F, } and F − = min { F, } .With this notation the two key hypotheses needed for the result are D := sup ≤ m ≤| Λ |− sup Y m +1 ∈ Λ m +1 X y m +2 ∈ Λ (cid:0) − e − βW + ( Y m | y m +1 ,y m +2 ) (cid:1) dy < ∞ , (B.6)and − B := inf ≤ m ≤| Λ |− inf Y m +1 ∈ Λ m +1 V ( Y m | y m +1 ) > −∞ . (B.7)Note that it follows from (B.7) that for any m and Y m +2 ∈ Λ m +2 , V ( Y m | y m +1 ) + W − ( Y m | y m +1 , y m +2 ) ≥ − B, (B.8)since if W ( Y m | y m +1 , y m +2 ) ≥ m replaced by m + 1. Weremark that in the spin language of Section 6.2 the condition, for translationinvariant systems, that P A | J A | < ∞ (see (6.13)) implies (B.8). Remark B.1.
These conditions look somewhat artificial for the general po-tentials we are considering here, but more natural in the case of pair inter-actions, when U ( Y m ) = P ≤ i = j ≤ m φ ( y i , y j ). Then D = sup y ∈ Λ X x ∈ Λ (cid:0) − e − βφ ( x,y ) (cid:1) and − B = inf x ∈ Λ inf Λ ′ ⊂ Λ X y ∈ Λ ′ , y = x φ ( x, y ) . (B.9)The next result was stated in [11] but only for the pair potentials ofRemark B.1; the proof was not given but was attributed to a private com-munication and a preprint. 38 heorem B.2. Suppose that (B.6) and (B.7) hold. Then for m ≤ N − , T m +1 − T m T m +2 ≤ ze βB D T m T m +1 . (B.10) Proof.
We make a preliminary calculation: e − β [ V ( Y m | x )+ W ( Y m | x,y )] = e − βV ( Y m | x ) (cid:2)(cid:0) e − βW ( Y m | x,y ) − (cid:1) + 1 (cid:3) ≥ e − β [ V ( Y m | x ) (cid:2) e − βW − ( Y m | x,y )] (cid:0) e − βW + ( Y m | x,y ) − (cid:1) + 1 (cid:3) ≥ e βB (cid:0) e − βW + ( Y m | x,y ) − (cid:1) + e − βV ( Y m | x ) , (B.11)where we have used (B.8). Now with this, T m +1 − T m T m +2 = z m +2 P ( z ) X X m ⊂ Λ X Y m ⊂ Λ X x,y ∈ Λ e − β [ U ( X m )+ U ( Y m )+ V ( Y m | y )] × (cid:2) e − βV ( X m | x ) − e − β [ V ( Y m | x )+ W ( Y m | x,y )] (cid:3) ≤ z m +2 P ( z ) X X m ⊂ Λ X Y m ⊂ Λ X x,y ∈ Λ e − β [ U ( X m )+ U ( Y m )+ V ( Y m | y )] × (cid:2)(cid:0) e − βV ( X m | x ) − e − βV ( Y m | x ) (cid:1) − e βB (cid:0) e − βW + ( Y m | x,y ) − (cid:1)(cid:3) . := R + R , (B.12)where R arises from the term (cid:0) e − βV ( X m | x ) − e − βV ( Y m | x ) (cid:1) and R from theterm − e βB (cid:0) e − βW + ( Y m | x,y ) − (cid:1) . We may average the formula for R given in(B.12) with the equivalent formula obtained by interchanging the X m and Y m summation variables to obtain R = − z m +2 P ( z ) X X m ⊂ Λ X Y m ⊂ Λ e − β [ U ( X m )+ U ( Y m )] × "X x ∈ Λ (cid:0) e − βV ( X m | x ) − e − βV ( Y m | x ) (cid:1) ≤ . (B.13)For R we can use (B.6) to estimate the sum over x and thus obtain R ≤ e βB D z m +2 P ( z ) X X m ⊂ Λ X Y m ⊂ Λ X y ∈ Λ e − β [ U ( X m )+ U ( Y m )+ V ( Y m | y )] = ze βB DT m T m +1 . (B.14)Now (B.10) follows from (B.13) and (B.14).39 eferences [1] M. Aizenman, Geometric analysis of φ fields and Ising models. I, II. Commun. Math. Phys. , 1–48 (1982).[2] T. Asano, Theorems on the partition functions of the Heisenberg ferro-magnets. J. Phys. Soc. Jap. , 350–359 (1970).[3] E. A. Bender, Central and local limit theorems applied to asymptoticenumeration. J. Comb. Theor. A , 91-111 (1973).[4] E. R. Canfield, Application of the Berry-Ess´een inequality to combina-torial estimates. J. Comb. Theor. A , 17–25 (1980).[5] E. R. Canfield, Asymptotic normality in enumeration. A chapter in Handbook of Enumerative Combinatorics. (Preliminary copy, privatelycommunicated.)[6] R. L. Dobrushin and S. B Shlosman, Large and moderate deviations inthe Ising model. Pp. 91–219 in
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