Virtile Reguli And Radiational Optaprints

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1 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES GEOFFREY GRIMMETT AND SVANTE JANSON Abstract. We study the radom graph G,λ/ coditioed o the evet that all vertex degrees lie i some give subset S of the oegative itegers. Subject to a certai hypothesis o S, the empirical distributio of the vertex degrees is asymptotically Poisso with some parameter bµ give as the root of a certai characteristic equatio of S that maximizes a certai fuctio ψ Sµ). Subject to a hypothesis o S, we obtai a partial descriptio of the structure of such a radom graph, icludig a coditio for the existece or ot) of a giat compoet. The requisite hypothesis is i may cases beig, ad applicatios are preseted to a umber of choices for the set S icludig the sets of respectively) eve ad odd umbers.. Itroductio Let S be a fixed oempty set of o-egative itegers. The purpose of this paper is to study the structure of radom graphs havig all their vertex degrees restricted to the set S. We call a graph a S-graph if all its vertex degrees belog to S. For example, if S = {s} is a sigleto, a S-graph is the same as a regular graph of degree s. We are ot goig to say aythig ew about this case.) Oe of our mai examples is the class of Euleria graphs, or eve graphs, give by the set of eve umbers S = 2Z 0, with Z 0 the set {0,, 2,... } of o-egative itegers. See Sectio 6 for further examples. More precisely, we will study the radom graph G,p;S defied as G,p coditioed o beig a S-graph, where G,p is the stadard radom subgraph of the labelled) complete graph K where two vertices are joied by a edge with probability p 0, ), ad these 2) evets, correspodig to the edges of K, are idepedet. I other words, G,p;S is a radom S-subgraph of K such that, if G is ay give subgraph of K that is a S-graph, the PG,p;S = G) = peg) p) 2) eg) PG,p is a S-graph),.) Date: December, 2007; revised Jue 28, Mathematics Subject Classificatio. 05C80, 05C07, 60C05. Key words ad phrases. Radom graph, eve graph, radom-cluster model.

2 2 GEOFFREY GRIMMETT AND SVANTE JANSON where eg) is the umber of edges of G. We are iterested i asymptotics as, ad we will tacitly cosider oly such that there exists a S-graph with vertices, i other words, such that the deomiator i.) is o-zero. Thus, a fiite umber of small may be excluded; moreover, if S cotais oly odd itegers, the has to be eve. It is easy to see that, apart from this parity restrictio, all large are allowed.) For example, with S the set of eve umbers, we obtai a radom eve graph, see Example 6.3. This is related to the radom-cluster model o the complete graph K [6; 7], ad was oe of our motivatios for the preset study. Remark.. The choice p = 2 gives a radom S-graph that is uiformly distributed over all S-graphs o labelled vertices. However, we will i this paper istead study the case whe p is of order / ad the average vertex degree is bouded for G,p, ad as we shall see later, for G,p;S too). It follows immediately from.) that two S-subgraphs of K with the same degree sequece are attaied with the same probability. Hece, the coditioal distributio of G,p;S give the degree sequece is uiform. We will therefore focus o studyig the radom degree sequece of G,p;S ; it is the possible to obtai further results o the structure of G,p;S by applyig stadard results o radom graphs with give degree sequeces to G,p;S coditioed o the degree sequece. For example, usig the results by Molloy ad Reed [4, 5] we obtai Theorem 3. below o existece of a giat compoet i G,p;S. 2. Mai theorem By symmetry, the labellig of the vertices ad thus the order of the degree sequece is ot importat, ad we shall therefore study the umbers of vertices with give degrees, rather tha the degree sequece itself. We itroduce some otatio. Let N be the set of sequeces = 0,,... ) of o-egative itegers j Z 0 with oly a fiite umber of o-zero terms j. Let N S := { N : j = 0 whe j / S}, the set of such sequeces supported o S. For a multi)graph G, let j G) be the umber of vertices of degree j i G, j Z 0, ad let G) := j G)) j=0 be the sequece of degree couts. Thus, G is a S-graph if ad oly if G) N S. Clearly, cf..), PG,p is a S-graph) = 2) eg). 2.) G: G) N S p eg) p) Usig a term that is stadard i statistical physics, we shall call this summatio the partitio fuctio, deoted as Z,p;S. Let P be the set of probability distributios o Z 0. I other words, P is the set of sequeces π = π 0, π,... ) of o-egative real umbers such

3 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 3 that j π j =. We regard P as a topological space with the usual topology of weak covergece deoted ); p it is well kow that this topology o P may be metrised by the total variatio distace d TV π, π ) := 2 π j π j. If G has = j jg) vertices, let π j G) := j G)/, the proportio of vertices of degree j, ad πg) := G)/ = π j G)) j=0. Note that πg) is the probability distributio of the degree of a radomly chose vertex i G. Let φ S be the expoetial geeratig fuctio of S, j φ S µ) := j S µ j j!. 2.2) Note that this is a etire fuctio of µ, ad that φ S µ) > 0 for µ > 0 while φ S 0) > 0 if ad oly if 0 S. Let Po S µ) be the distributio of a Poµ) distributed variable give that it belogs to S, i.e., LX µ X µ S) with X µ Poµ). Thus, recallig 2.2), Po S µ){k} = µk /k!, k S. 2.3) φ S µ) This coditioal distributio is always defied for µ > 0, ad i the case 0 S for µ = 0 too i which case it is a poit mass at 0). The mea of the Po S µ) distributio is EX µ X µ S) = kµ k φ S µ) k! k S Let λ > 0. We shall refer to the equatio = µφ S µ) φ S µ). 2.4) µφ S µ) φ S µ) = µ2 λ, 2.5) as the characteristic equatio of the set S for this value of λ), ad we write { Eλ) := µ 0 : µφ S µ) φ S µ) = µ2 }, 2.6) λ where we allow µ = 0 oly if 0 S. We further defie the auxiliary fuctio ψ S µ) := log φ S µ) µφ S µ) 2φ S µ) ad ote that whe µ Eλ), ψ S µ) equals the simpler fuctio 2.7) ψ S, µ; λ) := log φ S µ) µ2 2λ. 2.8) All logarithms i this paper are atural. We let c, C,... deote positive costats, geerally depedig o S ad λ or λ )) ad sometimes o other parameters too but ot o ), which may be idicated by argumets. We

4 4 GEOFFREY GRIMMETT AND SVANTE JANSON use C i for large ad c i for small costats.) > to avoid trivialities. We sometimes assume that Theorem 2.. Let λ λ > 0 ad suppose that Eλ) cotais a uique µ = µλ) that maximizes ψ S µ) or, equivaletly, ψ S, µ; λ)) over Eλ). The, the followig hold, as : p i) πg,λ/;s) Po S µ). I other words, for every j S, j G,λ/;S) p Po S µ){j} = µj /j! φ S µ). 2.9) ii) All momets of the radom distributio πg,λ/;s) coverge to the correspodig momets of Po S µ). I other words, if d,..., d is the degree sequece of G,λ/;S, ad X µ Poµ), the for every r 0, ), i= d r i = j=0 p I particular, eg,λ/;s) j r j G,λ/;S) j r Po S µ){j} = EXbµ r X bµ S). j=0 2.0) p 2 EX bµ X bµ S) = µφ S µ) 2φ S µ). 2.) iii) The error probabilities i i) decay expoetially: for every ε > 0, there exists a costat c = c ε, λ, S) > 0 such that, for all large, iv) We have that P d TV πg,λ/;s), Po S µ) ) ε ) e c. 2.2) log Z,λ /;S = log PG,λ / is a S-graph) ψ S µ) 2λ. 2.3) Remark 2.2. The reaso for takig large i 2.2) is as follows. Suppose, for example, that S is a ifiite set. The the left had side of 2.2) trivially equals for ay fixed ad sufficietly small ε depedig o ). Oe way aroud this, at least whe λ λ, is to replace the right had side of 2.2) by 2e c 2 for a suitable small c 2 > 0. The iequality the becomes trivial for small. More geerally, let E 0 λ) be the subset of Eλ) where ψ S µ) or ψ S, µ; λ)) is maximal: { } E 0 λ) := µ Eλ) : ψ S µ) = max ψ Sµ ). 2.4) µ Eλ) If E 0 λ) cotais a sigle elemet, we thus take that elemet as µλ); i particular, if Eλ) =, the Eλ) = E 0 λ) = { µλ)}. We shall see i

5 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 5 Sectio 4 that this is the ormal case: Eλ) is always fiite ad o-empty, ad E 0 λ) = except for at most a coutable umber of values of λ. Further results o µ ad the auxiliary fuctios are give i Sectio 4. Remark 2.3. The set E 0 λ) may cotai more tha oe elemet, see Example 6.8. I this case, the theorem may ot be applied, but the proof i Sectios 8 9 exteds to show that the radom degree distributio πg,λ/;s) approaches the fiite set F := {Po S µ) : µ E 0 λ)} i the sese that the aalogue of 2.2) holds for the distace to this set, i.e., P mi µ E 0 λ) d TV πg,λ/;s), Po S µ) ) ε) e c. 2.5) We ca regard the distributios i F as pure phases, i aalogy with the situatio for may ifiite systems of iterest i statistical physics, but for the fiite systems cosidered here this has to be iterpreted asymptotically. Thus, for large, the degree distributio of G,λ/;S is approximately give by oe of the pure phases, but we do ot kow which oe. It follows that if we let, oe of the followig happes for the radom degree distributio π = πg,λ/;s) regarded as a elemet of P): i) π coverges i probability to Po S µ) for some µ E 0 λ). ii) π coverges i distributio to some o-degeerate distributio o F. A mixture of two or more pure phases.) iii) There are oscillatios ad π does ot coverge i distributio; suitable subsequeces coverge as i i) or ii), but differet subsequeces may have differet limits. It is easy to show by a cotiuity argumet that all three cases may occur i Example 6.8 for suitable sequeces λ ) with λ λ 0 defied there). We do ot kow whether all three cases may occur for fixed λ with E 0 λ) >. We shall ot ivestigate the case E 0 λ) > further here. We close this sectio with a iformal explaatio of the results of Theorem 2., several applicatios of which are preseted i Sectio 6. Recall the partitio fuctio Z,λ/;S of 2.), cosidered as a summatio over suitable graphs. We wish to establish which graphs are domiat i this summatio. I so doig, we will treat certai discrete variables as cotiuous, ad shall study maxima by differetiatio ad Lagrage multipliers. Let π = π 0, π,... ) be a sequece of o-egative reals satisfyig i π i =, ad π i = 0 for i / S. We write ν = νπ) = i Let Zπ) represet the cotributio to the summatio of 2.) from graphs G havig, for each i, approximately π i vertices with degree i. The empirical) mea vertex-degree of such a graph is ν. Now, Zπ) is a summatio over simple graphs subject to costraits o the vertex degrees. It may be approximated by a similar summatio iπ i.

6 6 GEOFFREY GRIMMETT AND SVANTE JANSON over certai multigraphs, ad this is easier to express i closed form, as follows. The umber of ways of partitioig vertices ito sets V 0, V,... of respective sizes π i, i 0, is! π 0 )!π )!. Each vertex v V i will be take to have degree i, ad we therefore provide v with i half-edges. Each such half-edge will be coected to some other halfedge to make a whole edge. Sice half-edges are cosidered idistiguishable, we shall require the multiplicative factor { } i!) π i i S i order to treat them as labelled. The total umber of half-edges is 2N = i iπ i = ν, ad we assume for simplicity that N is a iteger. These half-edges may be paired together i ay of 2N )!! = 2N )2N 3) 3 = 2N)! 2 N N! ways, ad each such pairig cotributes λ ) N λ ) 2) N to the sum. We combie the above to obtai a approximatio to Zπ): { }! ) Zπ) i!) π 2N)! i λ N π 0 )!π )! 2 N λ ) 2) N. N)! i S By Stirlig s formula, as, log Zπ) 2 ν logλν/e) 2 λ π i logi! π i ). 2.6) i S We maximize the last expressio subject to i π i = to fid that π i = A µi, i S, 2.7) i! for some costat A ad some µ satisfyig µ = λν. 2.8) Thus π is the mass fuctio of the Po S µ) distributio ad, by 2.4) ad the defiitio of ν, ν = µφ S µ) φ S µ). 2.9) We combie this with 2.8) to obtai the characteristic equatio 2.5). If there exists a uique µ satisfyig the characteristic equatio, the we are doe. If there is more tha oe, we pick the value that maximizes the right had side of 2.6). That is to say, the expoetial asymptotics of

7 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 7 Z,λ/;S are domiated by the cotributios from graphs with π satisfyig 2.7) with µ chose to satisfy the characteristic equatio ad to maximize ψ S µ). Note from 2.6) that log Zπ) ψ Sµ) 2λ, 2.20) ad part iv) of Theorem 2.) is explaied. We make the above argumet rigorous i the forthcomig proof of Sectios 8 9, a substatial part of which is devoted to provig that the coditioal distributio of vertex degrees is cocetrated ear its mode. The half-edge method has bee used by may authors datig back to [4] or earlier, ad refereces will be icluded at appropriate places later. 3. The giat ad the core We show ext how to apply Theorem 2., i cojuctio with results of Molloy ad Reed [4, 5] ad Jaso ad Luczak [0, ], to idetify the sizes of the giat cluster ad the k-core of G,λ/;S. The proofs are deferred to Sectio 0. We cosider first the existece or ot of a giat compoet i the radom S-graph G,λ/;S as λ λ > 0 i.e., a compoet of size comparable to ). Let π = π 0, π,... ) be a vector of o-egative reals with sum, ad write ν = j jπ j. As explaied i [4; 5], if we cosider the radom graph with a give degree sequece d = d i ), i which there are + o))π j vertices with degree j, the quatity that is key to the existece of a giat compoet is Qπ) := jj 2)π j. j Subject to certai coditios, if Qπ) > 0, there exists a giat compoet, while there is o giat compoet whe Qπ) 0. We shall apply this with π j = Po S µ){j}, ad to that ed we itroduce some further otatio. Let, see 2.4), νµ) := j j Po S µ){j} = µφ S µ) φ S µ), 3.) Qµ) := j χ µ ξ) := j jj 2) Po S µ){j} = µ2 φ S µ) µφ S µ), φ S µ) 3.2) ) j Po S µ){j}ξ 2 ξ j ) = νµ)ξ 2 µξ φ S µξ φ S µ). 3.3) Note that χ µ 0) = χ µ ) = 0. Furthermore, the oly possibly egative term i the sums i 3.2) ad 3.3) for ξ 0, ) are those with j =, while the terms with j = 0 ad j = 2 always vaish ad the others are positive uless Po S µ) = 0.

8 8 GEOFFREY GRIMMETT AND SVANTE JANSON Let Γ,p;S be the compoet of G,p;S with the largest umber of vertices, ad let Γ 2),p;S be the secod largest. Break ties by ay rule.) We write vh) for the umber of vertices i a graph H. Theorem 3.. Suppose that S {0, 2}. Let λ λ > 0 ad suppose that E 0 λ) cotais a uique elemet µ 0. The, G,λ/;S has a giat compoet if ad oly if Q µ) > 0, i.e., if ad oly if µφ S µ) > φ S µ). More precisely, as, where vγ,λ/;s) vγ 2),λ /;S ) p γ 0, γ = γ µ) := j eγ,λ/;s) p 0, eγ 2),λ /;S ) p ζ 0, p 0, ξ j Po S µ){j} = φ S ξ µ) φ S µ), 3.4) ζ = ζ µ) := 2 ν µ) ξ 2) 3.5) with ξ = ξ µ) [0, ] give as follows: i) if Q µ) > 0 ad S, the ξ 0, ) is the uique solutio to χ bµ ξ) = 0 with 0 < ξ <, ad γ, ζ > 0; ii) if Q µ) > 0 ad / S, the ξ = 0 ad γ = /φ S µ) > 0, ζ = 2ν µ) > 0; iii) if Q µ) 0, the ξ = ad γ = ζ = 0. Remark 3.2. It is easily see, usig 3.3), that ξ equals the extictio probability of a Galto Watso process with offsprig distributio PX = j ) = j Po S µ){j} ν µ j µφ = S µ) ), j S \ {0}, j )! φ S µ) φ S µ) that is, the distributio Po S µ) where S := {k 0 : k+ S}. Note that φ S µ) = φ S µ).) Hece γ, the asymptotic relative size of Γ,λ /;S, equals by 3.4) the survival probability of a Galto Watso process with offsprig distributio Po S µ) ad iitial distributio Po S µ). Remark 3.3. If µ = 0, the Q µ) = 0 ad we are i Case iii) with o giat compoet; i fact, by Theorem 2., 0 / p so almost all vertices are isolated. I this case χ bµ ξ) = 0 for all ξ. If µ > 0, the χ bµ < 0 o 0, ξ) ad χ bµ > 0 o ξ, ) i all three cases, as follows from [, Lemma 5.5], which yields aother characterizatio of ξ. Remark 3.4. If / S, the Q µ) > 0 as soo as µ > 0. Hece we are i Case ii) if µ > 0 ad i Case iii) i µ = 0.

9 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 9 Remark 3.5. We have excluded the cases S {0, 2}, i.e., the trivial case S = {0} ad the cases {2} ad {0, 2} that are exceptioal; i the latter cases vγ,λ/;s) has a cotiuous limitig distributio ad thus ot a costat limit whe µ > 0, see Examples 6.2 ad 6.7. We ote also that Qµ) = 0 for all µ i the excluded cases. The k-core of a graph G is the largest iduced subgraph havig mimimum vertex degree at least k. The k-core of a Erdős Reńyi radom graph has attracted much attetio; see [0] ad the refereces therei. Theorem 2. may be applied i cojuctio with Theorem 2.4 of Jaso ad Luczak [0] to obtai the asymptotics of the k-core of G,λ/;S. Let K k),p;s deote the k-core of G,p;S. We shall require some further otatio i order to state our results for K k),λ. /;S Let k {2, 3,... }. Let µ 0, ad let W µ be a radom variable with the Po S µ) distributio. For r [0, ], let W µ,r be obtaied by thiig W µ at rate r so that, coditioal o W µ, W µ,r has the biomial distributio BiW µ, r). For k {2, 3,... }, let h µ,k r) = EW µ,r I {Wµ,r k}) = l PW µ,r = l), h µ,k r) = PW µ,r k). Theorem 3.6. Let λ λ > 0 ad suppose that E 0 λ) cotais a uique elemet µ. Let k 2, ad let, with ν = ν µ) as above, As : i) if r = 0, vkk),λ ) p /;S 0, if, further, k 3, the l=k r = sup{r : νr 2 = h bµ,k r)}. PK k),λ /;S ekk),λ ) p /;S 0; is empty) ; ii) if r > 0, ad i additio νr 2 < h bµ,k r) o some o-empty iterval r ε, r), the vkk),λ ) /;S ekk),λ ) /;S p h bµ,k r), p 2 h bµ,k r) = 2 ν r2. Remark 3.7. Let X µ be the Galto Watso process with offsprig distributio Po S µ), started with a sigle idividual o, ad let X µ be the modified process where the first geeratio has distributio Po S µ), cf. Remark 3.2. It may be see that r is the probability that the family tree of X µ cotais a ifiite subtree with root o ad every ode havig k childre. Similarly,

10 0 GEOFFREY GRIMMETT AND SVANTE JANSON h bµ,k r) equals the probability that X µ cotais a ifiite k-regular subtree with root o the root has k childre ad all other vertices have k ). It is easy to see heuristically that this yields the asymptotic probability that a radom vertex belogs to the k-core, see Pittel, Specer ad Wormald [6] for S = Z 0 ), but it is difficult to make a proof based o brachig process theory; see Riorda [7] where this is doe rigorously for aother radom graph model. 4. Roots of the characteristic equatio To avoid some trivial complicatios, we assume throughout this sectio that S {0}, thus excludig the trivial case S = {0} for which G,λ/;S comprises isolated vertices oly. Lemma 4.. φ S µ) C φ S µ) for all µ. Proof. By 2.2), φ Sµ) = j µj. j! j S We split the sum ito two parts. For j 4µ, jµ j 4µ j = 4φ S µ). j! j! j S, j 4µ j S For j > 4µ, Stirlig s formula implies jµ j jµ j e ) j 4 j e j j! j ad thus, for µ, j S, j>4µ jµ j j! j= e ) j 4 = C2 = C 3 φ S ) C 3 φ S µ). 4 Theorem 4.2. For each λ > 0, the set Eλ) is fiite ad o-empty. Proof. The characteristic equatio 2.5) may be writte as hµ) = 0 where hµ) = λµφ S µ) µ2 φ S µ). By Lemma 4., for µ, hµ) C λµ µ 2 )φ S µ), 4.) ad thus hµ) < 0 for µ > C 4 = max{c λ, }. Sice h is a etire fuctio, ad does ot vaish idetically by what we just have show, it has oly fiitely may zeros i each bouded subset of the complex plae, ad i particular i the iterval [0, C 4 ]. Hece Eλ) is fiite. To see that Eλ) is o-empty, let s be the smallest elemet of S. If s = 0, the 0 Eλ). If s > 0, the hµ) λsµ s /s! as µ 0, so hµ) > 0 for small positive µ. Sice further hµ) is egative for large µ, h possesses a zero o the positive real axis.

11 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES We have defied µ as the maximum poit of ψ S or ψ S, o Eλ). The ext theorem shows that, alteratively, it ca be defied as the maximum poit of ψ S, o [0, ) but ot of ψ S ). Furthermore, istead of ψ S,, we ca use the fuctio ψ S,2 µ; λ) := log φ S µ) + µφ S µ) 2φ S µ) log λφ S µ) µφ S µ) ), 4.2) that arises as follows. I Sectio 7, we will idicate the use of multigraphs i provig Theorem 2., of which we shall derive a multigraph equivalet at Theorem 7.3. With Z,λ/;S deotig the multigraph partitio fuctio, we shall see i the proof of Theorem 7.3 that ψ S,2 µ; λ) represets the cotributio to log Z,λ/;S from multigraphs with degree distributio close to Po S µ), see Remark 8.3. For this reaso, ψ S,2 is a more atural fuctio tha ψ S,, although it has a more complicated formula. We shall have to exclude the trivial case whe S = {s} is a sigleto; i this case ψ S,2 µ; λ) = 2 s logλs) ) logs!) is costat. It is easily see that ψ S,2 µ; λ) ψ S, µ, λ) for all µ ad λ > 0, with equality if ad oly if µ Eλ). We regard ψ S, ad ψ S,2 as fuctios of µ, with λ cosidered a fixed parameter. These fuctios are evidetly aalytic o 0, ). Note that if 0 S, the φ S 0) = ad ψ S µ), ψ S, µ; λ) ad ψ S,2 µ; λ) are cotiuous at µ = 0 with ψ S 0) = ψ S, 0; λ) = ψ S,2 0; λ) = 0. O the other had, if 0 / S, the φ S 0) = 0 ad ψ S, µ; λ), while a simple calculatio yields ψ S,2 µ; λ) 2 s logλs) ) logs!) as µ 0, where s = mi S. Theorem 4.3. The followig hold for every fixed λ > 0 ad j = or 2, except for j = 2 i the trivial case S =. i) Eλ) is the set of statioary poits of ψ S,j, possibly with 0 added: Eλ) 0, ) = { µ : d dµ ψ S,jµ; λ) = 0 }. ii) E 0 λ) is the set of global maximum poits of ψ S,j : { } E 0 λ) = µ : ψ S,j µ; λ) = max ψ S,jµ ; λ). µ 0 Proof. i): Differetiatio yields d dµ ψ S,µ; λ) = φ S µ) φ S µ) µ λ ad, after some simplificatios, d dµ ψ S,2µ; λ) = d µφ S µ) ) log dµ 2φ S µ) λφ S µ) µφ S µ) By the Cauchy Schwarz iequality, provided µ > 0 ad S 2, µ d µφ ) S µ) = µ d ) j S jµj /j! dµ φ S µ) dµ j S µj /j! 4.3) ). 4.4)

12 2 GEOFFREY GRIMMETT AND SVANTE JANSON = j S j2 µ j /j! ) j S µj /j! ) j S jµj /j! ) 2 j S µj /j! ) 2 > ) See Theorem 5.2 below for a more geeral result.) By 4.3) 4.5), for µ > 0, ψ S,j µ; λ) = 0 if ad oly if φ S µ)/φ Sµ) = µ/λ, i.e., the characteristic equatio 2.5) holds. ii): By 4.3) 4.5) ad Lemma 4., ψ S,j is decreasig for large µ. Furthermore, by the remarks prior to the theorem, ψ S,j is either cotiuous at 0 or teds to there. This implies that ψ S,j has a fiite maximum, attaied at oe or several poits i [0, ). It remais to show that the maximum poits belog to Eλ); it the follows that E 0 λ) equals the set of maximum poits. If µ > 0 is a maximum poit of ψ S,j, the ψ S,j µ) = 0 ad µ Eλ) by i). If 0 is a maximum poit ad 0 S, the 0 Eλ) by defiitio. Fially, if 0 / S, ad s := mi S > 0, the φ S µ)/φ Sµ) s/µ as µ 0, ad it follows from 4.3) 4.5) that ψ S,j > 0 for small µ; hece 0 is ot a maximum poit i this case. I fact, for j =, ψ S, 0) = whe 0 / S, as remarked above.) We defie µ λ) := mi E 0 λ) ad µ λ) := max E 0 λ); thus E 0 λ) = ad Theorem 2. applies) if ad oly if µ = µ, ad i that case µ = µ = µ. We have defied µ oly whe E 0 λ) = ; for coveiece we exted the defiitio to all λ > 0 by lettig µλ) be ay elemet of E 0 λ) for example µ λ) or µ λ)). Corollary 4.4. For every λ > 0 ad j =, 2, ψ S µλ)) = ψ S,j µλ); λ) = max µ 0 ψ S,jµ; λ). Theorem 4.5. If 0 < λ < λ 2, the µλ ) µλ 2 ), with equality oly if µλ ) = µλ 2 ) = 0. Proof. If µ < µ λ ) E 0 λ ), the ψ S, µ; λ ) ψ S, µ λ ); λ ) by Theorem 4.3ii), ad thus ψ S, µ; λ 2 ) = ψ S, µ; λ ) + µ2 ) 2 λ λ 2 < ψ S, µ λ ); λ ) + µ λ ) 2 ) 2 λ λ 2 = ψ S, µ λ ); λ 2 ), so µ is ot a global maximum poit of ψ S, µ; λ 2 ) ad, by Theorem 4.3ii) agai, µ / E 0 λ 2 ). Hece, µλ 2 ) µ λ 2 ) µ λ ) µλ ). Equality is possible oly if µ := µλ ) = µλ 2 ) Eλ ) Eλ 2 ), ad the the characteristic equatio 2.5) is satisfied with µ ad both λ ad λ 2 ; hece µ 2 /λ = µ 2 /λ 2 ad µ = 0.

13 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 3 Theorem 4.6. i) For every λ > 0, µλ ) µ λ) as λ λ ad µλ ) µ λ) as λ λ. ii) µλ) 0 as λ 0. iii) µλ) as λ. Proof. i): Let µ 0 := lim λ λ µλ ); the limit exist by the mootoicity i Theorem 4.5. For ay fixed µ, ψ S, µ; λ ) ψ S, µλ ); λ ) by Theorem 4.3 ad it follows by cotiuity that ψ S, µ; λ) ψ S, µ 0 ; λ). Hece, by Theorem 4.3 agai, µ 0 E 0 λ), ad Theorem 4.5 implies that µ 0 = µ λ). The secod statemet is proved similarly. ii): This is similar. Assume µ 0 := lim λ 0 µλ) > 0, ad let µ := µ 0 /2. The ψ S, µ ; λ) ψ S, µλ); λ) for all λ by Theorem 4.3 which cotradicts the fact that, by 2.8), ψ S, µ ; λ) ψ S, µλ); λ) = µλ)2 µ 2 2λ µ2 0 µ2 2λ + O) + O) as λ 0. iii): Assume µ 0 := lim λ µλ) <, ad let µ := µ 0 +. The ψ S, µ ; λ) ψ S, µλ); λ) for all λ by Theorem 4.3 ad it follows from 2.8) by cotiuity, lettig λ, that log φ S µ ) log φ S µ 0 ), a cotradictio sice φ S is strictly icreasig. Remark 4.7. For large λ, we have the estimates c 3 λ /2 µλ) C 5 λ, where the lower boud follows from 2.5) ad the upper from 4.). Examples 6. ad 6.6 show that both these orders of growth ca be attaied. We see from Theorem 4.6 that E 0 λ) > exactly whe µλ) is discotiuous, ad that all discotiuities are jump discotiuities: µ jumps from µ λ) to µ λ). I accordace with Remark 2.3, we iterpret these discotiuities as phase trasitios of G,λ/;S. More geerally, we say that we have a phase trasitio at each λ where µ is ot aalytic. See, further, Theorem 4.5.) We show that there is oly a coutable umber of phase trasitios with jump discotiuities, ad we ote from Example 6.9 that the umber may be ifiite. This otio of phase trasitio is differet from the questio whether there is a giat compoet or ot.) Theorem 4.8. The set Λ := {λ > 0 : E 0 λ) > } = {λ : µ λ) < µ λ)} of λ such that Theorem 2. does ot apply is at most coutable. Proof. By Theorem 4.5, the ope itervals µ λ), µ λ) ), λ > 0, are disjoit, ad thus at most a coutable umber of them are o-empty. It follows from Theorem 4.5 ad its corollaries that µ is the iverse fuctio of a cotiuous o-decreasig fuctio with graph {µ, λ) : µ [ µ λ), µ λ)]}; the exceptioal set Λ cosists of the values take by this fuctio o the itervals where it is costat.

14 4 GEOFFREY GRIMMETT AND SVANTE JANSON For µ > 0 we ca rewrite the characteristic equatio 2.5) as λ = ˆλµ) := µφ Sµ) φ. 4.6) S µ) Thus, Eλ) = {µ > 0 : ˆλµ) = λ} or {µ > 0 : ˆλµ) = λ} {0}. Note that our assumptio S {0} implies that φ S µ) > 0 for all µ > 0, so ˆλ is well-defied. Lemma 4.9. The fuctio ˆλ is positive ad aalytic o 0, ), with lim µ ˆλµ) = ad 0, 0 / S, 0, 0 S, S, lim ˆλµ) = µ 0, 0 S, / S, 2 S,, 0 S, / S, 2 / S. Proof. That ˆλ is aalytic ad positive is evidet. By Lemma 4., ˆλµ) c 4 µ for large µ, ad the behaviour as µ 0 follows by lookig at the first ozero terms i the Taylor expasios of φ S ad φ S. Lemma 4.0. ψ S µ) ad ˆλ µ) have the same sig for every µ > 0. Hece ψ S ad ˆλ have the same statioary poits ad are icreasig or decreasig o the same itervals. Proof. By 2.7) ad 2.8), ψ S µ) = ψ S, µ; ˆλµ)). Differetiatig ad usig 4.3) we obtai, for all µ > 0, ψ Sµ) = µ ψ S,µ, ˆλµ)) + λ ψ S,µ, ˆλµ))ˆλ µ) = 0 + µ2 2ˆλµ) 2 ˆλ µ). Theorem 4.. i) If ˆλ is decreasig o a iterval µ, µ 2 ) with 0 µ < µ 2, the there exists λ Λ with µ λ) µ < µ 2 µ λ). ii) Λ = if ad oly if ˆλ is icreasig o 0, ). I this case, for every λ > 0 either Eλ) = { µλ)} with µλ) 0 or Eλ) = {0, µλ)} with µλ) > 0. Proof. i): Suppose that µ λ) µ, µ 2 ) for some λ > 0. Takig a sequece λ λ, we have µλ ) µ λ) by Theorem 4.6, so, for large, µ < µλ ) < µ λ) < µ 2 ad hece λ = ˆλ µλ )) > ˆλ µ λ)) = λ, a cotradictio. Hece, µ λ) / µ, µ 2 ) for all λ > 0. Let λ 0 := sup{λ : µ λ) µ }. Thus, µ λ) µ 2 for λ > λ 0. By Theorem 4.6ii)iii), 0 < λ 0 <, ad by Theorem 4.6i) µ λ 0 ) µ ad µ λ 0 ) µ 2. I particular, µ λ 0 ) < µ λ 0 ) so λ 0 Λ. ii): If ˆλ is ot icreasig, the ˆλ µ) < 0 for some µ > 0 ad i) applies to some iterval µ ε, µ + ε) ad shows that Λ. If ˆλ is icreasig, the ψ S is strictly) icreasig o 0, ) by Lemma 4.0; if 0 S, the ψ S is cotiuous at 0 ad thus icreasig o [0, ) also.

15 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 5 Cosequetly, Eλ) cotais a uique µ = max Eλ) that maximizes ψ S. Further, whe ˆλ is icreasig, there is at most oe positive solutio to λ = ˆλµ), ad thus to 2.5), ad the result o Eλ) follows. We ext study whether µ = 0 is possible. Note that this is a rather degeerate case, whe Theorem 2. shows that G,λ/;S is very sparse with o p ) edges ad 0 / p, which is to say that o p )) vertices are isolated. Theorem 4.2. i) If 0 / S, the µ > 0 for every λ > 0. ii) If 0 S ad S, the µ > 0 for every λ > 0. iii) If 0 S ad / S, the there exists λ 0 > 0 such that µ = 0 for every λ < λ 0, but µ > 0 for every λ > λ 0. Proof. i): Trivial, sice 0 / Eλ) i this case. ii): I this case, φ S 0) = φ S 0) = ad thus 4.3) shows that d dµ ψ S,µ; λ) > 0 as µ 0; hece ψ S µ; λ) is icreasig for small µ ad 0 is ot a maximum poit. By Theorem 4.3ii), 0 / E 0 λ). iii): By Lemma 4.9, ˆλµ) teds to as µ, ad to either or as µ 0. Hece λ := if µ>0 ˆλµ) > 0. If λ < λ, there is thus o positive solutio to 2.5), so Eλ) = {0} ad µ = 0. The existece of λ 0 λ as asserted ow follows from Theorem 4.5 ad Theorem 4.6. I case iii), µ λ 0 ) = 0 by Theorem 4.6; it is possible both that E 0 λ 0 ) = {0} so that µ λ 0 ) = 0, ad that E 0 λ 0 ) > so that λ 0 Λ ad µ λ 0 ) > 0. We ca classify these subcases too. Theorem 4.3. Suppose that 0 S ad / S. i) If 2 S ad 3 / S S = {0, 2, s,... } with s 4, or {0, 2}), the µλ) = 0 for λ λ 0 = ad µλ) > 0 for λ >, with µ ) = 0 ad thus / Λ ad µλ) 0 as λ. ii) If 2 S ad 3 S, or if 2 / S S = {0, 2, 3,... } or {0, s,... } with s 3), the there exists λ 0 > 0 such that µλ) = 0 for λ < λ 0 ad µλ) > 0 for λ > λ 0, with µ λ 0 ) > 0 = µ λ 0 ) ad thus λ 0 Λ ad lim λ λ0 µλ) > 0. Proof. ii): If 2, 3 S, the the Taylor series φ S µ) = + 2 µ ad φ S µ) = µ + 2 µ yield ˆλµ) = 2 µ +... for small µ. If 2 / S, the /ˆλµ) 0 as µ 0 by Lemma 4.9. I both cases, /ˆλ is aalytic i a eighbourhood of 0 ad icreases o a iterval 0, µ 0 ), so ˆλ decreases there ad the result follows by Theorems 4.i) ad 4.2iii). i): Taylor expasios as i the proof of ii) show that ˆλµ) = + 3 µ whe 4 S) or ˆλµ) = + 2 µ whe 4 / S) so ˆλ icreases for small

16 6 GEOFFREY GRIMMETT AND SVANTE JANSON µ, say i a iterval 0, µ 0 ). By Lemma 4.0, ψ S icreases i 0, µ 0 ), ad sice ψ S is cotiuous at 0 we have ψ S µ) > ψ S 0) for 0 < µ < µ 0. However, we also have to cosider larger µ, ad we use Lemma 4.4 below which implies that λ := if µ µ0 ˆλµ) >. It follows that if λ, the Eλ) = {0}, ad i particular µ ) = 0. Similarly, if < λ < λ, the Eλ) = {0, µ} for the uique µ 0, µ 0 ) with ˆλµ) = λ; i this case, ψ S µ) > ψ S 0) ad we have µ = µ > 0. Fially, by Theorem 4.6, µλ) µ ) = 0 as λ. Lemma 4.4. Uder the assumptios 0, 2 S ad, 3 / S of Theorem 4.3i), ˆλµ) > for every µ > 0. Proof. By 4.6), the claimed iequality is equivalet to φ S µ) < µφ Sµ), where φ S µ) = k S µk /k )!. First, we use the trivial estimates ad φ Sµ) φ Z 0 \{,3} µ) = eµ 2 µ2 φ S µ) + 2 µ2. We may verify umerically by Maple, or otherwise) that e µ 2 µ2 < µ + 2 µ2 ) for 0 < µ < 3.38, ad thus the claim holds i this rage. For larger µ, we split the sum for φ S µ) ito two parts. With K := µ2, we have that µ k k )! µ + k S, k K k S, 4 k K = µ + K µ φs µ) 2 µ2) k µ µk k! µ + K µ k S, k 4 µφ S µ) 2 µ3. 4.7) For k > K we use the Cheroff boud for the Poisso distributio, see e.g. [2, Theorem 2. ad Remark 2.6]: k S, k K+ µ k k )! k=k µ k k! = eµ P Poµ) K ) µ k k! exp µ K logk/µ) + K µ ) eµ ) K. = 4.8) K For µ 3, K/µ = µ 2 /µ > 9/ 0 > e, so the sum i 4.8) is less tha. We combie this with 4.7) to obtai φ S µ) < µφ Sµ) for µ 3. Theorem 4.5. The set {λ : µ is ot aalytic at λ} of phase trasitios is at most coutable. Each phase trasitio is of oe of the followig types. i) A jump discotiuity: λ Λ ad µ λ) < µ λ). ii) A cotiuous phase trasitio with µ λ) = µ λ) = 0 but µλ ) > 0 for λ > λ. This ca happe oly at λ =, where it happes if ad oly if 0, 2 S but, 3 / S.

17 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 7 iii) A cotiuous phase trasitio with µ > 0; i this case, λ = ˆλµ) for some µ > 0 with ˆλ µ) = 0 but ˆλ icreasig i a eighbourhood of µ; thus µ is a iflectio poit of ˆλ. Examples of type i) ad ii) are give i Sectio 6. We do ot kow whether iii) actually occurs. Proof. Sice ˆλ µλ)) = λ whe µλ) > 0 ad ˆλ is aalytic, the implicit fuctio theorem shows that µ is aalytic at every poit where it is cotiuous ad positive ad ˆλ µ) 0. Hece we have oly the three give possibilities; i iii), ˆλ has to be icreasig i a eighbourhood of µ sice otherwise we would have a jump discotiuity by Theorem 4.i). The further characterizatio i ii) follows by Theorems 4.2 ad 4.3. The umber of jump discotiuities is coutable by Theorem 4.8, ad so is the umber of iflectio poits of ˆλ, while there is at most oe phase trasitio of type ii). I Case ii), by the proof of Theorem 4.3, ˆλµ) = +cµ 2 +oµ 2 ) as µ 0, so µλ) c λ as λ ad we have a square-root type sigularity. I Case iii), provided it happes at all, if the iflectio poit is µ 0, the ˆλµ) ˆλµ 0 ) cµ µ 0 ) m as µ µ 0 for some odd m 3 presumably m = 3), ad thus µλ) µ 0 c λ λ 0 ) /m as λ λ 0 = ˆλµ 0 ). Theorems 4. ad 4.5 show that phase trasitios, except the possible oe of type ii), occur whe ˆλ ceases to be icreasig at some poits, or at least almost ceases to be, i the form of a iflectio poit. Recall that ˆλ, so it icreases i the log ru.) We have o criterio for whe this happes, but it seems likely that it occurs wheever there are large gaps i S. Problems 4.6. Several ope problems remai. For example: i) Is there ever ay phase trasitio of type iii) i Theorem 4.5 with a iflectio poit of ˆλ)? ii) Does the set of phase trasitios lack accumulatio poits? I other words, if there is a ifiite umber of phase trasitios as i Example 6.9), ca we always order them i a icreasig sequece λ < λ 2 <... with λ? iii) Is E 0 λ) always or 2? I the latter case, is always Eλ) = 3, as i Example 6.8, with oe itermediate poit that is a miimum rather tha a maximum of ψ S, ad ψ S,2? 5. Mootoicity We begi with a geeral result that is a simple cosequece of stadard results, ad may be kow already. Recall that if X ad Y are two radom variables, we say that X is stochastically smaller tha Y, ad write X st Y, if PX > x) PY > x) for every real x; it is well-kow that this is

18 8 GEOFFREY GRIMMETT AND SVANTE JANSON equivalet to the existece of a couplig X, Y ) of X, Y ) with X Y a.s. See, for example, [3, Sectio IV.]. Lemma 5.. Let Y be a radom variable takig values i [0, ) with distributio fuctio F ad momet geeratig fuctio Mθ) = 0 e θy df that is fiite for all θ < θ, ad let, for θ < θ, Y θ have the cojugate or tilted) distributio fuctio F θ satisfyig df θ /df = e θy /Mθ). i) If f : [0, ) R is a o-decreasig fuctio such that E fy θ ) < for every θ < θ, the d dθ E fy θ) 0, with strict iequality except i the trivial case whe fy θ ) is almost surely costat. Whe the last holds for some θ, it holds for all θ < θ. ii) If θ θ 2 < θ the Y θ st Y θ2. Proof. i): For θ < θ, d dθ E fy θ) = d fy)e θy df dθ Mθ) yfy)e θy df fy)e θy df ) ye θy df ) = Mθ) Mθ) 2 = E Y θ fy θ ) ) EY θ ) E fy θ ) ) = Cov fy θ ), Y θ ) ) 0, sice, as is well-kow, the two o-decreasig fuctios fy θ ) ad Y θ of Y θ are positively correlated, for example by a calculatio of E [ fy θ ) fy θ ))Y θ Y θ )] 0 with Y θ a idepedet copy of Y θ. The same proof yields strict iequality if fy θ ) is ot a.s. costat. The fial remark of i) holds sice the Lebesgue Stieltjes measures of F θ ad F θ2 are absolutely cotiuous with respect to oe aother. ii): By i), PY θ > x) is a o-decreasig fuctio of θ for every x. Let, for µ > 0, X µ Poµ) ad X µ,s Po S µ). Applyig Lemma 5. to Y = X,S with µ = e θ, we obtai the followig. Theorem 5.2. i) If f : Z 0 R is a o-decreasig fuctio such that E fx µ ) < for every µ > 0, the d dµ E fx µ,s) = d dµ E fx µ ) X µ S ) 0, with strict iequality except i the trivial case whe f is costat o S. ii) If µ µ 2 the X µ,s st X µ2,s. This shows, i cojuctio with Theorem 2., that the asymptotic degree distributio of G,λ/;S is stochastically icreasig i µλ) ad thus, by Theorem 4.5, i λ. I particular, the asymptotic edge desity, which is give by E X bµλ),s = ν µλ)), is a icreasig fuctio of λ except that it is costat i the trivial case S = ). This holds for fiite too. Theorem 5.3. If 0 < p p 2 <, the eg,p ;S) st eg,p2 ;S).

19 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 9 Proof. This is aother applicatio of Lemma 5., sice it follows from.) that eg,p;s ) has the cojugate distributio Y p/ p) with Y = eg, ;S). 2 Ufortuately, if we cosider the etire radom graph G,p;S ad ot just the umber of its edges), it is i geeral ot stochastically icreasig i p. Example 5.4. Let = 4 ad let S = {0, 2} or the set of all eve umbers). The S-graphs are, igorig the labellig: i) E 4, the empty graph with o edges, ii) C 3 + E, a 3-cycle plus a isolated vertex, iii) C 4, a 4-cycle. We have PG,p;S = E 4 ) as p 0 ad PG,p;S = C 4 ) as p. Hece, if fg) is the umber of 3-cycles i G, the E fg,p;s ) = PG,p;S = C 3 + E ) teds to 0 both as p 0 ad p, so this expectatio is ot mootoe i p. Problem 5.5. Is the radom multigraph G,ν;S defied i Sectio 7 stochastically icreasig i ν? Its umber of edges is, by the same argumet as for Theorem 5.3.) For the existece of a giat compoet, we ote that the crucial quatity Qµ) i 3.2) is ot always mootoe i µ, ot eve i the classical case S = Z 0 whe Qµ) = µ 2 µ). Nevertheless, the coditio Qµ) > 0 is mootoe. Theorem 5.6. If µ µ 2 ad Qµ ) > 0, the Qµ 2 ) > 0. Moreover, assumig S {0, 2}, if λ λ 2 ad thus µλ ) µλ 2 ), the ξ µλ )) ξ µλ 2 )) ad γ µλ )) γ µλ 2 )). Hece, if G,λ /;S has a giat compoet, the so has G,λ2 /;S for all λ 2 λ, ad it is asymptotically) at least as large. Proof. The coditio Qµ) > 0 is equivalet to µφ S µ)/φ S µ) >. Sice φ S µ) = φ S µ), µφ S µ) φ S µ) = µφ S µ) φ S µ) = E X µ,s, 5.) which is o-decreasig by Theorem 5.2i). The mootoicity of ξ ad γ follows from the brachig process iterpretatios i Remark 3.2 together with the stochastic mootoicity Theorem 5.2ii) for both S ad S ) ad Theorem 4.5. Remark 5.7. Theorem 3. ad 5.) yield the curious relatio that, provided S {0, 2}, G,λ/;S has a giat compoet if ad oly if ν S µ) := E X bµ,s >, with µ = µλ) calculated for S. Cf. Remarks 3.2 ad 3.7. It follows similarly from Remark 3.7 that the existece ad size of a k-core, for ay fixed k 3, is mootoe i λ.

20 20 GEOFFREY GRIMMETT AND SVANTE JANSON 6. Examples Example 6.. S = Z 0, the Erdős Réyi radom graph. We have i this much studied case that φ S µ) = e µ. The characteristic equatio 2.5) becomes µ = µ 2 /λ, with solutios µ = 0, µ = λ. By 2.7), ψ S µ) = 2 µ, so ψ S 0) < ψ S λ). Therefore, ad i accordace with Theorem 4.2ii), µ = λ, so that the umber i of vertices of degree i satisfies i / λ i e λ /i!, i 0. This is a simple istace of Theorem 4.ii). There is o phase trasitio of G,λ/;S. We have ψ S, µ; λ) = µ µ 2 /2λ) ad ψ S,2 µ; λ) = 2 µ logλ/µ)+ ). Details of the applicatio of Theorem 3. to this well uderstood case may be foud i [5]. Similarly, the applicatio of Theorem 3.6 is described i [0]. Example 6.2. S = {s}, where s. We have φ S µ) = µ s /s! ad the characteristic equatio 2.5) becomes s = µ 2 /λ, with solutio µ = sλ. However, i this case, the value of µ is i fact immaterial, sice Po S µ) is a poit mass at s for every µ > 0. Moreover, the graph G,λ/;S is a radom regular graph with all vertices of degree s. It is immediate that Qµ) = ss 2), ad so there exists a giat compoet if s > 2, ad ot if s =. I fact, if s =, the graph cosists of isolated edges oly, while if s 3, it is well-kow that the graph is coected with probability tedig to, see Bollobás [5], Sectio VII.6. I the remaiig case s = 2, the graph cosists of cycles, of which the largest has a legth that divided by coverges to some o-degeerate distributio o [0, ], see, e.g., Arratia, Barbour ad Tavaré []; this is thus a exceptioal case where we do ot have covergece i probability of the proportio of vertices i the giat cluster, as i Theorem 3.. Example 6.3. S = 2Z 0, the eve umbers. I this case, µ 2k φ S µ) = = cosh µ. 2k)! k=0 The characteristic equatio 2.5) is so either µ = 0 or µ sih µ cosh µ = µ2 λ, λ = ˆλµ) = µ tah µ. 6.) Sice ˆλµ) = µ/ tah µ icreases strictly) from to for µ [0, ), it follows that: if λ, µ = 0 is the oly solutio, while if λ >, there is also a positive solutio. We have Therefore, ψ S µ) = logcosh µ) 2 µ tah µ. ψ Sµ) = sih2µ) 2µ 4 cosh 2 µ > 0

21 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 2 for µ > 0. Hece, ψ S µ) > ψ S 0) for µ > 0, whece µ is the uique positive solutio of 6.) whe λ >, cf. Lemma 4.0, Theorem 4.ii) ad Theorem 4.3i). We thus have a cotiuous phase trasitio at λ = with µ) = 0; there is a uique µ ad thus Theorem 2. applies) for every λ, ad µλ) is a cotiuous fuctio, but it is ot differetiable at λ =. This is the oly phase trasitio, ad Λ =. The asymptotic edge desity i.e., the umber of edges per vertex, see 2.)) is ν µ) = µ tah µ. 6.2) Sice / S, Theorem 3. shows that there is a giat compoet as soo as µ > 0, i.e., if λ >. I fact, it is easily see that ) µ Qµ) = µ tah µ tah µ. Oe may study a radom eve subgraph of a geeral graph G. It turs out that the radom eve subgraph with parameter p [0, 2 ] is related to the radom-cluster model o G with edge-parameter 2p ad cluster-weightig factor q = 2. Whe G is a plaar graph, the radom eve subgraph may be idetified as the dual graph of the +/ boudary of the Isig model o the Whitey) dual graph of G with a appropriate parameter-value. This relatioship is especially fruitful whe G is part of a plaar lattice such as the square lattice Z 2. See [6] for a geeral accout of the radom-cluster model, ad [7] for its relatioship with the radom eve subgraph ad the Isig model. Example 6.4. S = {, 3, 5,... }, the odd umbers. This time, φ S µ) = sih µ, ad the characteristic equatio is µ cosh µ sih µ = µ2 λ with ˆλµ) = µ tah µ. Sice 0 / S ad ˆλ is icreasig, the uique solutio µ is give as the uique positive solutio of µ tah µ = λ. Cf. Theorems 4.2i) ad 4.ii). There is o phase trasitio. This time, Qµ) = µ µ tah µ ). tah µ Thus Q µ) > 0 if ad oly if µ tah µ > ; sice µ tah µ = λ, it follows that there is a giat compoet for λ >, ad ot for λ. This also follows from Remark 5.7 ad 6.2).) I the critical case λ = we have µ tah µ = ad umerically µ.9968 ad asymptotic) edge desity ν µ) = µ 2 /λ = µ Example 6.5. S = {, 2, 3,... } = Z, graphs without isolated vertices. We have that φ S µ) = e µ, ad the characteristic equatio is µe µ e µ = µ2 λ.

22 22 GEOFFREY GRIMMETT AND SVANTE JANSON Sice 0 / S, we seek strictly positive solutios, which is to say that λ = ˆλµ) = µ e µ ). Sice µ e µ ) is icreasig o 0, ), there is a uique such solutio µ for every λ > 0. Cf. Theorem 4.ii). We have that µµ )eµ Qµ) = e µ. Thus Q µ) > 0 if ad oly if µ >, which is to say that λ > ˆλ) = e. There is a giat compoet whe λ > e, ad ot whe λ e. I the critical case λ = e, µ = ad the critical asymptotic) edge desity is ν µ) = µ 2 /λ = e/e ) Example 6.6. S = {0, }, matchigs. We have that φ S µ) = + µ, ad the characteristic equatio is µ + µ = µ2 λ. Either µ = 0 or λ = ˆλµ) = µ + µ), so the solutios for give λ are µ = 0 ad µ = 2 + λ + 4. Sice ˆλ is icreasig, Theorem 4.ii) applies ad shows that µ = 2 λ for all λ > 0. This ca also easily be verified directly, usig µ ψ S µ) = log + µ) 2 + µ) which yields ψ S µ) > 0 for µ 0 cf. Lemma 4.0). By Theorem 2., as, { } 0 p Po S µ){0} = φ S µ) = + µ = λ + λ 4. 2 Obviously there is o giat compoet. Ideed, Qµ) = Po S µ){} < 0 for µ > 0. Example 6.7. S = {0, 2}, isolated cycles. We have that φ S µ) = + 2 µ2, ad the characteristic equatio is µ 2 µ2 + = 2 µ2 λ. Therefore, either µ = 0 or λ = ˆλµ) = + 2 µ2, so that the solutios for a give λ are µ = 0 ad, whe λ >, µ = 2λ ). Agai, ˆλ is a icreasig fuctio, ad so is ψ S µ) = log + 2 µ2 ) 2 µ2 + 2 µ2, by Lemma 4.0 or direct calculatios. Thus, see Theorem 4.ii), { 0 whe λ, µ = 2λ ) whe λ >.

23 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 23 We thus have a cotiuous phase trasitio at λ =, of the same type as i Example 6.3, see Theorem 4.2iii) ad Theorem 4.3i). There is o other phase trasitio. It is easily see that Qµ) = 0 for all µ, which may be iterpreted as sayig that the radom graph is, i a certai sese, critical wheever λ >. If we remove the isolated vertices, ad coditio o the umber of remaiig vertices, we obtai a radom regular graph with degree 2. Hece, for µ > 0, i.e., for λ >, we see that the largest compoet behaves as for S = {2}, see Example 6.2, with covergece of vγ,λ/;s)/ to a distributio but ot to a costat. Example 6.8. S = {0, 3}. This time, φ S µ) = + 6 µ3, ad the characteristic equatio is 2 µ3 µ2 + = 6 µ3 λ. Either µ = 0, or λ = ˆλµ) = + 6 µ3 2 µ. 6.3) This is a covex fuctio of µ with a miimum of 3 2/3 at the poit µ = 3 /3. Hece, the characteristic equatio has o positive root whe λ < 3 2/3, oe such root if λ = 3 2/3, ad two such roots if λ > 3 2/3. We have ψ S µ) = log + 6 µ3 ) 4 µ3 + 6 µ3. Ulike the previous examples, ψ S is ot mootoe, cf. Lemma 4.0. I fact, ψ S 3 /3 ) = log + 2 ) 2 < 0 = ψ S0), so the correct root is µ = 0 rather tha 3 /3 ) whe λ = 3 2/3. The fuctio ψ has a miimum at µ = 3 /3 ; ψ decreases o [0, 3 /3 ] ad icreases o [3 /3, ). There exists thus a uique µ 0 > 3 /3 such that ψ S µ 0 ) = 0, ad we set λ 0 = µ3 0 )/µ 0. Numerically, µ ad λ 0 = ˆλµ 0 ) ) We deduce that µ = 0 for λ < λ 0 while, for λ > λ 0, µ is the largest root of 6.3). For λ = λ 0, there are two roots µ of 6.3) with the same value of ψ S µ), so we have a jump phase trasitio ad Theorem 2. does ot apply; see Theorems 4.2iii) ad 4.3ii). There is o other phase trasitio, ad Λ = {λ 0 }. Sice / S, by Theorem 3.ii), there exists a giat compoet wheever λ > λ 0. Ideed, 2 Qµ) = µ2 + > 0 6 µ3 for every µ > 0. The case S = {0, k}, k 4 is similar; we omit the details.

24 24 GEOFFREY GRIMMETT AND SVANTE JANSON Example 6.9. S = {, 2, 4, 8,... } = {2 j : j 0}. We claim that as j, ˆλ2 j x) = 2 j x 2 + o) ), 6.4) for every x 2/e, 4/e). I fact, this holds uiformly o every closed subiterval of 2/e, 4/e).) It follows that if a = 4/e ad ε > 0 is small ad fixed, the for large j, ˆλa ε)2 j ) a ε) 2 2 j ad ˆλa + ε)2 j ) = ˆλa/2 + ε/2)2 j+ ) a + ε) 2 2 j, so ˆλ drops by a factor of about 2 i the viciity of a2 j. Cosequetly, for all large j, there is a iterval I j a ε)2 j, a + ε)2 j) where ˆλ is decreasig, ad thus by Theorem 4. there exists λ Λ such that µ λ) max I j 2 j. Hece the set { µ λ) : λ Λ} is ubouded ad thus ifiite, so Λ is ifiite ad there is a ifiite umber of phase trasitios. To verify 6.4) we show that if µ = 2 j x a2 j, a2 j ) the φ S µ) = k S µk /k! ad φ S µ) = k S kµk /k )! are domiated by the terms with k = 2 j : φ S µ) = + o) ) µ 2j 2 j!, φ Sµ) = + o) ) µ 2j 2 j )! as j ; this yields that φ S µ) 2j /µ)φ S µ) ad ˆλµ) = µ φ S µ)/φ Sµ) µ 2 j /µ = µ2 2 j = 2j x 2. Fially, to show 6.5), we observe by Stirlig s iequality that, as k, ad thus, with k i = 2 i, µ k i+ / µ k i k i+! µ k k! = + o) ) 2πk) /2 eµ k ) k k i! = + o) ) 2 /2 eµ 2k i ) 2k i eµ k i ) k i = 2 /2 + o) ) eµ 4k i ) ki, 6.5) which for µ = 2 j x a2 j, a2 j ) is expoetially small if i j ad expoetially large if i < j. The estimate 6.5) for φ S µ) follows, ad a similar calculatio with k i = 2 i yields the result for φ S µ). 7. Multigraphs As explaied at the ed of Sectio 2, we shall cout multigraphs with certai properties, ad shall later relate our coclusios to simple graphs. Let G be the ifiite) set of all multigraphs o the vertex set {, 2,..., }, ad let G;S be the subset of S-multigraphs o {, 2,..., } we exted the defiitios above to multigraphs i the obvious way, otig that a loop couts two towards the degree of the vertex i questio).

25 RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES 25 Let ν 0. We defie a radom multigraph G,ν by takig Poν) edges betwee each pair of vertices ad Po 2ν) loops at each vertex, these radom umbers beig idepedet of oe aother. It is easily see that this is equivalet to assigig to each multigraph G G the probability where PG,ν = G) = wg)ν eg) e 2 ν/2, 7.) wg) := 2 l j 2 j! m j, with l the umber of loops of G, ad m j the umber of j-fold multiple edges icludig multiple loops). That is, m j = a j + b j where a j is the umber of distict pairs of vertices joied by exactly j parallel edges, ad b j is the umber of vertices havig exactly j loops. See, e.g., Jaso, Kuth, Luczak ad Pittel [9]. Note that the total umber of edges eg,ν) is Poisso-distributed with parameter ) 2 ν + 2 ν = 2 2 ν. We further defie the radom S-multigraph G,ν;S by coditioig G,ν o beig a S-multigraph. Thus, for ay multigraph G G;S, by 7.), P G,ν;S = G ) = Z,ν;S wg)ν eg), 7.2) where Z,ν;S := G G ;S wg)ν eg) = e 2 ν/2 PG,ν is a S-multigraph). 7.3) We shall assume, of course, that G;S. It is easy to see that this holds for all if S cotais some eve umber, but if all elemets of S are odd, the has to be eve. We tacitly assume this i the sequel. If the multigraph G G is simple, i.e. has o loops ad o multiple edges, the wg) = ad 7.) yields PG,ν = G) ν eg) PG,p = G) whe ν = p/ p), i.e., p = ν/ + ν). Hece, assumig this relatio betwee ν ad p, G,ν coditioed o beig simple has the same distributio as G,p. Coditioig further o beig S-graphs, we obtai the followig. Lemma 7.. If ν = p/ p), the G,p;S d = G,ν;S G,ν;S is simple ). We are iterested i the case p λ <, ad ote that p λ ad ν λ are equivalet. We shall also use the cofiguratio model for radom multigraphs with give vertex degrees itroduced by Bollobás [4], see Bollobás [5], Sectio II.4. See Beder ad Cafield [2] ad Wormald [8, 9] for related argumets.) To be precise, let us fix the vertex degrees to be some o-egative itegers d, d 2,..., d assumig tacitly that i d i is eve); equivaletly, we fix a degree sequece d = d i ). We attach d i half-edges or stubs) to vertex i.

26 26 GEOFFREY GRIMMETT AND SVANTE JANSON The total umber of half-edges is thus 2N := i= d i, ad a cofiguratio is oe of the 2N )!! = 2N)!/2 N N!) partitios of the set of half-edges ito N pairs. Each cofiguratio defies a multigraph i G by combiig each pair of half-edges to a edge; this multigraph has vertex degrees d,..., d ad N = 2 i d i edges. By takig a uiformly radom cofiguratio we thus obtai a radom multigraph G,d with the give degree sequece d. It is easily see that every multigraph G G with the give vertex degrees d,..., d arises from exactly wg) i= d i! cofiguratios. We obtai therefore that the cotributio to Z,ν;S i 7.3) from a set of multigraphs with give vertex degrees d,..., d S is give by summig ν N / i= d i! over all correspodig cofiguratios. I particular, sice the umber of cofiguratios is 2N )!!, the cotributio to Z,ν;S from all multigraphs with vertex degrees d,..., d S equals 2N )!! ν N i= d i!. 7.4) Moreover, for give d = d i ), the factor νn / i= d i! is a costat, so by 7.), the probability that G,ν belogs to ay give set of multigraphs with this degree sequece d is proportioal to the umber of correspodig cofiguratios. Cosequetly, if d i G) deotes the degree of vertex i i a multi)graph G, ad dg) := d i G)) i= for G G, we obtai the followig well-kow fact. This is aother reaso for the weights wg) i 7.).) Lemma 7.2. For ay give degree sequece d = d i ) ad ay ν > 0, the radom multigraph G,ν coditioed o havig degree sequece d has the distributio give by the cofiguratio model; i other words, G,ν dg,ν) = d ) d = G,d. As a cosequece, if every d i S, the same holds for G,ν;S. We are iterested i the case ν = λ /, with λ λ > 0. Theorem 7.3. The results of Theorem 2.i) iii) hold with G,λ/;S replaced by G,λ /;S. Furthermore, log PG,λ / is a S-graph) ψ S µ) 2 λ 7.5) ad log Z,λ /;S ψ S µ). We will prove Theorem 7.3 i the followig sectio, ad the obtai Theorem 2. as a cosequece usig Lemma 7. ad the followig techical result, which is proved i Sectio 9. Lemma 7.4. If λ λ > 0, the lim if P G,λ /;S is simple) > 0.