How frequently is a system of 2-linear Boolean equations solvable?

HO W FREQUENTL Y IS A SYSTEM OF 2 -LINEAR BOOLEAN EQUA TIONS SOL V ABLE? Boris Pittel and Ji-A Y eum Ohio State Univ ersit y Abstract. W e consider a random s ystem of equations x i + x j = b ( i,j ) (mo d 2) , ( x u ∈ { 0 , 1 } , b ( u,v ) = b ( v,u ) ∈ { 0 , 1 } ) , wi th the pairs ( i, j ) from E , a s ymmetric subset of [ n ] × [ n ] . E i s chosen uniformly at random among all such subsets of a given cardinality m ; al ternatively ( i, j ) ∈ E with a gi v en probabi lity p , indepe ndently of all other pai rs. Also, given E , Pr { b e = 0 } = Pr { b e = 1 } for e ach e ∈ E , indep endently o f all other b e ′ . It is well known that, as m pas ses through n/ 2 ( p pass es through 1 /n , resp.), the underlying random g raph G ( n, #edges = m ) , ( G ( n, Pr(edge) = p ) , re sp.) undergo es a rapi d transition, from ess entially a forest of many small trees to a graph with one large, m ul ticycli c, comp onent in a sea of small tree comp onents. W e should expe ct the n that the solv ability probabil ity dec reases precipitousl y i n the vicinity of m ∼ n/ 2 ( p ∼ 1 /n ), and indeed this probabili t y is of order (1 − 2 m/n ) 1 / 4 , for m < n/ 2 ( (1 − pn ) 1 / 4 , for p < 1 /n , resp.). W e show that in a near-critical phase m = ( n/ 2)(1 + λn − 1 / 3 ) ( p = (1 + λn − 1 / 3 ) /n , re sp.), λ = o ( n 1 / 12 ) , the syste m i s solv able with probability as ymptotic to c ( λ ) n − 1 / 12 , for som e expl icit function c ( λ ) > 0 . Mike Moll oy notic ed that the Boole an system with b e ≡ 1 is solv able iﬀ the underlying graph is 2 -colo rable, and ask ed whether this connection might be used to determine an order of probability of 2 -co lorability in t he near-critica l case. W e answ er Mik e’s question aﬃrmatively and sho w that probability of 2 -colorabi lity is . 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 , and asymptotic to 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 at a c ritical phase λ = O (1) , and for λ → −∞ . (Submitted to E lectronic Journal of Combinatorics on Septembe r 7, 2009.) 1. In tro ductio n. A syst em of 2 -linear e quations o ver GF (2) with n Bo olean v ariables x 1 , . . . , x n ∈ { 0 , 1 } is (1.1) x i + x j = b i,j (mo d 2) , b i,j = b j,i ∈ { 0 , 1 } ; ( i 6 = j ) . Here the uno rdered pa irs ( i, j ) corre sp ond to t he edge set of a given gra ph G on the v ert ex set [ n ] . The system (1.1) ce rtainly has a solution when G is a tree. It can b e ob tained by p ic king a n arbitrary x i ∈ { 0 , 1 } a t a r o ot i an d dete rmining the ot her x j recursively a long t he path s leading a w a y from the r o ot. There is, of c ourse, a t win so lution ¯ x j = 1 − x j , j ∈ [ n ] . Supp o se G is no t a tree, i.e. ℓ ( G ) := e ( G ) − v ( G ) ≥ 0 . If T is a tree span ning G , th en each o f additional ed ges e 1 , . . . , e ℓ ( G )+1 forms, tog ether with the edges of T , a single cycle C t , t ≤ ℓ ( G ) + 1 . 1991 Mathematics Subje ct Classiﬁc ation . 0 5C80, 60K35. Key wor ds and phr ases. Bo ole an eq uations, sol v abi lity , random graph, asym ptotics. Pittel’s res earch s upported i n part by NSF Grants DMS-0406024, DMS-0805996 Typeset by A M S -T E X 1 2 BORIS PITTEL AND JI-A YEUM Ob viously , a solution x j ( T ) of a subsystem of ( 1.1) ind uced by t he edge s of T is a solution of (1.1) provided tha t (1.2) b i,j = x i ( T ) + x j ( T ) , ( i, j ) = e 1 , ..., e ℓ ( G )+1 ; equiv alently (1.3) X e ∈ E ( C t ) b e = 0 (mo d 2) , t = 1 , ..., ℓ ( G ) + 1 . So, in tuitiv ely , the more edges G has t he less lik ely it is t hat t he system (1.1) has a solution. W e will deno te the n um b er of solutions b y S ( G ) . In this pap e r w e consider solv abilit y of a ran dom system (1.1) . Namely G is either the B ernoulli random graph G ( n, p ) = G ( n, Pr( edge) = p ) , or t he Erd˝ os- R ´ enyi random g raph G ( n, m ) = G ( n, # of edg es = m ) . F urth er, co nditioned on the edge set E ( G ( n, p )) ( E ( G ( n, m ) re sp.), b e ’s are indep ende n t, and Pr( b e = 1) = ˆ p , for all e . W e fo c us o n ˆ p = 1 / 2 and ˆ p = 1 . ˆ p = 1 / 2 is the ca se when b e ’s are “absolute ly random” . F o r ˆ p = 1 , b e ’s are all ones. Mik e Mollo y [17], who brought this case to our att ention, notice d tha t here (1.1) has a solution iﬀ the und erlying graph is b ipartite, 2-colorable in othe r words. It is well known that, as m passes throug h n/ 2 ( p passes t hrough 1 /n , resp.), the underlying random gra ph G ( n, m ) , ( G ( n, p ) , resp.) underg o es a ra pid t ransition, from essentially a forest of ma n y small tree s to a grap h with one larg e, m ulticyclic, comp on en t in a sea of small tree comp onents. B ollob´ as [4 ], [5] discov e red that, fo r G ( n, m ) , the phase transition windo w is within [ m 1 , m 2 ] , where m 1 , 2 = n/ 2 ± λn 2 / 3 , λ = Θ(ln 1 / 2 n ) . Luczak [14] w as able t o sho w that t he windo w is precisely [ m 1 , m 2 ] with λ → ∞ ho w ever slowly . (See Luczak et al [16], Pittel [19] for t he distribu tional results on the cr itical graphs G ( n, m ) and G ( n, p ) .) W e should e xp ect t hen tha t th e solv ability pro babilit y d ecrease s precipitously for m close to n / 2 ( p close to 1 /n resp.). Indee d, for a multi gr aph v ersion o f G ( n, m ) , Kolc hin [13] p ro v ed t hat this probability is asymptotic to (1.4) (1 − γ ) 1 / 4 (1 − ( 1 − 2 ˆ p ) γ ) 1 / 4 , γ := 2 m n , if lim su p γ < 1 . Se e Creignon and D aud´ e [9] for a simil ar result. Using the results from Pittel [19], w e show (se e App endix) that for the rando m graphs G ( n, γ n/ 2) and G ( n, p = γ /n ) , with lim sup γ < 1 , the corre sp onding probab ilit y is a symptotic to (1.5) (1 − γ ) 1 / 4 (1 − ( 1 − 2 ˆ p ) γ ) 1 / 4 exp  γ 2 ˆ p + γ 2 2 ˆ p (1 − ˆ p )  . The relations (1.4), (1 .5) make it plausible that , in the nearcrit ical p hase | m − n/ 2 | = O ( n 2 / 3 ) , t he so lv abilit y probability is of or der n − 1 / 12 . Ou r goal is to conﬁrm, rigorously , this conjectu re. RANDOM EQUA TIONS 3 T o form ulate our main result, we ne ed some notat ions. Let { f r } r ≥ 0 b e a sequence deﬁned b y a n implicit re curren ce (1.6) f 0 = 1 , r X k =0 f k f r − k = ε r , ε r := (6 r )! 2 5 r 3 2 r (3 r )!( 2 r )! . Equiv alently , t he fo rmal se ries P r x r f r , P r x r ε r (div ergent for all x 6 = 0 ) satisfy (1.7) X r x r f r ! 2 = X r x r ε r . It is n ot diﬃcu lt to show th at (1.8) ε r 2  1 − 1 r  ≤ f r ≤ ε r 2 , r > 0 . F or y , λ ∈ R , let A ( y , λ ) de note the sum of a co n v ergent series, (1.9) A ( y , λ ) = e − λ 3 / 6 3 ( y +1) / 3 X k ≥ 0  1 2 3 2 / 3 λ  k k !Γ[( y + 1 − 2 k ) / 3 ] . W e will write B n ∼ C n if lim n →∞ B n /C n = 1 , and B n . C n if lim su p n B n /C n ≤ 1 . Le t S n denote the ran dom num b er of so lutions o f (1.1) with t he u nderlying graph b eing either G ( n, m ) or G ( n, p ) , i. e. S n = S ( G ( n, m )) or S n = S ( G ( n, p )) , and the (condit ional) pro babilit y o f b e = 1 for e ∈ E ( G ( n, m )) ( e ∈ E ( G ( n, p )) resp.) b eing equal ˆ p . Theorem 1.1. (i) L et ˆ p = 1 / 2 . Sup p ose tha t (1.10) m = n 2 (1 + λn − 1 / 3 ) , p = 1 + λn − 1 / 3 n , | λ | = o ( n 1 / 12 ) . Then, for b oth G ( n, m ) and G ( n, p ) , (1.11) Pr ( S n > 0) ∼ n − 1 / 12 c ( λ ) , wher e (1.12) c ( λ ) :=                e 3 / 8 (2 π ) 1 / 2 X r ≥ 0 f r 2 r A (0 . 25 + 3 r, λ ) , λ ∈ ( −∞ , ∞ ); e 3 / 8 | λ | 1 / 4 , λ → −∞ ; e 3 / 8 4 · 3 3 / 4 λ 1 / 4 exp( − 10 λ 3 / 81) , λ → ∞ . (ii) L et ˆ p = 1 . Then, with c ( λ ) r eplac e d by c 1 ( λ ) := 2 − 1 / 4 e 1 / 8 c ( λ ) , (1.9) holds for b oth G ( n, m ) and G ( n , p ) if either λ = O (1) , o r λ → −∞ , | λ | = o ( n 1 / 12 ) . F or λ → ∞ , λ = o ( n 1 / 12 ) , Pr ( S n > 0) . n − 1 / 12 c 1 ( λ ) . 4 BORIS PITTEL AND JI-A YEUM Notes. 1. F or G ( n, m ) wit h λ → −∞ , and ˆ p = 1 / 2 , our re sult blend s, qualitatively , wit h the estimate (1.4) from [13] and [9] for a sub critical m ulti gra ph, and b ecomes t he e stimate (1.5) for the sub critic al graph s G ( n, m ) and G ( n, p ) . 2. The part (ii) answers M ollo y’s quest ion: t he critical gr aph G ( n, m ) ( G ( n, p ) resp.) is b ic hromatic (bipartite ) with probability ∼ c 1 ( λ ) n − 1 / 12 . V ery in terest ingly , the lar gest bipartite subgraph of t he crit ical G ( n, p ) can b e found in exp ec ted time O ( n ) , see Copp ersmit h e t al [8], Sc ott and S orkin [21] and refere nces ther ein. The case λ → ∞ of (ii) stron gly sug gests that t he sup erc ritical graph G ( n, p = c/ n ) , ( G ( n, m = cn/ 2) resp.), i. e. with lim inf c > 1 , is bichro- matic with exp on en tially small prob abilit y . In [8] this exp one n tial smallness was established for the condit ional probabilit y , given that the random graph has a gian t comp on en t. Here is a t ec hnical re ason wh y , for λ = O ( 1) at least, the asympto tic probabilit y of 2 -color abilit y is the a symptotic solv ability prob abilit y for (1.1) wit h ˆ p = 1 / 2 times 2 − 1 / 4 e 1 / 8 . Let C ℓ ( x ) ( C e ℓ ( x ) re sp.) deno te the e xp onential generating func - tions o f conn ected grap hs G (graphs G w ithout o dd cycles resp.) with exce ss e ( G ) − v ( G ) = ℓ ≥ 0 . It turns out that , for | x | < e − 1 (conv erg ence radius of C ℓ ( x ) , C e ℓ ( x ) ), and x → e − 1 , C e ℓ ( x )      ∼ 1 2 ℓ +1 C ℓ ( x ) , ℓ > 0 , = 1 2 C 0 ( x ) + ln  2 − 1 / 4 e 1 / 8  + o ( 1) , ℓ = 0 . Asymptotically , with in the facto r e ln  2 − 1 / 4 e 1 / 8  , this reduce s t he problem to t hat for ˆ p = 1 / 2 . Based on (1.5), w e conj ecture tha t g enerally , for ˆ p ∈ (0 , 1] , and the critical p , Pr( S n > 0) is that probabilit y for ˆ p = 1 / 2 times (1.11) (2 ˆ p ) − 1 / 4 exp  − (1 − ˆ p ) 2 2 + 1 8  . (F or ˆ p = 0 , Pr( S n > 0) = 1 obviously .) 3. While working o n this pro ject, we b e came aw ar e of a r ecent pap er [10] b y Daud´ e and Rav e lomanana. The y studied a close but diﬀerent case, when a syste m of m equations is c hosen unifo rmly at random among al l n ( n − 1) equations of the form (1.1) . In pa rticular, it is p ossible to hav e pairs of clear ly contradictory equations, x i + x j = 0 and x i + x j = 1 . F or m = O ( n ) t he probabilit y that none of these simplest con tradict ions o c curs is b ounded aw ay from zero. So, in tuitiv ely , the system t hey studied is close to ours with G = G ( n, m ) and ˆ p = 1 / 2 . Our asymptotic fo rm ula (1.9) , with tw o ﬁrst equ ations in (1 .10), in t his case is similar to Da ud ´ e -Ra v elomanana’s main theore m, but t here are some puzz ling diﬀerenc es. The exp onent series in their equation ( 2) is cert ainly misplaced; t heir claim do es not con tain our seque nce { f r } . As far as we can judge b y a pro of o utline in [10], our argu men t is quite diﬀerent. Still lik e [10], our analysis is based on the gener ating funct ions of spars e graphs disco v ered , to a gre at e xtent, by W righ t [23 ], [24 ]. W e gratefu lly cr edit D aud ´ e and Ra v elomanana for stressing impo rtance of W r igh t’s b ound s for the generating function C ℓ ( x ) . These b ounds play a substantial ro le in our argument a s well. RANDOM EQUA TIONS 5 4. W e sho uld men tion a larg e b o dy of w ork o n a relate d, b e tter kno wn, 2 − S AT problem, see for instan ce Bo llob´ as e t al [6], and referenc es ther ein. It is a prob lem of existenc e of a tru th-satisfying assignment for the v a riables in the conjun ction of m rand om disjunct iv e c lauses of a form x i ∨ x j , ( i, j ∈ [ n ] ). It is w ell kno wn, Ch v´ atal and Ree d [7], t hat the existence thresho ld is m/n = 1 . It w as prov ed in [6] that the phase transition windo w is [ m 1 , m 2 ] , with m 1 , 2 = n ± λ n 2 / 3 , | λ | → ∞ ho w ev er slo wly , and that the so lv abilit y prob abilit y is b ou nded aw ay from b o th 0 and 1 iﬀ m = n + O ( n 2 / 3 ) . 5. A natur al exten sion of the system (1.1) is a system of k -linear equations (1.12) X i ∈ e x i = b e (mo d 2) , where e run s ov e r a set E of (hype r)edge s of a k -uniform h yp ergrap h G , k ≥ 2 , on the v er tex se t [ n ] , Kolchin [13]. Supp ose G is chosen uniformly at r andom a mong all k -unifo rm graph s with a giv en num b er m o f ed ges, and, giv en G , the b e s ar e indep e ndent Bernoullis. It will b e interesting to st udy , for k > 2 , the limiting solv ability probab ilit y as a func tion of m/n . Se e [13] for so me thou gh t-prov oking results on t he b ehavi or of th e num b er of hyp ercycles in this random h yp ergrap h. The pap er is o rganized as follo ws. In the section 2 w e w ork on the G ( n, p ) and ˆ p = 1 / 2 c ase. Sp ec iﬁcally in the ( sub)sect ion 2.1 we e xpress the solv a bilit y proba bilit y , Pr ( S n > 0) , and its t runcat ed version, as a c o eﬃcient by x n in a p ow e r series based on the genera ting functio ns of th e sp arsely edges (co nnecte d) graphs. W e also est ablish p ositive corr elation b etw een solv ability and b oun dedness of a maximal “exce ss”, and dete rmine a prop er tru ncation of the latte r d ep end en t up o n t he b ehavior of λ . In the sect ion (2.2 ) we provide a nec essary information ab o ut th e g enerat ing function s an d the ir t runcat ed versions inv olv ed in the formula a nd the b o unds for Pr( S n > 0) . In the sect ion 2.3 we app ly complex analysis techniques to th e “co e ﬃcien t by x n ” formulas and obtain a sharp asympt otic es timate fo r Pr( S n > 0) for | λ | = o ( n 1 / 12 ) . In th e section 3 we t ransfer t he results o f the sect ion 2 t o the G ( n, m ) and ˆ p = 1 / 2 c ase . In the sec tion 4 we establish t he counterparts of the results from t he sect ions 2,3 for G ( n, p ) , G ( n, m ) with ˆ p = 1 . An e n umerat iv e ingredient of the a rgument is an analogue of W right’s fo rm ulas for the gene rating func tions o f t he co nnecte d graphs without o dd c ycles. In App endix we pro v e some auxilliary technical r esults, and an a symptotic for- m ula for Pr( S n > 0) in the sub c ritical case, i. e. when t he a v e rage v ert ex degree is less th an, and b ou nded aw ay from 1 . 2. Solv abilit y probabilit y: G ( n, p ) and ˆ p = 1 / 2 . 2.1. Represen ting b o unds for Pr ( S n > 0) as a co eﬃcien t o f x n in a p o we r series. 6 BORIS PITTEL AND JI-A YEUM Our ﬁ rst ste p is to comput e the pr obabilit y of the even t { S n > 0 } , condit ioned on G ( n, p ) . Given a grap h G = ( V ( G ) , E ( G ) ) , w e denot e v ( G ) = | V ( G ) | , e ( G ) = | E ( G ) | . Lemma 2.1.1. Given a gr aph G on [ n ] , let c ( G ) denote the total numb er of its c omp onents H i . Then Pr ( S n > 0 | G ( n, p ) = G ) = c ( G ) Y i =1  1 2  e ( H i ) − ( v ( H i ) − 1) =  1 2  X ( G ) , X ( G ) := e ( G ) − n + c ( G ) . Conse quently Pr ( S n > 0) = E "  1 2  X ( G ( n,p ) ) # . Pro of of Lemma 2.1.1 . Reca ll that , co nditioned o n G ( n, p ) , the edge v ar iables b e are m utually indep endent. So it is suﬃces to show that a system (1.1) for a connec ted graph H , with inde p ende n t b e , e ∈ E ( H ) , such that Pr( b e = 1) = 1 / 2 , is solv able with p robability (1 / 2) ℓ +1 , where ℓ = e ( H ) − v ( H ) . Let T b e a tree spanning H . Le t x ( T ) := { x i ( T ) } i ∈ V ( H ) b e th e so lution of the subsystem of (1 .1) corresp onding to v ( H ) − 1 ed ges of T , with x i 0 = 1 say , for a sp eciﬁe d “ro ot ” i 0 . x ( T ) is a solution o f t he wh ole system ( 1.1) iﬀ (2.1.1) b e = x i ( T ) + x j ( T ) , (( i, j ) = e ) , for ea c h of e ( H ) − ( v ( H ) − 1) = ℓ + 1 edge s e ∈ E ( H ) \ E ( T ) . By indep endenc e of b e ’s, the p robability that, condition ed on { b e } e ∈ E ( T ) , t he c onstraints (2.1.1) a re met is (1 / 2) ℓ +1 ,. ( It is crucial t hat Pr( b e = 0) = Pr( b e = 1) = 1 / 2 .) Hen ce th e uncondit ional solv ab ilit y probability for t he system (1.1) with the underlying graph H is (1 / 2) ℓ +1 as w e ll.  Note. F or a cyc le C ⊆ H , let b C = P e ∈ E ( C ) b e . Th e cond itions ( 2.1.1) are equiv alent to b C b eing e v en for the ℓ + 1 cyc les, ea c h formed by adding to T an edge in E ( H ) \ E ( T ) . Adding the equat ions (1.1) ov er the edges of any cycle C ⊆ H , we see that nec essarily b C is ev en to o. Th us our pro of eﬀect iv ely sho ws that Pr    \ C ⊆ H { b C is ev en }    =  1 2  ℓ ( H )+1 . Using Lemma 2.1.1, we express P ( S ( n, p ) > 0) as the co eﬃcient b y x n in a formal pow e r series. T o formulate t he r esult, intro duce C ℓ ( x ) , the exp one n tial genera ting funct ion of a sequenc e { C ( k , k + ℓ ) } k ≥ 1 , where C ( k , k + ℓ ) is the total n um b er o f connect ed gra phs H on [ k ] with exc ess e ( H ) − v ( H ) = ℓ . Of course, C ( k , k + ℓ ) = 0 un less − 1 ≤ ℓ ≤  k 2  − k . RANDOM EQUA TIONS 7 Lemma 2.1.2. Pr ( S n > 0) = N ( n , p ) [ x n ] exp   1 2 X ℓ ≥− 1  p 2 q  ℓ C ℓ ( x )   , (2.1.2) N ( n, p ) := n ! q n 2 / 2  p q 3 / 2  n . (2.1.3) Pro of of Lemma 2.1.2 . The pro o f mimic ks deriv at ion o f th e “c o eﬃcient-of x n - expression” for the largest comp on en t size distribution in [19 ]. Giv en α = { α k ,ℓ } , su c h tha t P k ,ℓ k α k ,ℓ = n , let P n ( α ) denote the probability that G ( n, p ) ha s α k ,ℓ comp on en ts H with v ( H ) = k and e ( H ) − v ( H ) = ℓ . T o compute P n ( α ) , we observ e t hat there a re n ! Q k ,ℓ ( k !) α k,ℓ α k ,ℓ ! w a ys t o partition [ n ] in to P k ,ℓ α k ,ℓ subsets, with α k ,ℓ subsets of ca rdinalit y k and “t yp e ” ℓ . F or each such partit ion, t here are Y k ,ℓ [ C ( k , k + ℓ )] α k,ℓ w a ys to build α k ,ℓ connec ted graphs H on t he cor resp ond ing α k ,ℓ subsets, with v ( H ) = k , e ( H ) − v ( H ) = ℓ . The pro babilit y tha t these grap hs a re induced subgraph s of G ( n, p ) is Y k ,ℓ h p k + ℓ q ( k 2 ) − ( k + ℓ ) i α k,ℓ =  p q 3 / 2  n Y k ,ℓ "  p q  ℓ q k 2 / 2 # α k,ℓ , as P k ,ℓ k α k ,ℓ = n . The p robability that no tw o v ert ices f rom tw o diﬀer en t sub sets are joined b y an edge in G ( n, p ) is q r , where r is t he total n umb er of all suc h pa irs, i. e. r = X k ,ℓ k 2  α k ,ℓ 2  + 1 2 X ( k 1 ,ℓ 1 ) 6 =( k 2 ,ℓ 2 ) k 1 k 2 α k 1 ,ℓ 1 α k 2 ,ℓ 2 = − 1 2 X k ,ℓ k 2 α k ,ℓ + 1 2   X k ,ℓ k α k ,ℓ   2 = − 1 2 X k ,ℓ k 2 α k ,ℓ + n 2 2 . Multiplying the pie ces, P n ( α ) = N ( n, p ) Y k ,ℓ 1 α k ,ℓ !  ( p/q ) ℓ C ( k , k + ℓ ) k !  α k,ℓ . 8 BORIS PITTEL AND JI-A YEUM So, using Lemma 2.1.1, (2.1.4) Pr( S n > 0) = N ( n, p ) X α Y k ,ℓ 1 α k ,ℓ !  (1 / 2) ℓ +1 ( p/q ) ℓ C ( k , k + ℓ ) k !  α k,ℓ . Notice that dropping fac tors ( 1 / 2) ℓ +1 on the r igh t, we ge t 1 instead of Pr( S n > 0) on the left, i.e. (2.1.5) 1 = N ( n, p ) X α Y k ,ℓ 1 α k ,ℓ !  ( p/q ) ℓ C ( k , k + ℓ ) k !  α k,ℓ . So, m ult iplying b oth sides of (2.1.4) b y x n N ( n,p ) and su mming ov e r n ≥ 0 , (2.1.6) X n x n Pr( S n > 0) N ( n, p ) = X P k,ℓ k α k,ℓ < ∞ Y k ,ℓ x k α k,ℓ α k ,ℓ !  (1 / 2) ℓ +1 ( p/q ) ℓ C ( k , k + ℓ ) k !  α k,ℓ = exp " 1 2 X ℓ ( p/ 2 q ) ℓ X k C ( k , k + ℓ ) x k k ! # = exp " 1 2 X ℓ ( p/ 2 q ) ℓ C ℓ ( x ) # . W e h asten to add th at the series on t he righ t, whence the o ne on the left, conv erge s for x = 0 only . Ind eed, using (2.1.5) inste ad of (2.1.4) , (2.1.7) exp " X ℓ ( p/q ) ℓ C ℓ ( x ) # = X n x n 1 N ( n, p ) = X n  xq 3 / 2 p  n n ! q n 2 / 2 = ∞ , for eac h x > 0 . Ther efore, set ting p/ 2 q = p 1 /q 1 , ( q 1 = 1 − p 1 ), X ℓ ( p/ 2 q ) ℓ C ℓ ( x ) = X ℓ ( p 1 /q 1 ) ℓ C ℓ ( x ) = ∞ , ∀ x > 0 , as w e ll.  Note. Sett ing p/q = w , x = y w , in ( 2.1.7), so that p = w/ ( w + 1 ) , q = 1 / ( w + 1) , w e obtain a well known (e xp onential) ide n tity , e. g. J anson et al [12], exp   X ℓ ≥− 1 w ℓ C ℓ ( y w )   = X n ≥ 0 y n n ! ( w + 1) ( n 2 ) ; the right expres sion (the left exp onent resp.) is a biv ariate genera ting function for graphs ( connect ed g raphs re sp.) G enumerated by v ( G ) and e ( G ) . Here is a similar iden tit y in v o lving gener ating func tions of connec ted graphs G with a ﬁxed p ositive excess, (2.1.8) exp   X ℓ ≥ 1 w ℓ C ℓ ( x )   = X r ≥ 0 w r E r ( x ) , RANDOM EQUA TIONS 9 where E 0 ( x ) ≡ 1 , and , for ℓ ≥ 1 , E ℓ ( x ) is the exp on en tial ge nerating funct ion of graphs G withou t tr ee comp onents and unicyclic comp onents, t hat ha v e excess ℓ ( G ) = e ( G ) − v ( G ) = ℓ , see [12]. In t he light o f Lemma 2.1.2, w e will n eed an expansion (2.1.9) exp   1 2 X ℓ ≥ 1 w ℓ C ℓ ( x )   = X r ≥ 0 w r F r ( x ) . Lik e E r ( x ) , ea c h p ow e r series F r ( x ) has nonnega tiv e co eﬃcie n ts, and c on v erges for | x | < e − 1 . By Lemma 2.1.2 and (2.1.8), (2.1.10) Pr( S n > 0) = N ( n, p ) X r ≥ 0  p 2 q  r [ x n ] n e H ( x ) F r ( x ) o ; H ( x ) := q p C − 1 ( x ) + 1 2 C 0 ( x ) . In terchange of [ x n ] and th e summation is just iﬁable as each o f th e funct ions o n the righ t has a p o w er series expansion with o nly nonnegat iv e co eﬃcients. Th at is, div ergenc e of P ℓ ( p/ 2 q ) ℓ C ℓ ( x ) in (2.1.6) do e s not imp ed e e v aluation of Pr ( S n > 0) . Indirect ly t hough this div er gence do es make it diﬃcult, if p ossible at a ll, t o obtain a suﬃcien tly sharp estimate of the terms in the ab o v e sum for r going to ∞ with n , nee ded t o de riv e an asymptotic form ula for tha t p robability . Th us w e ne ed to trunca te, one wa y or another , the div ergent series on the righ t in (2.1.6). On e of t he prop e rties of C ℓ ( x ) discov e red by W righ t [23 ] is tha t e ac h of t hese series c on v erges (div erges) for | x | < e − 1 (for | x | > e − 1 resp.). So, pic king L ≥ 0 , and rest ricting summation ran ge to ℓ ∈ [ − 1 , L ] , we deﬁnitely get a series co n v ergent for | x | < e − 1 . What is then a counterpart of Pr( S n > 0) ? P erusing the pro of of Lemma 2.1.2, w e easily see t he an sw er. Let G b e a gra ph with co mp onents H 1 , H 2 , . . . . Deﬁne E ( G ) , a ma xim u m e xcess of G , b y E ( G ) = max i [ e ( H i ) − v ( H i )] . It can b e easily seen that E ( G ) is monoto ne incre asing, i. e. E ( G ′ ) ≤ E ( G ′′ ) if G ′ ⊆ G ′′ . Le t E n = E ( G ( n, p )) . Lemma 2.1.3. (2.1.11) Pr ( S n > 0 , E n ≤ L ) = N ( n, p ) [ x n ] exp " 1 2 L X ℓ = − 1  p 2 q  ℓ C ℓ ( x ) # , The pro of of (2.1.11) is an o b vious mo diﬁcation o f t hat f or (2.1.2). If, using ( 2.1.11), w e are able t o estimate Pr( S n > 0 , E n ≤ L ) , then evidently w e will get a lower b ound of Pr( S n > 0) , via (2.1.12) Pr( S n > 0) ≥ Pr( S n > 0 , E n ≤ L ) . Crucially , the ev en ts { S n > 0 } and {E n ≤ L } are p ositively corr elated. 10 BORIS PITTEL AND JI-A YEUM Lemma 2.1.4. (2.1.13) Pr ( S n > 0) ≤ Pr ( S n > 0 , E n ≤ L ) Pr ( E n ≤ L ) . Note. The upsho t of (2.1.12) -(2.1.13) is that Pr( S n > 0) ∼ Pr( S n > 0 , E n ≤ L ) , prov ided that L = L ( n ) is just large enough to guara n tee t hat Pr( E n ≤ L ) → 1 . Pro of of Lemma 2.1.4. By Lemma 2.1.1, Pr( S n > 0 , E n ≤ L ) = E "  1 2  X ( G ( n,p ) ) 1 {E ( G ( n,p )) ≤ L } # , where X ( G ) = e ( G ) − n + c ( G ) . Notice that (1 / 2) X ( G ) is mono tone decrea sing. Indeed, if a grap h G 2 is obtained b y adding one edge to a grap h G 1 , then e ( G 2 ) = e ( G 1 ) + 1 , c ( G 2 ) ∈ { c ( G 1 ) − 1 , c ( G 1 ) } , so that X ( G 2 ) ≥ X ( G 1 ) . Hence , using induct ion on e ( G 2 ) − e ( G 1 ) , G 1 ⊆ G 2 = ⇒ X ( G 2 ) ≥ X ( G 1 ) . F urt hermore 1 {E ( G ) ≤ L } is also mono tone dec reasing. (F or e / ∈ E ( G ) , if e joins t w o v ertice s from t he same comp one n t of G t hen E ( G + e ) ≥ E ( G ) ob viously . If e joins tw o comp o nents, H 1 and H 2 of G , then the resulting c omp onent has an excess mo re t han or equal to max {E ( H 1 ) , E ( H 2 ) } , with equalit y when one of t w o comp on en ts is a tree.) No w no tice that each G on [ n ] is esse n tially a  n 2  -long t uple δ o f { 0 , 1 } -v alue d v ariables δ ( i,j ) , δ ( i,j ) = 1 mean ing that ( i, j ) ∈ E ( G ) . So, a graph fun ction f ( G ) can b e u nam bigiously writt en as f ( δ ) . Imp ortantly , a mon otone decrea sing ( in- creasing) gra ph function is a mon otone decre asing (increasing) func tion of t he co de δ . F or the random g raph G ( n, p ) , the comp onents of δ a re indep ende n t random v ariavbles. According to an FKG-t yp e inequa lit y , see Grimmett and Stirza k er [11] for instan ce, for a n y t w o decre asing (tw o incr easing) func tions f ( Y ) , g ( Y ) of a v ector Y with indep en den t comp o nents, E [ f ( Y ) g ( Y )] ≥ E [ f ( Y )] E [ g ( Y )] . Applying this inequalit y t o (1 / 2) X ( δ ) 1 {E ( δ ) ≤ L } , w e obt ain Pr( S n > 0 , E n ≤ L ) ≥ E "  1 2  X ( G ( n,p ) ) # E  1 {E ( G ( n,p )) ≤ L }  = Pr ( S n > 0) Pr( E n ≤ L ) .  RANDOM EQUA TIONS 11 Th us our next step is to determine how large E ( G ( n , p )) is ty pically , if (2.1.14) p = 1 + λn − 1 / 3 n , λ = o ( n 1 / 3 ) . F or p = c /n , c < 1 , it w a s sho wn in Pitte l [19] th at lim Pr ( G ( n, p ) do es n ot ha v e a cycle) = (1 − c ) 1 / 2 exp( c/ 2 + c 2 / 4) . F rom this result and mono tonicity o f E ( G ) , it follo ws that, for p in (2.1.14), lim Pr( E ( G ( n, p )) ≥ 0) = 1 . If λ → −∞ , the n we also hav e (2.1.15) lim Pr( E ( G ( n, p )) > 0) = 0 , that is E ( G ( n, p )) ≤ 0 with h igh prob abilit y (whp) . (The pro o f of (2.1.15 ) mimic ks Luczak’s pro of [14] of an analogous prop e rt y of G ( n, m ) , with n − 2 / 3 ( n/ 2 − m ) → ∞ .) F urt hermore, by Theo rem 1 in [16], and mon otonicity of E ( G ( n, p )) , it f ollo ws t hat E ( G ( n , p )) is b oun ded in probability (is O P (1) , in short), if lim sup λ < ∞ . Finally , su pp ose t hat λ → ∞ . Let L ( G ( n, m )) denote the tot al exce ss of the n um b er of edge s ov er the num b e r of v ert ices in the c om plex comp onents of G ( n, m ) , i. e . t he c omp one n ts that are ne ither trees nor unicyc lic. Accord ing to a limit theore m for L ( G ( n, m = ( n/ 2)(1 + λn − 1 / 3 ))) from [12], L ( G ( n, m )) / λ 3 → 2 / 3 , in pro babilit y . According to Luczak [14], whp G ( n, m ) has e xactly one complex comp on en t. So wh p E ( G ( n, m )) = L ( G ( n, m )) , i. e . E ( G ( n, m )) /λ 3 → 2 / 3 in probability , as w ell. No w, if m ′ = N p + O  p N pq  , N :=  n 2  , then m ′ = n 2 (1 + λ ′ n − 1 / 3 ) , λ ′ := λ  1 + O ( n − 1 / 6 )  . Therefor e, in probability , E ( G ( n , m ′ )) λ 3 → 2 3 , as w e ll. F rom a ge neral “ transfer principle” ([5 ], [1 5]) it follows th en th at E ( G ( n , p )) λ 3 → 2 3 , in probability , to o. This d iscussion justiﬁes t he fo llo wing c hoice of L : (2.1.16) L =      0 , if lim λ = −∞ , u → ∞ ho wev er slowly , if λ = O ( 1) , λ 3 , if λ → ∞ , λ = o ( n 1 / 12 ) . 12 BORIS PITTEL AND JI-A YEUM 2.2 Generating functio ns. First, some ba sic fac ts ab out the ge nerating function s C ℓ ( x ) and E ℓ ( x ) . In- tro du ce a tree function T ( x ) , the exp o nen tial g enerat ing function of { k k − 1 } , the counts of ro oted t rees on [ k ] , k ≥ 1 . It is w ell kno wn that the series T ( x ) = X k ≥ 1 x k k ! k k − 1 has con v ergen ce radius e − 1 , an d that T ( x ) = xe T ( x ) , | x | ≤ e − 1 ; in part icular, T ( e − 1 ) = 1 . ( This last fa ct has a probabilistic explan ation: { k k − 1 e k k ! } is the distribution of a to tal p rogeny in a branching pro cess with an immediate family size b eing P o isson ( 1 ) distr ibuted.) T ( x ) is a building blo ck fo r all C ℓ ( x ) . Namely , (Mo on [18], W right [23], Bagae v [1] resp.), C − 1 ( x ) = T ( x ) − 1 2 T 2 ( x ) , (2.2.1) C 0 ( x ) = 1 2  ln 1 1 − T ( x ) − T ( x ) − 1 2 T 2 ( x )  , (2.2.2) C 1 ( x ) = T 4 ( x )(6 − T ( x )) 24(1 − T ( x )) 3 , and ultimately , for all ℓ > 0 , (2.2.3) C ℓ ( x ) = 3 ℓ +2 X d =0 c ℓ,d (1 − T ( x )) 3 ℓ − d , W right [23]. Needless to say , | x | < e − 1 in all the formulas. One should righ tfully an ticipat e th ough that the b ehaviour of C ℓ ( x ) f or x ’s close to e − 1 is going t o determine a n asympto tic b e ha viour of Pr( S n > 0 , E n ≤ L ) . And so the ( d = 0) - term in (2.2.3) might w ell b e th e only te rm w e would need e v en tually . In this context, it is rema rk able tha t in a follow- up pa p er [24 ] W rig h t was able to sho w that (2.2.4) c ℓ (1 − T ( x )) 3 ℓ − d ℓ (1 − T ( x )) 3 ℓ − 1 ≤ c C ℓ ( x ) ≤ c c ℓ (1 − T ( x )) 3 ℓ , ( c ℓ := c ℓ, 0 > 0 , d ℓ := − c ℓ, 1 > 0) ( ∀ n ≥ 1) . (W e wr ite P j a j x j ≤ c P j b j x j when a j ≤ b j for a ll j .) In the same p ap er h e also demonstr ated existenc e of a constant c > 0 suc h that (2.2.5) c ℓ ∼ c  3 2  ℓ ( ℓ − 1)! , d ℓ ∼ c  3 2  ℓ ℓ ! , ( ℓ → ∞ ) . RANDOM EQUA TIONS 13 Later Bag aev and Dmit riev [2] sho w ed t hat c = (2 π ) − 1 . By no w t here hav e b een found ot her p ro ofs of this fact. Se e, fo r inst ance, Bender e t al [3] for an asympt otic expansion of c ℓ due to Me ertee ns, an d Lucz ak e t a l [1 6] for a rath er eleme n tary pro of ba sed on t he b eh a vior of the comp o nen t size distr ibution for th e critical G ( n, m ) . T urn to E r ( x ) , r ≥ 1 . It w as sh o wn in [12 ] t hat, analogously t o (2.2 .3), (2.2.6) E r ( x ) = 5 r X d =0 ε r,d (1 − T ( x )) 3 r − d , ε r,d = (6 r − 2 d )! Q d ( r ) 2 5 r 3 2 r − d (3 r − d ) !(2 r − d )! , where Q 0 ( r ) = 1 , an d, for d > 0 , Q d ( r ) is a p olynomial of d egree d . By Stirling’s form ula, (2.2.7) ε r := ε r, 0 ∼ (2 π ) − 1 / 2  3 2  r r r − 1 / 2 e − r , r → ∞ . F ormally diﬀerentiating b ot h sides of (2.1.8) wit h r esp ec t to w a nd equa ting c o ef- ﬁcients by w ℓ − 1 , w e get a recurr ence relation (2.2.8) rE r ( x ) = r X k =1 k C k ( x ) E r − k ( x ) . By ( 2.2.3) an d (2.2.6) , th e highe st p ow er o f ( 1 − T ( x )) − 1 on b oth sides of ( 2.2.8) is 3 r , and equating the tw o co eﬃcients w e get a recurr ence relat ion in v olving ε r and c r , (2.2.9) r ε r = r X k =1 k c k ε r − k , r ≥ 1 . With the se pre liminaries out of the wa y , we turn t o the formula ( 2.1.11) for Pr( S n > 0 , E n ≤ L ) . Notice upfront tha t, fo r L = 0 —arising wh en λ → −∞ —w e simply ha v e (2.2.10) P r( S n > 0 , E n ≤ 0) = N ( n, p ) [ x n ] e H ( x ) , H ( x ) = q p C − 1 ( x ) + 1 2 C 0 ( x ) . The next Lemma provides a counterpart of (2.1.10) and ( 2.2.10) f or L ∈ [1 , ∞ ) . Lemma 2.2.1. Given L ∈ [1 , ∞ ) , (2.2.11) Pr ( S n > 0 , E n ≤ L ) = N ( n, p ) ∞ X r =0  p 2 q  r [ x n ] n e H ( x ) F L r ( x ) o , wher e { F L r ( x ) } is deter mine d by a r e curr enc e r elation (2.2.12) r F L r ( x ) = 1 2 r ∧ L X k =1 k C k ( x ) F L r − k ( x ) , r ≥ 1 , 14 BORIS PITTEL AND JI-A YEUM and F L 0 ( x ) = 1 . (Her e a ∧ b := min { a, b } .) Pro of of Lemma 2.2.1. Clear ly (2.2.13) exp 1 2 L X ℓ =1 w ℓ C ℓ ( x ) ! = ∞ X r =0 w r F L r ( x ) , where F L r ( x ) a re some p o w er series, with no nnegat iv e co eﬃc ien ts, con v ergent for | x | < e − 1 . Th is iden tit y implies t hat exp L X ℓ =1 w ℓ C ℓ ( x ) ! = ∞ X r =0 w r F L r ( x ) ! 2 . Diﬀerentiating this with resp ect to w and replac ing exp  P L ℓ =1 w ℓ C ℓ ( x )  on th e left of the r esulting iden tity with  P ∞ s =0 w s F L s ( x )  2 , w e get , after m ult iplying b y w , ∞ X s =0 w s F L s ( x ) ! L X ℓ =1 ℓw ℓ C ℓ ( x ) ! = 2 ∞ X r =1 r w r F L r ( x ) . Equating t he co e ﬃcien ts by w r , r ≥ 1 , of th e tw o sides we obtain th e rec urrence (2.2.12). The re curren ce ( 2.2.12) yields a v ery usefu l infor mation a b out F L r ( x ) . Lemma 2.2.2. L et L > 0 . F or r ≥ 0 , (2.2.14) F L r ( x ) = 5 r X d =0 f L r,d (1 − T ( x )) 3 r − d , and, denoting f L r = f L r, 0 , g L r = − f L r, 1 (2.2.15) f L r (1 − T ( x )) 3 r − g L r (1 − T ( x )) 3 r − 1 ≤ c F L r ( x ) ≤ c f L r (1 − T ( x )) 3 r . F ur thermor e the le ading c o eﬃcients f L r , g L r satisfy a r e cur r enc e r elation r f L r = 1 2 r ∧ L X k =1 k c k f L r − k ; f L 0 = 1 , (2.2.16) r g L r = 1 2 r ∧ L X k =1 k c k g L r − k + 1 2 r ∧ L X k =1 k d k f L r − k ; g L 0 = 0 , (2.2.17) so, in p articular , f L r > 0 and g L r > 0 for r > 0 . Note. 1 . This Lemma and its pro of are similar t o th ose for th e gene rating function s E r ( x ) obtaine d in [10]. RANDOM EQUA TIONS 15 Pro of of Lemma 2.2.2. (a) W e pr o v e (2.2.14 ) by induct ion on r . ( 2.2.14) holds fo r r = 0 as F L 0 ( x ) ≡ 1 an d f L 0 , 0 = f L 0 = 1 . F urther, by (2.2.12 ) a nd (2 .2.3), F L 1 ( x ) = 1 2 C 1 ( x ) = 1 2 5 X d =0 c 1 ,d (1 − T ( x )) 3 − d , i. e. (2.2.14) holds for r = 1 t o o. Assume that r ≥ 2 and t hat ( 2.2.14) holds for for r ′ ∈ [1 , r − 1] . Then , by ( 2.2.12), ( 2.2.3) a nd inductiv e assu mption, F L r ( x ) = 1 2 r r ∧ L X k =1 k C k ( x ) F L r − k ( x ) = 1 2 r r ∧ L X k =1 k 3 k +2 X d =0 c k ,d (1 − T ( x )) 3 k − d 5( r − k ) X d 1 =0 f L r − k , d 1 (1 − T ( x )) 3( r − k ) − d 1 = 1 2 r r ∧ L X k =1 k X d ≤ 3 k +2 , d 1 ≤ 5( r − k ) c k ,d f L r − k , d 1 (1 − T ( x )) 3 r − ( d + d 1 ) . Here 0 ≤ d + d 1 ≤ 3 k + 2 + 5 ( r − k ) = 5 r − 2( k − 1) ≤ 5 r , so (2.2.14) h olds f or r as w ell. (b) Plug ging (2.2.14) a nd (2 .2.3) into (2.2.12) we ge t 5 r X d =0 f L r,d (1 − T ( x )) 3 r − d = r ∧ L X k =1 k 2 r 3 k +2 X d 1 =0 c k ,d 1 (1 − T ( x )) 3 k − d 1 5( r − k ) X d 2 =0 f L r − k , d 2 (1 − T ( x )) 3( r − k ) − d 2 . Equating the c o eﬃcients b y (1 − T ( x )) − 3 r (b y (1 − T ( x ) ) − 3 r +1 resp.) on the right and on the left, w e obtain ( 2.2.16) ((2.2.17) resp.). (c) F o r r = 0 , ( 2.2.15) holds trivially . F or r ≥ 1 , induct iv ely w e hav e: by ( 2.2.4) (upp e r b oun d) and (2.2.12), (2.2.16), F L r ( x ) ≤ c 1 2 r r ∧ L X k =1 k c k (1 − T ( x )) 3 k f L r − k (1 − T ( x )) 3( r − k ) = 1 (1 − T ( x )) 3 r 1 2 r r ∧ L X k =1 k c k f L r − k = f L r (1 − T ( x )) 3 r ; 16 BORIS PITTEL AND JI-A YEUM furthe rmore, by (2.2.4) (lo w er b ound ), (2.2.12 ) and (2.2.16)-(2 .2.17), F L r ( x ) ≥ c 1 2 r r ∧ L X k =1 k  c k (1 − T ( x )) 3 k − d k (1 − T ( x )) 3 k − 1  F L r − k ( x ) ≥ c 1 2 r r ∧ L X k =1 k c k (1 − T ( x )) 3 k " f L r − k (1 − T ( x )) 3( r − k ) − g L r − k (1 − T ( x )) 3( r − k ) − 1 # − 1 2 r r ∧ L X k =1 k d k (1 − T ( x )) 3 k − 1 · f L r − k (1 − T ( x )) 3( r − k ) = f L r (1 − T ( x )) 3 r − 1 (1 − T ( x )) 3 r − 1 " 1 2 r r ∧ L X k =1 k c k g L r − k + 1 2 r r ∧ L X k =1 k d k f L r − k # = f l r (1 − T ( x )) 3 r − g L r (1 − T ( x )) 3 r − 1 .  T o make the b ound (2.2.15) work w e ne ed to hav e a close lo ok at th e sequen ce { f L r , g L r } r ≥ 0 . F irst of all, it follo ws from ( 2.2.16) th at f L r ≤ f r := f ∞ r , g L r ≤ g r := g ∞ r . That is f r and − g r are the c o eﬃcients by (1 − T ( x )) − 3 r and (1 − T ( x )) − 3 r +1 in the expan sion (2.2.13) for F r ( x ) := F ∞ r ( x ) . Now, using ( 2.2.13) for L = ∞ and (2.1.8), w e see t hat   X r ≥ 0 w r F r ( x )   2 = X r ≥ 0 w r E r ( x ) . So, equating t he c o eﬃcients b y w r , r ≥ 0 , we get r X k =0 F k ( x ) F r − k ( x ) = E r ( x ) . Plugging ( 2.2.6) and ( 2.2.14) (with L = ∞ ), and c omparing co eﬃc ien ts b y (1 − T ( x ) ) − 3 r ( (1 − T ( x )) − 3 r +1 , resp.), w e obta in r X k =0 f k f r − k = ε r, 0 ; 2 r X k =0 f k g r − k = − ε r, 1 . In particular, f r ≤ 1 2 ε r, 0 , g r ≤ − 1 2 ε r, 1 . RANDOM EQUA TIONS 17 Consequently , using (2.2.6) for r ≥ 2 and d = 0 , f r = 1 2 ε r, 0 − 1 2 r − 1 X k =1 f k f r − k ≥ 1 2 ε r, 0 − 1 2 r − 1 X k =1 1 2 ε k , 0 1 2 ε r − k , 0 ≥ ε r, 0 2   1 − 1 4 r − 1 X j =1  r j  − 1  r j  2 r 2 j  3 r 3 j   6 r 6 j    ≥ ε r, 0 2   1 − 1 4 r − 1 X j =1  r j  − 1   ≥ ε r, 0 2 (1 − 1 /r ) , that is (2.2.18) ε r, 0 2 (1 − 1 /r ) ≤ f r ≤ ε r, 0 2 ∼ 1 2 √ 2 π  3 2  r r r − 1 / 2 e − r , ( r → ∞ ) , see (2.2.7). F u rthermor e, using (2.2.6) for r > 0 and d = 1 , (2.2.19) g r ≤ b  3 2  r r r +1 / 2 e − r . And on e can prov e a matching lo w er b oun d for g r . Hence , lik e ε r , f r , g r grow essentially as r r , to o fast for F r ( x ) = F ∞ r ( x ) to b e useful for asymptot ic estimates. The next Lemma (last in t his sub section) sho ws that, in a pleasing con trast , f L r , g L r grow m uc h slo w er when r ≫ L . Lemma 2.2.3. Ther e exists L 0 such that, for L ≥ L 0 , (2.2.20) f L r ≤ b  3 L 2 e  r , g L r ≤ b r  3 L 2 e  r , ∀ r ≥ 0 . Pro of o f Lem ma 2.2 .3. (a) It is immediat e from (2.2.18) , (2.2.19) t hat, for some absolute constant A and all L > 0 , f L r = f r ≤ A  3 L 2 e  r , g L r = g r ≤ Ar  3 L 2 e  r 0 ≤ r ≤ L. Let us pr o v e existence an integer L > 0 , with a p rop erty: if for some s ≥ L an d all t ≤ s , (2.2.21) f L t ≤ A  3 L 2 e  t , g L t ≤ At  3 L 2 e  t , then f L s +1 ≤ A  3 L 2 e  s +1 , g L s +1 ≤ A ( s + 1)  3 L 2 e  s +1 . 18 BORIS PITTEL AND JI-A YEUM By (2.2.16) , ( 2.2.21), and (2.2.5), t here exist s an a bsolute c onstant B > 0 suc h that ( s + 1 ) f L s +1 ≤ AB  3 L 2 e  s +1 L 1 / 2 L X k =1  k L  k . A fu nction ( x/L ) x attains its minimum on [0 , L ] at x = L/e , and it is easy t o sho w that ( x/L ) x ≤ ( e − x , x ≤ L/e , e − ( L − x )(3 − e ) / 2 , x ≥ L/e . Since s + 1 ≥ L , we obta in the n f L s +1 ≤ AB  1 1 − e − 1 + 1 1 − e − (3 − e ) / 2  · L − 1 / 2  3 L 2 e  s +1 ≤ A  3 L 2 e  s +1 , if w e cho ose L ≥ L 1 := B 2  1 1 − e − 1 + 1 1 − e − (3 − e ) / 2  2 . Lik ewise, b y (2.2.17), (2.2.21) and (2.2.5) , ( s + 1 ) g L s +1 ≤ AB ( s + 1)  3 L 2 e  s +1 L 1 / 2 L X k =1  k L  k + AB ′ ( s + 1 )  3 L 2 e  s +1 L 1 / 2 L X k =1  k L  k , so that g L s +1 ≤ A ( s + 1)  3 L 2 e  s +1 , if w e cho ose L ≥ L 2 := ( B + B ′ ) 2  1 1 − e − 1 + 1 1 − e − (3 − e ) / 2  2 . Th us, pic king L = max { L 1 , L 2 } = L 2 , w e can acco mplish the inductive step, from s ( ≥ L ) t o s + 1 , showing that , for this L , (2.2.20) holds f or all t .  Com bining (2.2.10), Lemma 2.2.1, Lemma 2.2.2, w e b ound Pr( S n > 0 , E n ≤ L ) . Prop osition 2 .2 .4. L et L ∈ [0 , ∞ ) . Then Σ 1 ≤ Pr ( S n > 0 , E n ≤ L ) ≤ Σ 2 . Her e (2.2.22) Σ 1 = N ( n, p ) X r ≥ 0  p 2 q  r [ x n ]  f L r e H ( x ) (1 − T ( x )) 3 r − g L r e H ( x ) (1 − T ( x )) 3 r − 1  , Σ 2 = N ( n, p ) X r ≥ 0  p 2 q  r [ x n ] f L r e H ( x ) (1 − T ( x )) 3 r , RANDOM EQUA TIONS 19 and (2.2.23) f L r    = f r , r ≤ L, ≤ b  3 L 2 e  r , r ≥ L, g L r    = g r , r ≤ L, ≤ b r  3 L 2 e  r , r ≥ L, with f r , g r satisfying the c onditions (2.2.18)-(2.2 .19). Note. The relations (2.2.22)-(2 .2.23) inde ed cov er t he case L = 0 since in this case f 0 = 1 , g 0 = 0 and f L r = g L r = 0 for r > 0 . 2.3 Asymptotic formula for Pr ( S n > 0) . The Prop osition 2.2.4 ma k es it c lear tha t w e nee d to ﬁnd a n asymptot ic form ula for (2.3.1) N ( n , p ) φ n,w , φ n,w := [ x n ] e H ( x ) (1 − T ( x )) w , w = 0 , 3 , 6 . . . Using N ( n, p ) = n ! q n 2 / 2 ( pq − 3 / 2 ) n and Stirling’s formula for n ! , w ith so me work w e obtain (2.3.2) N ( n, p ) = √ 2 π n exp  − n 3 2 + n 2 / 3 λ 2 − n 1 / 3 λ 2 2 + λ 3 3 + 5 4 + O ( n − 1 / 3 (1 + λ 4 ))  . The big-Oh t erm here is o (1) if | λ | = o ( n 1 / 12 ) , wh ic h is the co ndition o f T heorem 1.1. T urn to φ n,w . S ince t he function in questio n is analytic for | x | < e − 1 , φ n,w = 1 2 π i I Γ e H ( x ) x n +1 (1 − T ( x )) w dx, where Γ is a simple close d c on tour enclosing th e origin and lying in the disc | x | < e − 1 . B y ( 2.1.10), (2 .2.1)-(2.2.2), t he funct ion in ( 2.3.1) dep ends on x only t hrough T ( x ) , whic h satisﬁes T ( x ) = xe T ( x ) . This sug gests in tro du cing a new v ariable of in tegrat ion y , suc h that y e − y = x , i. e. y = T ( x ) = X k ≥ 1 x k k ! k k − 1 , | x | < e − 1 . Pic king a simple closed co n tour Γ ′ in the y -plane su c h that its image unde r x = y e − y is a simple c losed contour Γ within th e disc | x | < e − 1 , a nd using ( 2.2.1)- (2.2.2), w e obtain (2.3.3) φ n,w = 1 2 π i I Γ ′ y − n − 1 e ny exp  κ ( y ) − y 4 − y 2 8  (1 − y ) 3 / 4 − w dy , κ ( y ) := q p  y − y 2 2  ; 20 BORIS PITTEL AND JI-A YEUM q /p ∼ n , so y − n e ny e κ ( y ) w ould ha v e fully acc ounted for asympto tic b ehavior of the in t egral, ha d it not b e en for the fac tor (1 − y ) 3 / 4 − w . On ce Γ ′ is pic k ed, it can b e replac ed b y an y circ ular contour y = ρe iθ , θ ∈ ( − π , π ] , ρ < 1 . (The c ondition ρ < 1 is dict ated by t he fac tor (1 − y ) 3 / 4 − w .) And (2 .3.3) b ecomes (2.3.4) φ n,w = 1 2 π I ( w ) , I ( w ) := Z π − π e h ( ρ,θ ) exp  − ρe iθ / 4 − ρ 2 e i 2 θ / 8  (1 − ρe iθ ) 3 / 4 − w dθ , h ( ρ, θ ) = q p  ρe iθ − ρ 2 e i 2 θ 2  + nρe iθ − n ( ln ρ + iθ ) . Let us cho ose ρ < 1 in such a wa y that , a s a func tion of θ , | e h ( ρ,θ ) | at tains its maxim um at θ = 0 . No w | e h ( ρ,θ ) | = e f ( ρ,θ ) , with f ( ρ, θ ) = Re h ( ρ, θ ) = q p ρ cos θ − q 2 p ρ 2 cos 2 θ + nρ co s θ − n ln ρ, so that f ′ θ ( ρ, θ ) = 2 q p ρ 2 sin θ  cos θ − 1 + np/q 2 ρ  . Then f ′ θ ( ρ, θ ) > 0 ( < 0 r esp.) for θ < 0 ( θ > 0 resp.) if (2.3.5) ρ < 1 2 (1 + np / q ) . Let us se t ρ = e − an − 1 / 3 , where a = o ( n 1 / 3 ) , sinc e w e w ant ρ → 1 . No w 1 2 (1 + np / q ) > 1 + λ 2 n − 1 / 3 , ρ ≤ 1 − an − 1 / 3 + a 2 2 n − 2 / 3 ; so (2.3.5) is obviously sat isﬁed if (2.3.6) a + λ 2 ≥ a 2 . (2.3.6) is trivially met if λ ≥ 0 . F or λ < 0 , | λ | = o ( n 1 / 3 ) , (2.3.6) is met if a ≥ | λ | . In all c ases we will assume that lim inf a > 0 . Wh y do w e w an t a = o ( n 1 / 3 ) ? Bec ause, as a func tion of ρ , h ( ρ, 0) atta ins its minim um at np/ q ∼ 1 , if λ < 0 is ﬁxed , and in this case n p/q < 1 , and th e minim um p oin t is 1 if λ ≥ 0 . So our ρ is a reasona ble approximation of the sa ddle p oint of | h ( ρ, θ ) | , dep e ndent on λ , chosen fr om amon g th e fe asible v alues, i. e. those st rictly b elo w 1 . Charact eristically ρ is v ery c lose t o 1 , t he singular p oin t of the factor ( 1 − y ) 3 / 4 − w , whic h is esp ec ially inﬂuential for larg e w ’s. Its presence rules out a “pain-free ” app lication o f g eneral to ols such as W ats on’s Le mma. Under ( 2.3.6), | f ′ θ ( ρ, θ ) | ≥ a 2 n 2 / 3 | sin θ | , RANDOM EQUA TIONS 21 and sign f ′ θ ( ρ, θ ) = − sign θ , so that (2.3.7) f ( ρ, θ ) ≤ f ( ρ, 0) − a 2 n 2 / 3 Z | θ | 0 sin z dz = f ( ρ, 0) − an 2 / 3 sin 2 ( θ / 2) ≤ f ( ρ, 0) − aπ − 2 n 2 / 3 θ 2 = h ( ρ, 0) − a ( π ) 2 n 2 / 3 θ 2 . Let us break the integral I ( w ) in (2.3.4) into tw o part s, I 1 ( w ) for | θ | ≤ θ 0 , and I 2 ( w ) for | θ | ≥ θ 0 , where θ 0 = π n − 1 / 3 ln n. Since f ( ρ, θ ) is d ecreasing with | θ | , an d | 1 − ρe iθ | ≥ 1 − ρ , it follo ws from (2.3.7) that (2.3.8) | I 2 ( w ) | ≤ b  1 − e − an − 1 / 3  − w e f ( ρ,θ 0 ) ≤  a 1 n − 1 / 3  − w e h ( ρ, 0) exp( − a ln 2 n ); a 1 := n 1 / 3  1 − e − an − 1 / 3  . T urn to I 1 ( w ) . Th is t ime | θ | ≤ θ 0 . F irst, let us write ρe iθ = e − sn − 1 / 3 , s = a − it, t := n 1 / 3 θ ; so | s | ≤ a + π ln n . The secon d (easy) exp onent in the integrand of I 1 ( w ) is asymptotic to − 3 / 8 , or more prec isely , (2.3.9) − 1 4 e − sn − 1 / 3 − 1 8 e − 2 sn − 1 / 3 = Q 2 ( a ) + O ( | t | n − 1 / 3 ) , Q 2 ( a ) := − 1 4 e − an − 1 / 3 − 1 8 e − 2 an − 1 / 3 . Determina tion of a usable asympto tic form ula for h ( ρ, θ ) is more lab orious. It is conv enient to set q / p = ne − µn − 1 / 3 ; th us µ = n 1 / 3 ln np q ≥ n 1 / 3 ln(1 + λn − 1 / 3 ) ≥ λ (1 − λn − 1 / 3 / 2) , and µ − λ = O ( n − 2 / 3 + n − 1 / 3 λ 2 ) . Using the new p arameter s s and µ w e t ransform the f orm ula ( 2.3.4) fo r h ( ρ, θ ) to h ( ρ, θ ) = n  e − ( µ + s ) n − 1 / 3 − 1 2 e − ( µ +2 s ) n − 1 / 3 + e − sn − 1 / 3 + sn − 1 / 3  . Appro ximating th e thr ee exp o nen ts b y the 4 -th d egree T aylor p olyno mials, w e obtain (2.3.10) h ( ρ, θ ) = n  3 2 − n − 1 / 3 µ 2 + n − 2 / 3 µ 2 4 − n − 1 µ 3 12  + Q 1 ( µ, a ) +  µs 2 2 + s 3 3  + O  D 1 ( t )  ; Q 1 ( µ, a ) := n − 1 / 3  ( µ + a ) 4 4! − ( µ + 2 a ) 4 4!2 + a 4 4!  ; D 1 ( t ) := n − 1 / 3 | t | ( | λ | + a + ln n ) 3 + n − 2 / 3 ( | λ | + a + ln n ) 5 . 22 BORIS PITTEL AND JI-A YEUM (Explanation : the second summan d in D 1 ( t ) is the ap pro ximation error b ound for each o f the T aylor p olynomials; t he ﬁrst summand is the c ommon b ou nd o f | ( µ + a ) 4 − ( µ + s ) 4 | , | ( µ + 2 s ) 4 − ( µ + 2 a ) 4 | , an d | s 4 − a 4 | , times n − 1 / 3 .) And w e notice immediate ly that b ot h Q 1 ( µ, a ) and D 1 ( t ) are o (1) if, in a ddition to | λ | = o ( n 1 / 12 ) , w e require that a = o ( n 1 / 12 ) as w e ll, a condition w e assume from no w on. Obvi ously O ( D 1 ( t )) absorbs the re mainder term O ( | t | n − 1 / 3 ) from (2.3.9) . F urt hermore, since n − 1 / 3 µ = ln  np q  = n − 1 / 3 λ − n − 2 / 3 λ 2 2 + n − 1  λ 3 3 + 1  + O ( n − 4 / 3 (1 + λ 4 )) , for the c ubic p olynomial of n − 1 / 3 µ in (2.3.10) w e ha v e n  3 2 − n − 1 / 3 µ 2 + n − 2 / 3 µ 2 4 − n − 1 µ 3 12  = n 3 2 − n 2 / 3 λ 2 + n 1 / 3 λ 2 2 − λ 3 2 − 1 2 + O ( n − 1 / 3 (1 + λ 4 )) . Observe tha t t he ﬁrst t hree summands are those in the exp onent of t he form ula (2.3.2) for N ( n, p ) times ( − 1) .) Th erefore , u sing ( 2.3.2) f or N ( n, p ) , (2.3.11) N ( n, p ) exp  h ( ρ, θ ) − 1 4 ρe iθ − 1 8 ρ 2 e i 2 θ  =  1 + O ( D 1 ( t ))  √ 2 π n · exp  − λ 3 6 + 3 4 + Q ( µ, a ) + µs 2 2 + s 3 3  ; Q ( µ, a ) := Q 1 ( µ, a ) + Q 2 ( a ) + O ( n − 1 / 3 (1 + λ 4 )) , and Q ( µ, a ) = o (1 ) as λ, a = o ( n 1 / 12 ) . In particu lar, u sing (2.3.8) , ( 2.3.10) for θ = 0 , i. e. s = a , w e see that (2.3.12) N ( n, p ) | I 2 ( w ) | ≤ b n 1 / 2 ( a 1 n − 1 / 3 ) − w e − a ln 2 n exp  − λ 3 6 + µa 2 2 + a 3 3  . F urt hermore, switching integration from θ to t = n 1 / 3 θ , the contribution of t he remainder term O ( D 1 ( t )) to N ( n , p ) I 1 ( w ) is O ( δ n,w ) , δ n,w := n − 1 / 12 ( a + ln n ) 3 / 4 e − λ 3 / 6 ( a 1 n − 1 / 3 ) w Z ∞ −∞     exp  µs 2 2 + s 3 3      D 1 ( t ) dt. (Explanation : n − 1 / 12 = n 1 / 2 n − 1 / 3 n − 1 / 4 , with n − 1 / 4 coming from n − 1 / 4 ( a + π ln n ) 3 / 4 , an u pp er b ound of | 1 − ρe iθ | 3 / 4 , for | θ | ≤ θ 0 .) No w     exp  µs 2 2 + s 3 3      = exp  µa 2 2 + a 3 3 −  µ 2 + a  t 2  , where, see ( 2.3.6), µ 2 + a = λ 2 + a + O ( n − 2 / 3 + n − 1 / 3 λ 2 ) > 0 , RANDOM EQUA TIONS 23 since lim inf a > 0 , a nd a ≥ | λ | if λ < 0 . Hence, see (2.3.10 ) for D 1 ( t ) , we hav e δ n,w ≤ b ∆ n,w , where (2.3.13) ∆ n,w := n − 1 / 12+ w/ 3 · a − w 1 exp  − λ 3 6 + µa 2 2 + a 3 3  · ( a + ln n ) 3 / 4  n − 1 / 3 ( | λ | + a + ln n ) 3 µ/ 2 + a + n − 2 / 3 ( | λ | + a + ln n ) 5 ( µ/ 2 + a ) 1 / 2  . The denominator s µ/ 2 + a , ( µ/ 2 + a ) 1 / 2 come from th e integrals Z ∞ −∞ | t | k exp h −  µ 2 + a  t 2 i dt = c k ( µ/ 2 + a ) − ( k +1) / 2 , ( k ≥ 0) , for k = 0 , 1 . C learly ∆ n,w absorbs the b ound (2 .3.12). Th us, switc hing from θ to s = a − in 1 / 3 θ , it remains t o ev aluate sharply (2.3.14) − i (2 π ) − 1 / 2 n 1 / 6 exp  − λ 3 6 + 3 4 + Q ( µ, a )  · s 2 Z s 1 exp  µs 2 2 + s 3 3  (1 − e − sn − 1 / 3 ) 3 / 4 − w ds ; here s 1 = a − in 1 / 3 θ 0 , s 2 = a + i n 1 / 3 θ 0 , and t he in tegral is ov er th e vertical line segment connect ing s 1 and s 2 . Lastly w e need t o e stimate an er ror coming from replacing (1 − e − sn − 1 / 3 ) 3 / 4 − w with a ge n uinely palatable ( sn − 1 / 3 ) 3 / 4 − w . Using | sn − 1 / 3 | ≥ | 1 − e − sn − 1 / 3 | ≥ | 1 − e − an − 1 / 3 | , ( s = a − it ) , | x u − 1 | ≤ u | x − 1 | , ( u ≥ 1 , | x | ≤ 1) , w e ha v e: for u ≥ 1     1 (1 − e − sn − 1 / 3 ) u − 1 ( sn − 1 / 3 ) u     ≤ 1 | 1 − e − an − 1 / 3 | u      1 − 1 − e − sn − 1 / 3 sn − 1 / 3 ! u      ≤ u | 1 − e − an − 1 / 3 | u      1 − 1 − e − sn − 1 / 3 sn − 1 / 3      ≤ b u | sn − 1 / 3 | | 1 − e − an − 1 / 3 | u ≤ u ( a + | t | ) n − 1 / 3 | 1 − e − an − 1 / 3 | u . Also, for s in question, | 1 − e − sn − 1 / 3 | ≥ 0 . 5 | sn − 1 / 3 | . So | (1 − e − sn − 1 / 3 ) 3 / 4 − ( sn − 1 / 3 ) 3 / 4 | = | sn − 1 / 3 | 3 / 4       1 − e − sn − 1 / 3 sn − 1 / 3 ! 3 / 4 − 1       ≤ b | sn − 1 / 3 | 3 / 4+1 ≤ b n − 7 / 12 ( a + | t | ) 7 / 4 . 24 BORIS PITTEL AND JI-A YEUM Com bining these t w o estimates, w e hav e: for w ∈ { 0 , 3 , 4 , . . . } , | (1 − e − sn − 1 / 3 ) 3 / 4 − w − ( sn − 1 / 3 ) 3 / 4 − w | ≤ b ( w + 1) n − 7 / 12 ( a + | t | ) 7 / 4 ( a 1 n − 1 / 3 ) w ; see (2.3.8) for a 1 . Conse quen tly , rep lacing (1 − e − sn − 1 / 3 ) 3 / 4 − w in (2.3.14) with ( sn − 1 / 3 ) 3 / 4 − w incurs a n additive error of or der ( w + 1) n − 1 / 12+ w/ 3 · a − w 1 exp  − λ 3 6 + µa 2 2 + a 3 3  · n − 1 / 3 a 5 / 4 , at most; t h us the e rror is easily O (( w + 1) ∆ n,w ) , see (2.3.13) for ∆ n,w . While these bo unds will suﬃce for λ = O (1) , the case λ → ∞ require s a s harp er approximation of (1 − e − sn − 1 / 3 ) 3 / 4 − w for w = O ( λ 3 ) . W e write (2.3.15) (1 − e − sn − 1 / 3 ) 3 / 4 − w = ( s n − 1 / 3 ) 3 / 4 − w exp " (3 / 4 − w ) ln 1 − e − sn − 1 / 3 sn − 1 / 3 # = ( s n − 1 / 3 ) 3 / 4 − w exp  Q 3 ( w , a ) + O ( D 3 ( w , t ))  ; Q 3 ( w , a ) := (3 / 4 − w ) ln 1 − e − an − 1 / 3 an − 1 / 3 ; D 3 ( w , t ) := ( w + 1) tn − 1 / 3 . Notice that Q 3 ( a, w ) → 0 as w a = O ( λ 3 n 1 / 12 ) = o ( n 1 / 3 ) , and D 3 ( t, w ) → 0 a s w ln n = o ( n 1 / 3 ) . The expression ( 2.3.14) therefore b ecomes (2.3.16) − i (2 π ) − 1 / 2 n − 1 / 12+ w/ 3 exp  − λ 3 6 + 3 4 + Q ( µ, w , a )  · s 2 Z s 1 exp  µs 2 2 + s 3 3  s 3 / 4 − w ds + O (( w + 1) ˜ ∆ n,w ); Q ( µ, w , a ) := Q ( µ, a ) + Q 3 ( w , a ); ˜ ∆ n,w := n − 5 / 12+ w/ 3 · a − w exp  − λ 3 6 + µa 2 2 + a 3 3  ( a + ln n ) 3 / 4 . Finally , aft er this r eplacement we can ext end the integration to ( a − i ∞ , a + i ∞ ) , since t he att endant add itiv e e rror is easily shown to b e ab sorb ed by ( w + 1 )∆ n,w for all w , and by ( w + 1) ˜ ∆ n,w if w = O ( λ 3 ) . Lemma 2 .3.1. Supp ose that λ = o ( n 1 / 12 ) . L et a ≥ | λ | b e such that li m a > 0 , a = o ( n 1 / 12 ) . Then, denoting µ = n 1 / 3 ln( np/q ) , (2.3.17) N ( n, p ) [ x n ] e H ( x ) (1 − T ( x )) w = − i (2 π ) − 1 / 2 e 3 / 8+ o (1) n − 1 / 12+ w/ 3 e − µ 3 / 6 a + i ∞ Z a − i ∞ s 3 / 4 − w exp  µs 2 2 + s 3 3  ds + O (( w + 1) R n,w ) , RANDOM EQUA TIONS 25 with R n,w ≤ ∆ n,w for al l w , and R n,w = ∆ n,w ∧ ˜ ∆ n,w if w a and w ln n ar e b oth o ( n − 1 / 3 ) . F ur thermor e, shifting the inte gr a tio n line to { s = b + it : t ∈ ( −∞ , ∞ ) } do es not change the value of the inte gr al as long as b ∧ ( µ/ 2 + b ) r em ains p ositive. Pro of of Lemma 2.3.1. W e o nly ha v e to explain preserv a tion o f t he in t egral, and wh y e − λ 3 / 6 can b e replaced with e − µ 3 / 6 . Given such a b , pick T > 0 a nd in tro d uce tw o ho rizontal line se gments, C 1 , 2 = { s = α ± iT : α ∈ [ a, b ] } , the top segment and the b o ttom seg men t b eing resp ec tiv ely right and left oriented. On C 1 ∪ C 2 , Re  µs 2 2 + s 3 3  = µα 2 2 + α 3 3 − T 2  µ 2 + α  , and µ 2 + α ≥ µ 2 + ( a ∧ b ) > 0 . Therefor e lim T →∞ Z C 1 ∪ C 2 s 3 / 4 − w exp  µs 2 2 + s 3 3  ds = 0 . As for e − λ 3 / 6 ∼ e − µ 3 / 6 , this fo llo ws fr om (2.3.18) | λ 3 − µ 3 | ≤ b λ 2 ( n − 2 / 3 + n − 1 / 3 λ 2 ) = ( n − 1 / 3 λ ) 2 + n − 1 / 3 λ 4 → 0 .  In t he co n text of the crit ical ra ndom gra ph G ( n, m ) , the in tegra l app ea ring in (2.3.17) w as e ncountered and studied in [12]. F ollo wing [12], intro duce (2.3.19) A ( y , µ ) = e − µ 3 / 6 2 π i a + i ∞ Z a − i ∞ s 1 − y exp  µs 2 2 + s 3 3  ds. W e know that this in teg ral is well d eﬁned, and do es not dep e nd on a , if a > 0 and a > − µ/ 2 . It w as sho wn in [12] tha t (1 ) (2.3.20) A ( y , µ ) = e − µ 3 / 6 3 ( y +1) / 3 ∞ X k =0 (3 2 / 3 µ/ 2) k k !Γ(( y + 1 − 2 k ) / 3 ) , (2) A ( y , µ ) ≥ 0 for y > 0 , A ( y , µ ) > 0 fo r y ≥ 2 , and ( 3) (2.3.21) A ( y , µ ) ∼      (2 π ) − 1 / 2 | µ | 1 / 2 − y , µ → −∞ , e − µ 3 / 6 2 y/ 2 Γ( y / 2) µ 1 − y/ 2 , µ → ∞ . W e will also n eed tw o b ounds (2.3.22) A ( y , µ ) ≤ b e 2 | µ | 3 / 3 (2 / 3) y +1 3 Γ  y +1 3  , A ( y , µ ) ≤ b ( a + µ/ 2) − 1 / 2 a 1 − y exp  − µ 3 6 + µa 2 2 + a 3 3  , 26 BORIS PITTEL AND JI-A YEUM (the secon d b ound holding for y ≥ 1 and a + µ/ 2 > 0 ), a nd an asymptotic formula: if µ → ∞ , y → ∞ , and y = O ( µ 3 ) , t hen (2.3.23) A ( y , µ ) ∼ ( 2 π ) − 1 / 2 ( y ξ − 2 + µ + 2 ξ ) − 1 / 2 ξ 1 − y exp  − µ 3 6 + µξ 2 2 + ξ 3 3  , where ξ = ξ ( y , µ ) is a unique p ositive ro ot of µξ 2 + ξ 3 = y . Also, if y = O ( λ 3 ) , t hen (2.3.24) A ( y , µ ) ≤ b µ − 1 / 2 ξ 1 − y exp  − µ 3 6 + µξ 2 2 + ξ 3 3  , (See App endix fo r a pro of of (2.3.22) and ( 2.3.23)-(2.3.24 ).) With A ( y , µ ) , w e write (2.3.17) mor e comp actly: (2.3.25) N ( n, p ) [ x n ] e H ( x ) (1 − T ( x )) w = (2 π ) 1 / 2 e 3 / 8+ o (1) A (1 / 4 + w , µ ) n − 1 / 12+ w/ 3 + O (( w + 1) ∆ n,w ) . Let us use (2.3 .25) for asymptot ic ev alua tion o f Pr( S n > 0 , E n ≤ L ) giv en by (2.2.10)-(2 .2.11). Case | λ | = O (1) . According to (2 .1.16), we can p ic k L → ∞ as slo wly as we wish. W e pic k L = ln 1 / 4 n . As a ﬁrst step , let us estimate the ov e rall con tribut ions, R (1) n and R (2) n , of the remainder s O (( w + 1) R n,w ) to t he b o unds Σ 1 and Σ 2 in Prop osition 2 .2.3. In t his case w e cho ose a = ( L ) 1 / 3 for eac h w , and R n,w = ∆ n,w . Co nsider R (2) n ﬁrst. B y (2.2.19) and ( 2.3.13), and dro pping (3 r + 1)( np/ 2 q ) r = (3 r + 1)(1 / 2 + o (1 )) r factor , (2.3.26) R (2) n ≤ b n − 1 / 12 ·  n − 1 / 3 ln 15 / 4 n  · exp  − λ 3 6 + µa 2 2 + a 3 3  · ∞ X r =0 f L r a − 3 r 1 . No w a 1 ∼ a , so b y (2.2.23) an d (2.2.18), ∞ X r =0 f L r a − 3 r 1 ≤ b X r ≤ L  3 r 2 ea 3 1  r + X r >L  3 L 2 ea 3 1  r ≤ X r ≥ 0  2 e  r < ∞ . So (2.3.26) be comes R (2) n ≤ b n − 1 / 12 · n − 1 / 3 e ln 3 / 4 n = n − 5 / 12+ o (1) . RANDOM EQUA TIONS 27 F urt her, by ( 2.2.22) a nd g L 0 = 0 , Σ 2 − Σ 1 = N ( n, p ) X r > 0  p 2 q  r [ x n ] g L r e H ( x ) (1 − T ( x )) 3 r − 1 . Therefor e | R (1) n | ≤ b R (2) n + n − 1 / 12 ·  n − 1 / 3 ln 15 / 4 n  · exp  − λ 3 6 + µa 2 2 + a 3 3  · ∞ X r =1 g L r a − 3 r +1 1 . So, u sing the b ounds (2 .2.19) and (2 .2.23) for g L r , we conclude t hat | R (1) n | ≤ 2 R (2) n . Th us, for L = ln 1 / 4 n , (2.3.27) Σ ∗ 1 + O ( n − 1 / 3+ o (1) ) ≤ Pr( S n > 0 , E n ≤ L ) (2 π ) 1 / 2 e 3 / 8 n − 1 / 12 ≤ Σ ∗ 2 + O ( n − 1 / 3+ o (1) ) . where (2.3.28) Σ ∗ 2 = X r ≥ 0  np 2 q  r f L r A (1 / 4 + 3 r, µ ) , Σ ∗ 1 =Σ ∗ 2 − n − 1 / 3 X r > 0  np 2 q  r g L r A ( − 3 / 4 + 3 r, µ ) . Let u s hav e a close lo ok at Σ ∗ 1 and Σ ∗ 2 . W rite Σ ∗ 2 = X r ≤ L  np 2 q  r f r A (1 / 4 + 3 r , µ ) + X r >L  np 2 q  r f L r A (1 / 4 + 3 r , µ ) = Σ ∗ 21 + Σ ∗ 22 . By f L r ≤ f r , (2.2.18), ( 2.3.22), a nd Stirling’s fo rm ula for Γ( r ) = ( r − 1) ! , Σ ∗ 22 ≤ b e 2 | µ | 3 / 3 X r >L  1 2 + O  | λ | n − 1 / 3   r  3 2  r r r − 1 / 2 e − r  2 3  r Γ − 1 ( r ) ≤ e 2 | µ | 3 / 3 X r >L  2 3  r ≤ b  2 3  L . F urt her, since unifor mly for r ≤ L , (1 + λn − 1 / 3 ) r = exp  O ( L | λ | n − 1 / 3 )  = 1 + O ( n − 1 / 3 ln 1 / 4 n ) , w e ha v e Σ ∗ 21 = (1 + o (1 )) X r ≤ L f r 2 r A (1 / 4 + 3 r , µ ) . And, analogously t o Σ ∗ 22 , X r >L f r 2 r A (1 / 4 + 3 r , µ ) ≤ b  2 3  L . Therefor e Σ ∗ 2 ∼ X r ≤ L f r 2 r A (1 / 4 + 3 r , µ ) → X r ≤ L f r 2 r A (1 / 4 + 3 r , µ ) . Also, by the deﬁnit ion of Σ ∗ 1 in (2 .3.25), it follo ws that | Σ ∗ 1 − Σ ∗ 2 | is O ( n − 1 / 3 ) . Hence 28 BORIS PITTEL AND JI-A YEUM Prop osition 2 .3 .2. F or | λ | = O (1) , Pr ( S n > 0 , E n ≤ L ) (2 π ) 1 / 2 e 3 / 8 n − 1 / 12 ∼ c ( µ ) := X r ≥ 0 f r 2 r A (1 / 4 + 3 r , µ ) , and µ (= λ + O ( n − 1 / 3 )) c a n b e r eplac e d with λ , as c ( x ) is p o sitive and c ontinuous for al l x . Case λ → ∞ , λ = o ( n 1 / 12 ) . Acc ording t o ( 2.1.16), we select L = αλ 3 , α > 2 / 3 . Th is t ime we use a re ﬁned v ersion of (2 .3.24), with the e xp onential fact or sneaking b ehind the sum op erat ion for r ≤ αλ 3 , which allows us t o c ho ose a ( ≤ 2 λ ) dep e ndent o n r for r ≤ αλ 3 . Also, for t hose r an d a , r a = O ( λ 4 ) = o ( n 1 / 3 ) and r ln n = O ( λ 3 ln n ) = o ( n 1 / 3 ) ; so R n, 3 r = ˜ ∆ n, 3 r in th is range. F o r r > αλ 3 w e select a = λ , a nd her e R n, 3 r = ∆ n, 3 r . (So , a = o ( n 1 / 12 ) through out.) B y (2 .2.20), exp  − λ 3 6 + λa 2 2 + a 3 3  X r >αλ 3 ( r + 1) (1 / 2 + o (1)) r f L r ≤ b exp  2 λ 3 3  X r >αλ 3 ( r + 1 )  3 αλ 3 (1 + o ( 1)) 4 eλ 3  r ≤ b λ 3 exp  2 λ 3 3 + λ 3 α ln 3 α (1 + o (1)) 4 e  ≤ b λ 3 exp  λ 3 α ln 3 α (1 + o ( 1)) 4  , and, pushing α down to 2 / 3 , w e can make the co eﬃc ien t of λ 3 in t he exp one n t arbitrar ily close to 2 3 · ln 1 2 = − 0 . 46 . . . . According to (2.2.18) an d (2.3 .13), it rema ins to b ound X r ≤ αλ 3 r + 1 ( r + 1) 1 / 2 min a ≤ 2 λ  exp  − µ 3 6 + µa 2 2 + a 3 3   3 r 4 ea 3  r  ; (w e ha v e replace d a 1 = n 1 / 3 (1 − e − an − 1 / 3 ) with a , since for r ≤ αλ 3 , a 3 r 1 = a 3 r e O ( λ 4 n − 1 / 3 ) ∼ a 3 r , and λ 3 / 6 with µ 3 / 6 , se e (2.3.18). So w e need to ﬁnd min a ≤ 2 λ Φ( r , a ) , (2.3.29) Φ( r , a ) := − µ 3 6 − r ln  3 r 4 ea 3  + µa 2 2 + a 3 3 . Φ( r , a ) att ains its absolute minim u m at ξ ( r ) , a unique p o sitiv e r o ot of (2.3.30) µξ + ξ 2 = 3 r ξ ≤ 3 αλ 3 ξ , RANDOM EQUA TIONS 29 i. e. ξ ( r ) < 2 λ if α is suﬃciently close to 2 / 3 from ab o v e. F u rther φ ( r ) := Φ( r , ξ ( r ) ) atta ins its maxim um at ¯ r , a r o ot of φ ′ ( y ) = ln 3 y 4 − 3 ln ξ ( y ) = 0 , i. e. (2.3.31) ¯ r = 4 81 µ 3 , ¯ a := ξ ( ¯ r ) = µ 3 ( < 2 λ ) . Consequently (2.3.32) φ ( ¯ r ) = e xp  − 10 µ 3 81  ∼ exp  − 10 λ 3 81  . It is e asy t o show that (2.3.33) φ ′′ ( ¯ r ) = − 27 µ 3 ∼ − 27 λ 3 , and, with some work, th at φ ′′ ( r ) < 0 a lw a ys. A standar d ap plication of Laplac e metho d yields X r ≤ αλ 3 ( r + 1) 1 / 2 min a ≤ λ  exp  − µ 3 6 + λa 2 2 + a 3 3   3 r 4 ea 3  r  ≤ b λ 3 exp  − 10 λ 3 81  . Therefor e, consu lting ( 2.3.16) fo r ˜ ∆ n,w and (2.3.13) for ∆ n,w , w e b o und R (2) n , the total con tribution of th e remainders ( w + 1 ) R n,w to t he sum Σ 2 in ( 2.2.20): R (2) n = X r ( r + 1) R n, 3 r ≤ b n − 1 / 12 exp  − 10 λ 3 81   n − 1 / 3 λ 3 . 75 + n − 1 / 3 ln 3 / 4 n  + n − 1 / 12 λ 3 e − 0 . 27 λ 3  n − 1 / 3 λ 4 + n − 1 / 3 ln 4 n  ≤ b  λ − 1 / 4 + n − 1 / 3+ o (1)  n − 1 / 12 exp  − 10 λ 3 81  . (2.3.34) As for R (1) n , th e total c on tribut ion of the remainders ( w + 1) R n,w to Σ 1 in (2.2.20), it is O ( R (2) n ) , just like t he λ = O (1) case. S o w e arrive at t he co un terp art of (2.3.27)-(2 .3.28), with  λ − 1 / 4 + n − 1 / 3+ o (1)  exp  − 10 λ 3 81  taking p lace of n − 1 / 3+ o (1) . F u rther, ag ain w e split Σ ∗ 2 = Σ ∗ 21 + Σ ∗ 22 . T o b ou nd Σ ∗ 22 w e use the sec ond b ound for A (1 / 4 + 3 r , µ ) with a ≡ λ (2.3.21) , and t he b ound (2.2.20) for f L r . Ju st lik e R (2) n , w e obtain Σ ∗ 22 ≤ b λ 1 / 2 exp  − λ 3 6 + λa 2 2 + a 3 3  · X r >αλ 3 (1 / 2 + o (1 )) r f L r ≤ b λ 1 / 2 e − 0 . 46 λ 3 . (2.3.35) 30 BORIS PITTEL AND JI-A YEUM T o ev alua te sh arply Σ ∗ 21 , w e use (2.3.23 ) to a pprox imate A ( y , µ ) for εµ ≤ y , y = O ( λ 3 ) , and (2 .3.24) to b ound A ( y , µ ) for y ≤ εµ , ε > 0 suﬃcien tly small. In v oking (2.2.18) as w ell, w e hav e X εµ ≤ r ≤ αλ 3  np 2 q  r f r A (1 / 4 + 3 r, µ ) ∼ 1 4 π X εµ ≤ r ≤ αλ 3 ξ 3 / 4 e φ ( r )  r ((3 r + 1 ) ξ − 2 + µ + 2 ξ )  1 / 2 ; here φ ( r ) := min a Φ( r , a ) = Φ( r, ξ ) , see (2.3.28 )-(2.3.29) for Φ( r , a ) and ξ = ξ ( r ) . W e know tha t φ ( r ) attains its pronounc ed maxim um at ¯ r = (4 / 27) λ 3 , i. e. w e ll within [ εµ, αλ 3 ] . Using (2.3.31) -(2.3.33), by Laplac e met ho d, X εµ ≤ r ≤ αλ 3  np 2 q  r f r A (1 / 4 + 3 r , µ ) ∼ 1 4 π ¯ ξ 3 / 4  ¯ r ( (3 ¯ r + 1) ¯ ξ − 2 + µ + 2 ¯ ξ )  1 / 2  2 π − φ ′′ ( ¯ r )  1 / 2 ∼ 1 4(2 π ) 1 / 2 3 3 / 4 λ 1 / 4 exp  − 10 λ 3 81  . Applying (2.3.24), it is not diﬃcult to show th at X r ≤ ε µ  np 2 q  r f r A (1 / 4 + 3 r , µ ) ≪ λ 1 / 4 exp  − 10 λ 3 81  . So Σ ∗ 12 := X r ≤ αλ 3  np 2 q  r f r A (1 / 4 + 3 r, µ ) ∼ 1 4(2 π ) 1 / 2 3 3 / 4 λ 1 / 4 exp  − 10 λ 3 81  , hence (see ( 2.3.35)) (2.3.36) Σ ∗ 2 ∼ 1 4(2 π ) 1 / 2 3 3 / 4 λ 1 / 4 exp  − 10 λ 3 81  , as w e ll. And , analogously t o the λ = O (1) c ase, for Σ ∗ 1 deﬁned in (2.3.28), (2.3.37)   Σ ∗ 1 − Σ ∗ 2 | ≪ λ 1 / 4 exp  − 10 λ 3 81  . Prop osition 2 .3 .3. F or λ → ∞ , λ = o ( n 1 / 12 ) , Pr ( S n > 0 , E n ≤ L ) ∼ e 3 / 8 4 · 3 3 / 4 λ 1 / 4 exp  − 10 λ 3 81  . Pro of of Prop o sitio n 2.3.3 The pro babilit y is asympt otic to the expression in (2.3.36) time s (2 π ) 1 / 2 e 3 / 8 n − 1 / 12 .  Lastly , RANDOM EQUA TIONS 31 Case λ → −∞ , | λ | = o ( n 1 / 12 ) . According to (2 .1.16), we c an pic k L = 0 . By Prop osition 2.2.4 and (2.3.17 ) for w = 0 , an d a ≥ | λ | , a = o ( n 1 / 12 ) , we ha v e Pr( S n > 0 , E n ≤ 0) = N ( n , p ) [ x n ] e H ( x ) = (2 π ) 1 / 2 n − 1 / 12 e 3 / 8 A (1 / 4 , µ ) + O ( ∆ n, 0 ) . Notice that  µa 2 2 + α 3 3      a = | λ | = λ 3 6 + o (1) , since λ 3 − µ 3 = o (1) . Sett ing a = λ in (2.3.13 ), we obtain ∆ n, 0 ≪ n − 1 / 12 · n − 1 / 3 · n − 1 / 3  | λ | 3 . 75 + ln 3 . 75 n  . And, b y (2.3.21), A (1 / 4 , µ ) ∼ (2 π ) − 1 / 2 | µ | 1 / 2 − 1 / 4 ∼ (2 π ) − 1 / 2 | λ | 1 / 4 . Prop osition 2 .3 .4. Supp ose λ → −∞ , | λ | = o ( n 1 / 12 ) . Then Pr ( S n > 0 , E n ≤ 0) ∼ e 3 / 8 | λ | 1 / 4 . Since in e ac h o f t he t hree cases our L is such that lim Pr ( E n ≤ L ) = 1 , Prop osition s 2.3.2 -2.3.4 c om bined wit h t he relat ions (2.1.1 2) and (2.1.13) , prov e the part of Theore m 1.1 ab out G ( n, p ) , ˆ p = 1 / 2 . 3. Solv abilit y probabilit y: G ( n, m ) and ˆ p = 1 / 2 . Our ta sk is t o sh o w t hat the result for the near-cr itical G ( n, p ) , p = (1 + λn − 1 / 3 ) /n , λ = o ( n 1 / 12 ) , implies t he an alogous c laim for G ( n , m ) , m = ( n/ 2) (1 + λn − 1 / 3 ) . Denoting N =  n 2  , (3.1) p = m N + O ( m 1 / 2 N − 1 ) = 1 + n − 1 / 3 λ ′ n , λ ′ = λ + O ( n − 1 / 6 )) . Ob viously λ ′ = o ( n 1 / 12 ) , so Pr( S ( G ( n, p ) > 0) → 0 . Since an even t { S ( G ) > 0 } is mono tone (increasing) with G , a ge neral “ p -to- m ” result, Bollob´ as [5], Lucz ak [15], implies that Pr( S ( G ( n, m ) > 0) → 0 , 32 BORIS PITTEL AND JI-A YEUM to o. How ever w e wan t t o pro v e a shar p f orm ula (3.2) Pr( S ( G ( n, m ) > 0) ∼ c ( λ ) n − 1 / 12 , so that t he pr obabilities in ques tion ca n b e as small as exp  − 10 81 ( n 1 / 12 − o (1) ) 3  = exp  − 10 81 n 1 / 4 − o (1)  . It turns out that in our case t he arg umen t in [5], [15 ] can b e sharp e ned to yield (3.2). T o start, r ecall the classic entrop y b ound Pr(Bin( N , p ) ≥ k ) ≤ exp[ N H ( k / N )] , k > N p, Pr(Bin( N , p ) ≤ k ) ≤ exp[ N H ( k / N )] , k < N p, where H ( x ) : x ln( p/x ) + (1 − x ) ln( q / (1 − x )) . Appro ximating H ( x ) b y its second de gree T ayl or p o lynomial p lus a re mainder term, w e o btain: u niformly for p ≤ 1 / 2 , and ω ≤ a ( N p ) 1 / 6 , a > 0 b eing ﬁxed, (3.3) Pr  Bin( N , p ) ≥ N p + ω p N pq  ≤ b e − ω 2 / 2 , Pr  Bin( N , p ) ≤ N p − ω p N pq  ≤ b e − ω 2 / 2 , (The b o unded factor implicit in ≤ b notat ion de p ends on a .) Given m and ω ≤ m 1 / 6 , in t ro duc e p 1 < p 2 : (3.4) N p 1 + ω p N p 1 = m = ⇒ p 1 = (4 N ) − 1  p 4 m + ω 2 − ω ) 2 , N p 2 − ω p N p 2 = m = ⇒ p 2 = (4 N ) − 1  p 4 m + ω 2 + ω ) 2 . Then (3.5) N p 2 ω 6 > N p 1 ω 6 = m ω 6  p 1 + ω 2 / 4 m − ω / (2 √ m  2 ≥ a := ( √ 2 − 1) 2 , as ω / 2 √ m ≤ 0 . 5 m − 1 / 3 ≤ 1 . No w, using e ( G ) to den ote th e n umber of e dges in a graph G , e ( G ( n, p )) = Bin( N , p ) . So , by ( 3.3)-(3.5), (3.6) Pr( e ( G ( n, p 1 )) > m ) ≤ Pr( e ( G ( n, p 1 )) ≥ N p 1 + ω p N p 1 q 1 ) ≤ b e − ω 2 / 2 , Pr( e ( G ( n, p 2 )) < m ) ≤ Pr( e ( G ( n, p 2 )) ≤ N p 2 − ω p N p 2 q 2 ) ≤ b e − ω 2 / 2 . Since Pr( S ( G ( n, p )) > 0) = N X µ =0 Pr( e ( G ( n, p )) = µ ) Pr( S ( G ( n, µ )) > 0) , RANDOM EQUA TIONS 33 and Pr( S ( G ( n , µ )) > 0 ) decr eases with µ , w e ha v e Pr( S ( G ( n, p 1 )) > 0) ≥ Pr( e ( G ( n , p 1 )) ≤ m ) Pr( S ( G ( n, m )) > 0) ≥ (1 − O ( e − ω 2 / 2 )) Pr( S ( G ( n, m )) > 0) , and Pr( S ( G ( n, p 2 )) = 0) ≥ Pr( e ( G ( n, p 2 )) ≥ m ) Pr( S ( G ( n, m )) = 0)) =(1 − O ( e − ω 2 / 2 )) Pr( S ( G ( n, m )) = 0) . Therefor e (3.7) Pr( S ( G ( n, p 1 )) > 0) 1 − O ( e − ω 2 ) ≥ Pr( S ( G ( n, m ) ) > 0) ≥ Pr( S ( G ( n, p 2 )) > 0) − O ( e − ω 2 / 2 ) 1 − O ( e − ω 2 / 2 ) . No w, b y (3.4), p 1 , 2 = m N  1 + O ( ω m − 1 / 2 )  = 1 + λ 1 , 2 n − 1 / 3 n , λ 1 , 2 = λ + O ( ω m − 1 / 2 + n − 2 / 3 ) , so, a s | λ | = o ( n 1 / 12 ) , λ 3 1 , 2 = λ 3 + O  λ 2 ( ω m − 1 / 2 + n − 2 / 3 )  + O  ( ω m − 1 / 2 ) 3 + n − 2  = λ 3 + o ( ω n − 1 / 3 ) . That is, λ 3 1 , 2 − λ 3 → 0 . Henc e, (3.8) Pr( S ( G ( n, p 1 , 2 )) > 0) ∼ c ( λ ) n − 1 / 12 . Also ω 2 ≫ | λ | 3 if ω = n 1 / 8 , which is c ompatible with the re striction ω ≤ n 1 / 6 . F or this c hoice of ω , the relations ( 3.7)-(3.8) imply: for λ = o ( n 1 / 12 ) , Pr( S ( G ( n, m )) > 0 ) ∼ c ( λ ) n − 1 / 12 . This completes t he pr o of of The orem 1.1 for ˆ p = 1 / 2 .  4. Solv abilit y ( 2 -c olorabilit y) probabilit y: G ( n, p ) , G ( n , m ) and ˆ p = 1 . Consider the G ( n, p ) case. W e kno w that the system x i + x j ≡ 1 (mo d 2) , ( i, j ) ∈ E ( G ) is solv able iﬀ t he gr aph G has no o dd cyc les. S o a counterpart of (2.1.11) is (4.1) Pr( S n > 0 , E n ≤ L ) = N ( n, p ) [ x n ] e xp " L X ℓ = − 1  p q  ℓ C e ℓ ( x ) # , 34 BORIS PITTEL AND JI-A YEUM ( S n = S ( G ( n, p )) , E n = E ( G ( n, p )) ), wh ere C e ℓ ( x ) is t he exp onential generating function of graphs G without o dd cycles , with an excess E ( G ) = ℓ . And again the ev ents { S n > 0 } and {E n ≤ L } are p ositiv ely correlat ed, i. e . Pr( S n > 0 , E n ≤ L ) ≤ Pr( S n > 0) ≤ Pr( S n > 0 , E n ≤ L ) Pr( E n ≤ L ) . Th us the generating fun ctions C e ℓ ( x ) take a center stage . Ob viously C e − 1 ( x ) = C − 1 ( x )  = T ( x ) − 1 2 T 2 ( x )  . F urt hermore, while C 0 ( x ) = 1 2  ln 1 1 − T ( x ) − T ( x ) − 1 2 T 2 ( x )  , for C e 0 ( x ) we hav e (4.2) C e 0 ( x ) = 1 4  ln 1 1 − T 2 ( x ) − T 2 ( x )  . Indeed, we e n umerat e the connec ted un icyclic grap hs with an ev en cycle, i. e. forests of an even num b er of ro o ted tre es, who se r o ots form an undirect ed cycle. So C e 0 ( x ) = X ev en j ≥ 4 ( j − 1) ! 2 T j ( x ) j ! , whic h simpliﬁes to (4.2) . Comparing C e 0 ( c ) a nd C 0 ( x ) we see that, for | x | < e − 1 , x → 1 , i. e . fo r x dominant asympt otically , (4.3) C e 0 ( x ) = 1 2 C 0 ( x ) + 1 8 − 1 4 ln 2 + O ( | T ( x ) − 1 | ); in p articular, C e 0 ( x ) ∼ ( 1 / 2) C 0 ( x ) . W e wan t to show th at t his pa ttern p e rsists for ℓ > 0 , namely (4.4) C e ℓ ( x ) ∼ 1 2 ℓ +1 C ℓ ( x ) , ( | x | < e − 1 , x → e − 1 ) . Comparing (2 .1.11) and (4.1), a nd recalling th e diﬀeren t roles p la y ed by C 0 ( x ) an d { C ℓ ( x ) } ℓ> 0 in the ana lysis of the ˆ p = 1 / 2 makes it t ranspar en t, h op efully , th at fo r ˆ p = 1 w e should ha v e Pr( G ( n, p ) is 2-colo rable) = Pr( S n > 0) ∼ 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 . Let u s pr o v e (4.4). First RANDOM EQUA TIONS 35 Prop osition 4.1. Given n and m ≤ N :=  n 2  , let C ( n , m ) denote the total numb er of c onne cte d gr aphs on [ n ] with m e d ges, and let C e ( n, m ) denote the total numb er of c onne cte d gr aphs without o dd cycles. Then (4.5) C e ( n, m ) ≤ 1 2 m +1 − n C ( n, m ) . Conse quently (4.6) C e ℓ ( x ) ≤ c 1 2 ℓ +1 C ℓ ( x ) , ℓ ≥ − 1 . Pro of of Prop o sition 4.1. W e b egin with a simple claim. Lemma 4.2. L et T b e a tr e e o n the vertex set [ n ] . L et X ( T ) denote the total numb er of p a ths in T of an even e dge-length 2 at le ast. Then X ( T ) ≥ X ( P n ) , wher e P n is a p ath on [ n ] , and (4.7) X ( P n ) =  n ( n − 2) 4  . Pro of of Lemma 4.2. Pic k a v ert ex v ∈ [ n ] , and in t ro duc e V 0 ( T ) and V 1 ( T ) the se t of vertices reachable from v by pat hs of e v en length 2 at least, an d o dd length re sp ect iv ely; in p articular v ∈ V 0 . No w ev ery tw o v ert ices from V i ( T ) , ( i = 0 , 1) , are connect ed b y a n e v en p ath, while th ere is no e v en path co nnecting v 0 ∈ V 0 ( T ) and v 1 ∈ V 1 ( T ) . Hence X ( T ) =  | V 0 ( T ) | 2  +  | V 1 ( T ) | 2  . It follo ws t hat X ( T ) at tains its minim um wh en | V 0 ( T ) | = ⌊ n/ 2 ⌋ and | V 1 ( T ) | = ⌈ n/ 2 ⌉ , or th e oth er wa y aro und, i. e . when T = P n , and the minim um v alue is X ( P n ) =  ⌊ n/ 2 ⌋ 2  +  ⌈ n/ 2 ⌉ 2  =  n ( n − 2 ) 4  .  Armed with this Lemma , w e will d eriv e a recur rence ine quality fo r C e ( n, m ) . First we re call a recurre nce equality fo r C ( n, m ) , [23], [3]: fo r n ≥ 3 , n − 1 ≤ m ≤ N , (4.8) mC ( n, m ) = ( N − m + 1 ) C ( n, m − 1) + 1 2 X n 1 + n 2 = n, m 1 + m 2 = m − 1  n n 1  n 1 n 2 C ( n 1 , m 1 ) C ( n 2 , m 2 ) . Explanation. The left hand side of (4.8) is th e total n um b er of the conn ected ( n, m ) graphs with a mark ed edge. Each one of th ese graphs with a marked edge can b e obtained in one of tw o, mutually exclusiv e wa ys. F irst w a y is insert ing a mark ed edge in t o a con necte d gr aph o n [ n ] with m − 1 ed ges, which acc ounts fo r the ﬁrst 36 BORIS PITTEL AND JI-A YEUM term on the right hand side of ( 4.6); indeed N − m + 1 is t he tot al num b er of unorder ed pairs of v ertice s not connec ted b y an edge in a given connec ted graph with m − 1 edg es. Se cond w a y is to start with a connect ed ( n 1 , m 1 ) g raph and a connec ted ( n 2 , m 2 ) g raph, having m − 1 e dges in t otal, and to add a mark ed edg e that joins t w o conne cted gra phs; n 1 n 2 is the t otal num b er of w a ys to selec t tw o “contact” p oin ts, representing each of t w o graphs. Let us see if t here is a similar recursiv e formula for C e ( n, m ) . Clearly , if a mar k ed edge joins t w o co nnecte d grap hs, none of these t wo graphs may hav e an o d d cycle. So we deﬁn itely hav e th e “ C e ( · , · ) ” counterpart of the se cond te rm on the righ t hand side o f (4.8 ). As for a p ote n tial c ounterpart of t he ﬁ rst t erm, a diﬃcu lt y is that an ad ditional m -th edge is not allo w ed to form an o dd c ycle wit h any of the m − 1 edge s already present. And so the total num b er of admissible op tions dep e nds on the struct ure of a ( n, m − 1) grap h G in quest ion. (F o r suc h a gr aph to b e conn ected , it is nece ssary that m ≥ n .) Ho wev er w e can b oun d the num b e r of options. G is sp anned by a tr ee T on [ n ] , and none o f the m − 1 − ( n − 1) = m − n edges of G \ T co mpletes an o dd cycle b y joining t he ends of an ev en path in T . B y Lemma 4.2, the tot al n u m b er of those even pa ths is ⌈ n ( n − 2) / 4 ⌉ , at le ast. Hence the t otal num b e r of op tions for the m -edge is N − ( m − 1) − ⌈ n ( n − 2) / 4 ⌉ , at mo st. And it is str aigh tforward t hat, f or n ≥ 3 and b y m ≥ n , N − ( m − 1) −  n ( n − 2 ) 4  ≤ 1 2 ( N − ( m − 1)) . So C e ( · , · ) satisﬁes a re cursiv e ine quality : for n ≥ 3 , n − 1 ≤ m ≤ N , (4.9) mC e ( n, m ) ≤ 1 2 ( N − m + 1) C e ( n, m − 1) + 1 2 X n 1 + n 2 = n, m 1 + m 2 = m − 1  n n 1  n 1 n 2 C e ( n 1 , m 1 ) C e ( n 2 , m 2 ) . ( C e ( ν, µ ) := 0 if ν = 0 , or µ / ∈ [ ν − 1 ,  ν 2  ] .) W e will use (4.9) a nd induct ion to prov e th e b ou nd ( 4.5). T o t his end, we deﬁne a lexicogra phical order, ≺ , on { ( n, m ) : n ≥ 1 , n − 1 ≤ m ≤  n 2  } as follo ws: de noting ℓ = m − n , ( n 1 , m 1 ) ≺ ( n 2 , m 2 ) ⇐ ⇒ ℓ 1 < ℓ 2 , or ℓ 1 = ℓ 2 and n 1 < n 2 . The o rder ≺ is total, and (1 , 0) is the minimal element. The ind uctiv e basis ho lds, since C e (1 , 0 ) = C (1 , 0) = 1 , and C e (2 , 1 ) = C (2 , 1) = 1 . Supp ose tha t, for so me n ≥ 2 and m ∈ [ n − 1 ,  n 2  ] , C e ( ν, µ ) ≤ 1 2 µ − ν +1 C ( ν, µ ) , ∀ ( ν , µ ) ≺ ( n, m ) . Since ( n , m − 1) ≺ ( n, m ) , the inductiv e assumption implies t hat 1 2 ( N − m + 1) C e ( n, m − 1) ≤ 1 2 ( N − m + 1) 1 2 m − 1 − n +1 C ( n, m − 1 ) = 1 2 m − n +1 ( N − m + 1) C ( n, m ) . (4.10) RANDOM EQUA TIONS 37 F urt her, for the double sum in (4.9) , m 1 − n 1 + 1 ≥ 0 , m 2 − n 2 + 1 ≥ 0 , and ( m 1 − n 1 + 1 ) + ( m 2 − n 2 + 1 ) = m − 1 − n + 2 = m − n + 1 , so that m i − n i + 1 ≤ m − n + 1 = ⇒ m i − n i ≤ m − n, i = 1 , 2 . So, for n 1 , n 2 > 0 , we h a v e ( n i , m i ) ≺ ( n, m ) and theref ore, by the induct iv e assumption, (4.11) 2 Y i =1 C e ( n i , m i ) ≤ 2 Y i =1 1 2 m i − n i +1 C ( n i , m i ) = 1 2 m − n +1 2 Y i =1 C ( n i , m i ) . Com bining (4.9)-(4.11) , and the re currenc e equation (4.8) for C ( · , · ) , w e obtain mC e ( n, m ) ≤ 1 2 m − n +1 ( N − m + 1) C ( n, m − 1) + 1 2 m − n +1 1 2 X n 1 + n 2 = n, m 1 + m 2 = m − 1  n n 1  2 Y i =1 n i C ( n i , m i ) = 1 2 m − n +1 mC ( n, m ) . Th us th e b ound (4.3) ho lds for ( n, m ) to o . The pro of of Prop osit ion 4 .1 is c om- plete.  By Prop osition 4.1 and and the for m ula (4.1), we hav e (4.12) Pr( S n > 0 , E n ≤ L ) ≤ N ( n, p ) [ x n ] exp " q p C 1 ( x ) + C e 0 ( x ) + 1 2 L X ℓ =1  p 2 q  ℓ C ℓ ( x ) # . Since C e 0 ( x ) is asympto tic to (1 / 2) C 0 ( x ) + ln(2 − 1 / 4 e 1 / 8 ) as x → e − 1 , only a t rivial c hange in t he pro of of Theore m 1.1 (i) is needed to show that (4.13) Pr( S n > 0) . 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 , ( | λ | = o ( n 1 / 12 ) . W e omit th e deta ils. F ur thermore , since for λ → −∞ we use L = 0 , the sums P L ℓ =1 in ( 4.1), (4.12) disapp ear, a nd w e obt ain an asymptot ic equality Pr( S n > 0) ∼ 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 , ( | λ | = o ( n 1 / 12 , λ → −∞ ) . T o complete t he pro of of (ii) , ( case λ = O (1) ), we need to prov e (4.4) for ea c h ﬁxe d ℓ > 0 . Rec all W righ t’s form ula (4.14) C ℓ ( x ) = (1 − T ( x )) − 3 ℓ " 2 ℓ X d =0 c ℓ,d (1 − T ( x )) d # , ( ℓ > 0 ) , Let us ﬁ nd a similar formula for C e ℓ ( x ) , ℓ > 0 . 38 BORIS PITTEL AND JI-A YEUM Prop osition 4 .3 . F or ℓ > 0 , (4.15) C e ℓ ( x ) = (1 − T 2 ( x )) − 3 ℓ " 8 ℓ − 1 X d =0 c e ℓ,d (1 − T ( x )) d # , wher e (4.16) c e ℓ, 0 = 2 2 ℓ − 1 c ℓ, 0 . Conse quently, for | x | < e − 1 and x → e − 1 , C e ℓ ( x ) ∼ 1 2 ℓ +1 C ℓ ( x ) . Pro of of Prop o sitio n 4 .3. W e use t he ideas of W right’s original pr o of of (4.14), and t he impr o v ements sugges ted by Ste panov [22 ], (cf. [12], Section 9). Giv en a co nnecte d graph G on [ n ] , with an excess ℓ = e ( G ) − v ( G ) > 0 , w e apply a “p runing” algo rithm wh ic h suc cessiv ely delete s v ertices of degree 1 . Ob viously the exc ess is preserved, and so for a t erminal graph (cor e) ¯ G we hav e e ( ¯ G ) − v ( ¯ G ) = ℓ . ¯ G inhe rits all the cycles of G , and thus ¯ G has only ev en cycles iﬀ G do es. A minim um deg ree of ¯ G is 2 at le ast, an d—since ℓ ( ¯ G ) = ℓ > 0 —a maxim um d egree is 3 at le ast. Next w e apply a “c ancellation” algorithm t o ¯ G : at each st ep, w e d elete a v ert ex of degree 2 , sp licing together the t w o edges it formerly touched. The excess is pre serv ed again. On ce all th e v ertice s o f degree 2 are gone, w e get a con nected m ultig raph (kernel) ˜ G , with p ossible lo ops and para llel edges, and a minimum vertex degree 3 at least. Thus 2 e ( ˜ G ) ≥ 3 v ( ˜ G ) , e ( ˜ G ) − v ( ˜ G ) = ℓ, and so (4.17) v ( ˜ G ) ≤ 2 ℓ, e ( ˜ G ) ≤ 3 ℓ. Notice that the largest num b ers of v ert ices and the edg es in the k erne l ar e 2 ℓ a nd 3 ℓ resp ectively , and the co rresp on ding k ernel is a 3 -regu lar multigraph. (I n [12] graphs G with su c h k ernels were called clean. It is these clean gra phs t hat are most p opulous asymptot ically among all conne cted gra phs on [ n ] with exce ss ℓ .) No w that w e hav e a reduced n um b er v ( ˜ G ) of v ert ices, w e relab el them using indices from [ v ( ˜ G )] and pr eserving t he ord er of t heir old indices f rom [ n ] . Under t his ru le, it follows from (1 1) that t he n umber of kernels ˜ G fo r the c ollection of a ll conn ected graphs G on [ n ] with exce ss ℓ is a fu nction of ℓ only! A key element of W r igh t’s argume n t was t he f ollo wing identit y . Let M b e a connec ted m ult igraph on a vertex set [ ν ] , with µ i indistinguishable lo ops at v ertex i , a nd µ ij indistinguishable parallel edges jo ining i an d j , ( i, j ∈ [ ν ] , i 6 = j ). Let h n,M denote the total num b er o f the c onnect ed simple gra phs G on [ n ] , with minim um degree 2 at least and maxim um degree 3 at le ast (core-type gr aphs, in short) , such that ˜ G = M . Let ting (4.18) H M ( z ) = X n h n,M n ! z n , RANDOM EQUA TIONS 39 w e ha v e (4.19) H M ( z ) = κ ν ! z ν (1 − z ) µ · K M ( z ) , µ := X i µ i + X i 0 , t he n umbe rs of 2 -degre e v ert ices of G on µ ij parallel edges of M m ust all b e of the same p arit y , he nce µ e ij = µ ij or µ o ij = µ ij . Su b ject to this condition, ho w many choices for ( µ e , µ o ) do we hav e ? F or each ( i, j ) su c h that µ ij > 0 , deﬁn e b ij = b j i =  1 , if µ e ij = µ ij , 0 , if µ e ij = 0 . If C is a cycle in M , then the pa rit y of a c ycle in G t hat canc els to C is th e parity of b ( C ) := P ( i,j ) ∈ C b ij . Hence b ( C ) m ust b e e v en for all cycles C , and w e nee d to chec k this c ondition only for simple cycles t hat do not u se parallel edges. Let T = T ( M ) b e a tree on [ ν ] t hat sp ans M . Pick µ e ij for all ν − 1 pairs ( i, j ) such that ( i, j ) ∈ E ( T ) , i. e . one of µ ij parallel edges is in E ( T ) . Let µ ij > 0 an d e = ( i, j ) / ∈ E ( T ) . The n e complet es a cycle C with a path in T that conne cts i and j . The condition “ b ( C ) is ev en” d etermines µ e ij uniquely . Henc e a choice of ν − 1 v alue s o f µ e ij determine s uniquely the remaining µ e ·· . Arguing as in the p ro of of Lemma 2 .1.1, we se e t hat the co ndition “ b ( C ) is even” will hold for all other cycles C . Thus w e hav e ha v e 2 ν − 1 c hoices for ( µ e , µ o ) . F or eac h of t hose choices, (4.28) b ecomes (4.29) K ( µ e , µ o ) ( z ) = Y i ∈ [ ν ] z 3 µ i · Y 1 ≤ i 0 , (4.31) H e M ( z ) = κ ν ! 2 ν − 1 (1 − z 2 ) µ P M ( z ) , wher e P M ( z ) is a p olynomia l of de gr e e 4 µ − 3 ν = µ + 3 ℓ at most, and P M (1) = 1 . This Le mma directly implies Corollary 4.5. (4.32) C e ℓ ( x ) = X M : e ( M ) − v ( M )= ℓ H e M ( x ) , wher e (4.33) H e M ( x ) = κ ν ! 2 ν − 1 (1 − T 2 ( x )) µ P M ( T ( x )) , Using ℓ < µ ( M ) ≤ 3 ℓ , ν ( M ) = µ ( M ) − ℓ , w e deduce from (4 .32)-(4.33) th at (4.34) C e ℓ ( x ) = (1 − T 2 ( x )) − 3 ℓ " 8 ℓ − 1 X d =0 c e ℓ,d (1 − T ( x )) d # , where (4.35) c e ℓ, 0 = 2 2 ℓ − 1 (2 ℓ )! X µ meets (4 . 19) κ ( µ ) . So, by t he second line in (4.22), c e ℓ, 0 = 2 2 ℓ − 1 c ℓ, 0 . The pro o f of Prop o sition 4 .3 is complete .  Comparing ( 4.14) and ( 4.34)-(4.35) , and using T ( e − 1 ) = 1 , w e obtain: for ℓ > 0 , (4.36) C e ℓ ( x ) = 1 2 ℓ +1 C ℓ ( x ) + O  | 1 − T ( x ) | − 3 ℓ +1  , ( | x | < e − 1 , x → e − 1 ) . And w e re call, ( 4.3), that (4.37) C e 0 ( x ) = 1 2 C 0 ( x ) + ln( 2 − 1 / 4 e 1 / 8 ) + O ( | T ( x ) − 1 | ) . No w, b y (4.1), for a ﬁxed L > 0 , Pr( S n > 0 , E n ≤ L ) = N ( n, p ) I Γ x − n − 1 exp " L X ℓ = − 1  p q  ℓ C e ℓ ( x ) # dx, where Γ is within t he disc | x | < e − 1 . As in S ection 2.3, we switc h to y by x = y e − y , and choose in the y -plane th e circular contour Γ ′ y = e − an − 1 / 3 + iθ , a > 0 b eing ﬁxed this time. Observe that , for each 1 ≤ ℓ ≤ L ,  p q  ℓ | 1 − T ( y e − y ) | − 3 ℓ +1 ≤ b n − ℓ | 1 − y | − 3 ℓ +1 ≤ b n − 1 / 3 , RANDOM EQUA TIONS 43 and, lik e wise in (4.3) th e remainder term O ( | T ( y e − y ) − 1 | ) is O ( n − 1 / 3 ) . And of course C e − 1 ( y e − y ) = C − 1 ( y e − y ) . On the basis of ( 4.36)-(4.37) , it ca n b e shown then that Pr( S n > 0 , E n ≤ L ) ∼ 2 − 1 / 4 e 1 / 8 N ( n, p ) I Γ ′ ( y e − y ) − n − 1 exp " 1 2 L X ℓ = − 1  p 2 q  ℓ C ℓ ( y e − y ) # d ( y e − y ) , where no w Γ ′ can b e r eplaced b y a circular c on tour of an arbitrarily small ra dius. Going back to the x -plan e, w e rec ognize (see (2.1.11) ) th e v alue of th e r esulting in tegral as 2 − 1 / 4 e 1 / 8 Pr( S n > 0 , E n ≤ L ) | ˆ p =1 / 2 , for λ = O ( 1) nee dless to say . B y ( 2.1.13) t he lat ter probabilit y is at least Pr( S n > 0) | ˆ p =1 / 2 · Pr( E n ≤ L ) . Letting n → ∞ , and using the part (i) for Pr( S n > 0) | ˆ p =1 / 2 , w e get lim inf Pr( S n > 0 , E n ≤ L ) 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 ≥ lim inf Pr( E n ≤ L ) . Since Pr( S n > 0) ≥ Pr( S n > 0 , E n ≤ L ) , an d E n = O P (1) , let ting L ↑ ∞ enables us to c onclude t hat Pr( S n > 0) & 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 . T oget her with (4.14) this pro v es that Pr( S n > 0) ∼ 2 − 1 / 4 e 1 / 8 c ( λ ) n − 1 / 12 . The pro of of Theore m 1.1 (ii) is no w co mplete.  Ac kno wledgemen t. W e are gratef ul to the p articipants of a grad uate stu- dent w orkshop a t Ohio Sta te Universit y for he lpful c ommen tarie s during numerous discussions of v arious pha ses of this study . W e thank Mike Mo llo y for p osing a 2 -colorability problem of the critica l grap h G ( n, m ) , a nd G reg So rkin for an e n- courage men t whe n w e needed it most. ´ Ak os Seres and Saleh T anv ee r, members of the sec ond auth or’s Ph.D Committ ee, pr o vided v aluab le comments o n the pro ject. References 1. G . N. Bagaev, R andom gr aphs with de gr e e of c onne cte dness 2 (Russian) , Dis cret. Analiz 2 2 (1973), 3–14. 2. G . N. Bagaev and E. F. Dmitriev , Enumer ation of conn e cte d lab ele d bip artite gr aphs (Ru s- sian) , Dokl. Ak ad. Nauk BSSR 28 (1984 ), 1061–1063. 3. E . A. Bender, E. R. Ca nﬁeld and B. D. McKay, The asymptotic numb er of lab ele d c onne cte d gr aphs with a given numb er of vertic es and e dges , Random Structures and Algorithms 1 (1990), 127–169. 4. B. Bollob´ as, The evolution of r andom gr aphs , T rans. Amer. Math. So c. 286 (1984), 257–274. 44 BORIS PITTEL AND JI-A YEUM 5. , R andom Gr aphs, Se c ond Edition , Cambridge University Press, 2001. 6. B. Bol lob´ as, C. Borgs, J . T. Chay es, J. H. Kim and D. B. Wilso n, The sc aling window of the 2-SA T tr ansition , Random Structures and Algorithms 8 (2001), 201–256. 7. V. Chv´ atal and B. Reed, Mick gets some (the odds ar e on his side) , 33 th Annual Symp osi um on F oundations of Computer Sci ence, (Pittsburgh, P A, 1992), IEEE Comput. So c. Press , Los Alamitos, CA, pp. 620–627. 8. D. Copp ersmith, D. Gamarnik, M. T. Ha jiaghayi and G. B. Sorkin, R andom MAX SA T, r andom MAX CUT, and their phase tr ansitions , Random Structures and Al gorithms 24 (2004), 502–545. 9. N. Creig non and H. Daud´ e, Smo oth and sharp thr esholds for r andom k -XOR-CNF satisﬁa- bility , Theor. Inform. Appl. 37 (2003), 127–1 47. 10. H. Daud´ e and V. Rav elomanana, R andom 2-XORSA T at the satisﬁability thr eshold , LA TIN 2008: Theoretic al Informatic s, 8th Latin American Sym pos ium Pro ce edings, (20 08), pp. 12- 23. 11. G. R. Grimmett and D. R. Stirzaker, Pr ob ability and R andom Pr o c esses: Pr oblems and So- lutions , Clarendon Press, Oxford, 1992. 12. S. Janson, D. Knuth , T. Luczak and B. Pittel, The birth of the giant c omp onent , Random Structures and Algorithms 4 (1993), 233-358 . 13. V. F. Kolchin, R andom Gr aphs , Cambridge University Pres s, 1999. 14. T. Luczak, Comp onent b ehavior ne ar the critic al p oint of the r andom gr aph pr o c ess , Random Structures and Algorithms 1 (1990), 287–310 . 15. , On the equ ivalenc e of two ba sic mo dels of r andom gr aphs , Random Graphs ’8 7 (M. Karo´ nski, J. Jaw orski a nd A. Ruci ´ nski, e ds.), Pozna´ n, 1987, Pro ceedi ngs, pp. 151–158. 16. T. Luczak, B. P ittel and J. Wierman, The structur e of a r andom gr aph ne ar the p oint of the phase transi tion , T rans. Amer. Math. So c. 341 (1994), 721– 748. 17. M. Molloy, Personal c ommunic ation (200 7). 18. J. W . Mo on, Counting L abelled T r e es , Canad. Math. Congres s, Montreal, 197 0. 19. B. Pittel, A r andom gr aph with a sub critic al numb er of e dges , T rans. Amer. Math. So c. 309 (1988), 51–75. 20. , On the lar gest c ompo nent of the r andom gr aph at a ne ar critic al stage , J. Comb. Theory B 8 2 (2001), 237–269. 21. A. D. Scott a nd G. B. Sorkin, Solving sp arse r andom instanc es of MAX CUT and MAX 2- CSP in line ar exp ecte d time , Combinatorics, Probabil ity and Computing 15 (2006), 281–315. 22. V. E. Stepanov, On some fe autur es of the structur e of a r andom gr aph ne ar a critic al p oint , Theory Probab. Appl. 32 (1988), 573–59 4. 23. E. M. W right, The numb er of c onne cte d sp arsely e dge d gr aphs , J. G raph Theory 1 (1977), 317–330. 24. , The numb er of c onne cte d sp arsely e dge d gr aphs. III. Asymptotic r esults , J. Graph Theory 4 (1980), 393–407. App endix. Pro of of (2 .3.22). (i) F or the seco nd b o und, we u se ( 2.3.19) and, setting s = a + it ,     s 1 − y exp  µs 2 2 + s 3 3      ≤ a 1 − y exp  µa 2 2 + a 3 3  exp  − t 2 ( a + µ/ 2)  . So A ( y , µ ) ≤ ( 2 π ) − 1 r π a + µ/ 2 exp  − µ 3 6 + µa 2 2 + a 3 3  .  (ii) F or the ﬁr st b ound, we use (2.3.20) , i. e. (A.1) e µ 3 / 6 3 ( y +1) / 3 A ( y , µ ) = ∞ X k =0 (3 2 / 3 µ/ 2) k Γ( k + 1)Γ( ( y + 1 − 2 k ) / 3) , RANDOM EQUA TIONS 45 and the ine qualities (1 / 2) a + b +2 a + b + 2 ≤ Γ( a + 1) Γ( b + 1) Γ( a + b + 2) ≤ a a b b ( a + b ) a + b ≤ 1 , ( a ≥ 0 , b ≥ 0 ) , whic h follo w from a classic formula Z 1 0 x a (1 − x ) b dx = Γ( a + 1) Γ( b + 1) Γ( a + b + 2) , and max x ∈ [0 , 1] x a (1 − x ) b = a a b b ( a + b ) a + b , Z 1 0 x a (1 − x ) b dx ≥ (1 / 2 ) b Z 1 / 2 0 x a dx + (1 / 2) a Z 1 1 / 2 (1 − x ) b dx = (1 / 2) a + b +1  1 a + 1 + 1 b + 1  . Break th e sum in (A.1) into Σ 1 , Σ 2 , an d Σ 3 , fo r { k ≥ 2 : ( y + 1 − 2 k ) / 3 ≥ 1 } , { k ≥ 1 : ( y + 1 − 2 k ) / 3 ≤ 0 } , and { k = 0 , 1 : or ( y + 1 − 2 k ) / 3 ≥ 1 } , r esp ec tiv ely . (Recall that Γ(0) = ∞ .) F or Σ 1 , 1 Γ( k + 1)Γ( ( y + 1 − 2 k ) / 3)) = Γ(2 k / 3) Γ( k + 1) · 1 Γ(( y + 1 − 2 k ) / 3)Γ( 2 k / 3) ≤ 2 ( y +1) / 3 Γ(( y + 1) / 3) · Γ(2 k / 3)Γ( k / 3 + 1) Γ( k + 1) · 1 Γ( k / 3 + 1) ≤ 2 ( y +1) / 3 Γ(( y + 1) / 3) · 2 (2 k / 3) 2 k/ 3 − 1 ( k / 3 ) k / 3 k k − 1 · 1 Γ( k / 3 + 1) ≤ 6 2 ( y +1) / 3 Γ(( y + 1) / 3) · (2 2 / 3 / 3) k Γ( k / 3 + 1) . Therefor e (A.2) | Σ 1 | ≤ 6 2 ( y +1) / 3 Γ(( y + 1) / 3 X k ≥ 0 ( | µ | 3 / 6) k / 3 Γ( k / 3 + 1) ≤ b ( | µ | 3 ∨ 1 ) 2 ( y +1) / 3 Γ(( y + 1) / 3) · e | µ | 3 / 6 . F or Σ 2 , w e use Γ( z ) Γ(1 − z ) = π sin( π z ) = ⇒ 1 | Γ(( y + 1 − 2 k ) / 3) | ≤ Γ(1 + (2 k − y − 1 ) / 3) , and Γ(1 + ( 2 k − y − 1) / 3) Γ( k + 1) ≤ 1 Γ(1 + ( y + 1) / 3) · Γ(2 k / 3 + 2) Γ( k + 1) = 1 Γ(1 + ( y + 1) / 3) · Γ(2 k / 3 + 2) Γ( k / 3 + 1) Γ( k + 1) · 1 Γ( k / 3 + 1) ≤ 1 Γ(1 + ( y + 1) / 3) · Γ( k + 3) Γ( k + 1) · ( k / 3 ) k / 3 (1 + 2 k / 3) 1+2 k/ 3 ( k + 1) k +1 ≤ 1 Γ(1 + ( y + 1) / 3) · ( k + 2) 2 (2 2 / 3 / 3) k Γ( k / 3 + 1) . 46 BORIS PITTEL AND JI-A YEUM Therefor e (A.3) | Σ 2 | ≤ b ( | µ | 9 ∨ 1 ) 1 Γ(( y + 1) / 3) · e | µ | 3 / 6 . And it is not diﬃcult to show that (A.4) | Σ 3 | ≤ b | Σ 1 | + | Σ 2 | . The relations (A.1)-(A.4) imply that A ( y , µ ) ≤ b e | µ | 3 / 2 (2 / 3) y +1 3 Γ  y +1 3  .  Pro of o f (2.3 .23)-(2.3.2 4). Aga in we use (2.3.19). Let u s cho ose a = ξ , where ξ = ξ ( y , µ ) is a maxim um p oin t of Ψ( a ; y , µ ) := − y ln a + µa 2 2 + a 3 3 , a ∈ (0 , ∞ ) , i. e. a p osit iv e ro ot of (A.5) Ψ (1) a ( a ; y , µ ) = µa + a 2 − y a = 0 . A ro ot exists and is unique, sinc e Ψ a (0+; y , µ ) = −∞ , Ψ( ∞ ; y , µ ) = ∞ and Ψ (2) a ( a ; y , µ ) = µ + 2 a + y a 2 > 0 , ( a ≥ 0) . Observe that µξ 2 /y is b o unded aw ay fro m z ero. If not, then, b y ( A.5), µ 3 ξ 6 y 3 → 0 , y 2 ξ 6 → 1 , whic h implies that µ 3 /y → 0 , con trad icting y = O ( λ 3 ) = O ( µ 3 ) . Break t he in tegr al in (2.3.19) into I 1 o v er | t | ≤ µ − 1 / 2 y 1 / 7 and I 2 o v er | t | ≥ µ − 1 / 2 y 1 / 7 . Argu ing as the part (i) of th e pre vious pro of, we b ound (A.5) | I 2 | ≤ b ξ e xp[Ψ( ξ ; y , µ )] Z | t |≥ µ − 1 / 2 y 1 / 7 exp  − t 2 ( ξ + µ/ 2)  dt ≤ b ξ e xp[Ψ( ξ ; y , µ )] ( ξ + µ/ 2) 1 / 2 · e − y 2 / 7 / 2 . T urn to I 1 . S ince Ψ (3) s ( s ; y , µ ) = − 2 y s 3 = O ( y ξ − 3 ) , RANDOM EQUA TIONS 47 w e ha v e Ψ( s ; y , µ ) = Ψ( ξ ; y , µ ) − t 2 2 ( µ + 2 ξ + y ξ − 2 ) + O  y ξ − 3 µ − 3 / 2 y 3 / 7  , and y ξ − 3 µ − 3 / 2 y 3 / 7 = y 3 / 7 − 1 / 2  µξ 2 y  3 / 2 ≤ b y − 1 / 14 . Consequently (A.6) I 1 ∼ ξ exp[Ψ( ξ ; y , µ )] Z | t |≤ µ − 1 / 2 y 1 / 7 exp  − t 2 2 ( µ + 2 ξ + y ξ − 2 )  dt ∼ ξ exp[Ψ( ξ ; y , µ )]  2 π µ + 2 ξ + y ξ − 2  1 / 2 . Since y ξ − 2 = O ( µ ) , (A.5)-(A.6) imply that I 1 ≫ I 2 , hence A ( y , µ ) ∼ e − µ 3 / 6 (2 π ) − 1 I 1 , whic h pro v es (2.3.23) .  If w e drop the condition y → ∞ , then t he in tegral in ( 2.3.19) is of order ξ e xp[Ψ( ξ ; y , µ )] ∞ Z −∞ exp  − t 2 ( ξ + µ/ 2)  dt = ξ exp[Ψ( ξ ; y , µ )]  2 π ξ + µ/ 2  1 / 2 , whic h pro v es (2.3.24) .  Pro of of (1.5). The system (1.1) is so lv able iﬀ fo r ev e ry c ycle C of G , (A.7) X e ∈ E ( C ) b e = O ( mo d 2 ) . If b e ∈ { 0 , 1 } are indep end en t ra ndom v ar iables with Pr( b e = 1) = ˆ p , the co ndition (A.7) is me t with proba bilit y (1 + ( 1 − 2 ˆ p ) | C | ) / 2 , Kolc hin [13 ]. Consider G = G ( n, p = γ /n ) , γ < 1 . Let X ns denote the num b er of c ycles o f length s whic h are “ bad”, i. e . do not meet the condition (A.7). W e nee d t o ﬁnd the limiting distribution of X n = P s ≥ 3 X ns the total n um b er of “bad” cycles. T o this end, observ e that, wit h probabilit y approac hing 1 , the cycles G ( n, p ) ma y ha v e are th ose in th e unicyclic c omp one n ts. Let us c all the m u-cycles. The exp ecte d n um b er of all cyc les o f le ngth k ≥ 3 is  n k  ( k − 1)! 2 p k ≤ γ k 2 k . So lim A →∞ lim n →∞ Pr( G ( n, p ) h as a cycle of length ≥ A ) = 0 . 48 BORIS PITTEL AND JI-A YEUM Let Y ns b e th e total num b er of all u -cycles of length s . In [19] it w as pr o v en t hat, for γ ﬁxed , { Y ns } s ≤ A conv erge s in dist ribution to { P oisson ( σ s ) } s ≤ A , where the P oissons are indep ende n t and σ s = T s ( γ e − γ ) 2 s , s ≥ 3 . As γ < 1 , we hav e T ( γ e − γ ) = γ , b ec ause T ( x ) = xe T ( x ) , f or x < e − 1 . No w a u-cycle of len gth s is bad with probabilit y π s = 1 − (1 − 2 ˆ p ) s 2 . Consequently { X ns } s ≤ A conv erge s to { Poisson( π s σ s ) } s ≤ A , whence X n conv erge s to Poisson  P s ≤ A π s σ s  . Th erefore , lim n →∞ Pr { ther e are no d bad u -cycles of leng th A at most } = e − P s ≤ A π s σ s . It remains to notice that X s ≥ 3 π s σ s = X s ≥ 3 1 − (1 − 2 ˆ p ) s 2 γ s 2 s = 1 4 ln 1 − γ (1 − 2 ˆ p ) 1 − γ − γ 2 ˆ p − γ 2 2 ˆ p (1 − ˆ p ) . That the same formula holds for G ( n, m = γ n/ 2) follo ws th en in a standard w a y .  Ohio St a te University, Columbus, Ohio, US A E-mail addr ess : bgp@math.o hio-state.ed u, yeum@math.o hio-state.ed u

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