Cycles of given lengths in unicyclic components in sparse random graphs
Marc Noy, Vonjy Rasendrahasina, Vlady Ravelomanana, Juanjo Rué
aa r X i v : . [ m a t h . C O ] D ec CYCLES OF GIVEN LENGTHS IN UNICYCLIC COMPONENTS IN SPARSERANDOM GRAPHS
MARC NOY, VONJY RASENDRAHASINA, VLADY RAVELOMANANA, AND JUANJO RU´E
Abstract.
Let L be subset of { , , . . . } and let X ( L ) n,M be the number of cycles belonging to unicycliccomponents whose length is in L in the random graph G ( n, M ). We find the limiting distribution of X ( L ) n,M in the subcritical regime M = cn with c < / M = n (cid:0) µn − / (cid:1) with µ = O (1). Depending on the regime and a condition involving the series P ℓ ∈ L z ℓ / (2 ℓ ), weobtain in the limit either a Poisson or a normal distribution as n → ∞ . MSC : Primary 05A16, 01C80; Secondary 05C10
Keywords:
Random graphs, random variables, analytic combinatorics. Introduction
A graph is unicyclic if it is connected and has a unique cycle. We say that a cycle in a graph is isolated if it is the unique cycle in a unicyclic connected component. Let G ( n, M ) be the random graphwith n vertices and exactly M edges drawn uniformly at random from the set of (cid:0) n (cid:1) possible edges.This is the model introduced in the seminal paper of Erd˝os and R´enyi [3], in which each graph has thesame probability (cid:18)(cid:0) n (cid:1) M (cid:19) − . We are interested in the number of isolated cycles in G ( n, M ) whose lengths are restricted to takecertain values. More precisely, let N > = { , , . . . } and L a subset of N > . We denote by X ( L ) n,M the random variable equal to the number of isolated cycles in G ( n, M ) whose lengths lie in L . Ourmain result gives the limiting distribution of X ( L ) n,M for various values of M , corresponding to theso-called subcritical and critical regimes. Depending on the regime and a condition involving thegenerating function λ ( z ) = P ℓ ∈ L z ℓ / (2 ℓ ), we obtain in the limit as n → ∞ either a Poisson or anormal distribution.The number of cycles in G ( n, M ) has been studied since the appearance of [3]. When M = cn ,Erd˝os and R´enyi showed [3, Theorem 3b] that the number of cycles of length k converges to a Poissonlaw with parameter (2 c ) k / (2 k ). Let X n, M be the random variable equal to the number of isolatedcycles in G ( n, M ). When M = cn and c < /
2, asymptotically almost surely (that is, with probabilitytending to 1 as n → ∞ ) all cycles are isolated. As a consequence we havelim n →∞ E [ X n,cn ] = X k > (2 c ) k k = 12 log 11 − c − c − c . We next recall the different regimes for sparse random graphs (see for instance [7, 1]). The followingresults hold asymptotically almost surely (shortened to a.a.s.). • Subcritical regime . When M = cn with c < /
2, the connected components of G ( n, M ) areeither trees or unicyclic graphs. • Barely subcritical regime . When M = n (cid:0) − µn − / (cid:1) with µ → ∞ and µ = o (cid:0) n / (cid:1) , • Critical regime . This is when M = n (cid:0) µn − / (cid:1) and µ = O (1). In this regime theconnected components of G ( n, M ) are trees, unicyclic graphs, and complex components. Acomplex component is obtained from a connected cubic multigraph K by performing thefollowing operations: first replace edges in K by induced paths of any length so that to obtaina simple graph C , and then attach rooted trees to the vertices of C . • Supercritical regime . When M = cn with c > /
2, there exists a unique component L oflinear size and the remaining components are either trees or unicyclic graphs. The ‘Symmetry rinciple’ (see [7, Section 5.6]) says that in this case G ( n.M ) \ L in some sense ‘looks like’ asubcritical random graph with suitable parameters.In the barely subcritical regime Kolchin showed that if r = log n − log µ , then the normalizedrandom variable ( X n,M − r ) / √ r tends in distribution to a Gaussian law (see [8, Theorem 1.1.15]). Inthe critical regime, Flajolet, Knuth and Pittel [4, Corollary 6] showed that E [ X n,M ] ∼ log n . By theso-called symmetry property [7, Theorem 5.24], X n,M properly normalized should also be Gaussianwhen M = n (1 + µn − / ) and µ → ∞ with µ = o ( n / ).Some results have been obtained fixing a set L of positive integers as possible cycle lengths. Fol-lowing [4], define an L -cycle as an isolated cycle whose length is in L . Let X ( L ) n, M be the number of L -cycles in G ( n, M ). It is shown in [4, Corollary 7] that if lim n →∞ Mn = λ <
1, then the probabilitythat a graph (or multigraph) with n vertices and M edges has no L -cycle is equal to √ − λ exp X l > ,ℓ/ ∈ L λ ℓ l + O (cid:16) n − / (cid:17) = exp − X l > ,ℓ ∈ L λ ℓ l + O (cid:16) n − / (cid:17) . (1)Our results concern the distribution of the random variables X ( L ) n,M . In particular, we obtain fulllimiting distributions both in the subcritical and the critical regimes. Theorem 1.1.
Let L ⊆ N > and set λ L ( z ) = P ℓ ∈ L z ℓ ℓ , considered as a function of one complexvariable in the unit disk | z | < . Let X ( L ) n,M be the random variable equal to the number of L -cycles in G ( n, M ) . Then the following holds: (A) (Subcritical regime). Let c = c ( n ) be such that < lim sup n →∞ c < / and M = cn . Then X ( L ) n, M λ L (2 c ) d −→ Poisson (1) , as n → ∞ . (2)(B) (Barely subcritical regime). Let M = n (1 − µn − / ) with lim µ = + ∞ and µ = o ( n / ) . Thentwo situations may happen: if lim n →∞ λ L ( Mn ) < + ∞ , then X ( L ) n, M λ L (cid:0) Mn (cid:1) d −→ Poisson (1) , as n → ∞ . (3) Otherwise, if lim n →∞ λ L ( Mn ) = + ∞ , then X ( L ) n, M − λ L (cid:0) Mn (cid:1)q λ L (cid:0) Mn (cid:1) d −→ N (0 , , as n → ∞ . (4)(C) (Critical regime). Let M = n (1+ µn − / ) , with µ = O (1) . Let α be the unique positive solutionof µ = α − α . Then two situations may happen: if lim n →∞ λ L ( e − αn − / ) < + ∞ , then X ( L ) n, M λ L (cid:0) e − αn − / (cid:1) d −→ Poisson (1) , as n → ∞ . (5) Otherwise, if lim n →∞ λ L (cid:16) e − αn − / (cid:17) = + ∞ , then X ( L ) n, M − λ L (cid:16) e − αn − / (cid:17)q λ L (cid:0) e − αn − / (cid:1) d −→ N (0 , , as n → ∞ . (6)Points (A), (B) and (C) in Theorem 1.1 are the contents of Theorems 3.1, 3.2 and 3.4 given inthe Section 3. We remark that in the previous statement there is no discontinuity between equations(2)–(3)–(5) and equations (4)–(6): the Taylor expansion of the term e − αn − / in the statement for thecritical regime is equal to 1 − αn − / + o ( n − / ), which coincides with the term 1 − µn − / in thebarely subcritical region.The proofs are based on estimating coefficients of generating functions by means of Cauchy integralsalong suitable contours and applying the saddle-point method. Remarks.
Observe that (1) follows directly from (2). Let us mention that technical refinements ofour techniques would provide similar results for the region just before the supercritical regime, namely = n (1 + µn − / ) when µ → ∞ , µ = o ( n / ). We do not include the analysis of this region becausethe computations become too involved.Finally, one may wonder why in the previous theorem we do not have a corresponding result for thesupercritical regime. The reason is that in this case our techniques, based on the detailed structureof G ( n, p ) together with saddle-point estimates for the associated generating functions, do not applyin this situation. Given the Symmetry principle mentioned above, one should expect the number of L -cycles in the supercritical regime follows a limit Poisson law as in the subcritical regime, but thetools provided by the Symmetry principle do no seem precise enough to prove such a statement.2. Preliminaries and notation
All graphs considered in this paper are labelled. The size of a graph is the number of vertices. The excess of a graph G is the number of vertices minus the number of edges. In G ( n, M ) the excess is M − n .2.1. Analytic combinatorics of graphs.
We use the language of analytic combinatorics as in [5].Given a generating function A ( x ) = P n > a n x n , we write [ x n ] A ( x ) = a n . If A ( x ) = P n > a n x n and B ( x ) = P n > b n x n , we write A ( x ) (cid:22) B ( x ) if there exists n such that [ x n ] A ( x ) [ x n ] B ( x ) for n > n .All the generating functions that appear in this work are exponential generating functions of the form P n > a n x n /n !, or EGF for short (see [5, Chapter 2]).We denote by T ( x ) and W − ( x ) the EGF of rooted and unrooted labelled trees, respectively. It iswell known that T ( x ) = xe T ( x ) = ∞ X n =1 n n − x n n ! , W − ( x ) = T ( x ) − T ( x ) . (7)The EGF W ( x ) of unicyclic graphs (connected graphs with n vertices and n edges) is given by (seefor instance [6, Equation (3.5)]) W ( x ) = X k > T ( x ) k k = −
12 log (1 − T ( x )) − T ( x )2 − T ( x ) . (8)We write λ ( t ) = P k > t k k = − log(1 − t ) − t/ − t /
4, so that W ( x ) = λ ( T ( x )).2.2. From Poisson parametrizations to central limit theorems.
We include the following resultby Kolchin that provides an approximation to a normal law by a Poisson parametrization.
Theorem 2.1 ([8, Theorem 1.1.15]) . Let k = λ n + ρ n √ λ n . If (1 + ρ n ) /λ n → as n → ∞ then e − λ n λ kn k ! = 1 √ πλ n e − ρ n / (cid:18) ρ n − ρ n √ λ n + O (cid:18) ρ n λ n (cid:19)(cid:19) . Proof of Theorem 1.1
We present separately the proof for each regime in Theorem 1.1. The main idea in all proofs isto encode the typical structure of random graphs in the regime under consideration using generatingfunctions and then obtain large power estimates by means of saddle point bounds.3.1.
Subcritical regime.
In this regime, the connected components of G ( n, M ) are a.a.s. a set ofacyclic graphs (a forest) together with a set of unicyclic graphs. We exploit this property in order toget the following result which refines the first statement in Theorem 1.1: Theorem 3.1.
Let c such that < c < / , and M = cn . Let L ⊆ N > and λ L ( z ) = P ℓ ∈ L z ℓ ℓ . Thenthe random variable X ( L ) n, M equal to the number of L -cycles satisfies Pr h X ( L ) n, M = k i = e − λ L (2 c ) λ L (2 c ) k k ! (cid:0) O (cid:0) n − (cid:1)(cid:1) . Moreover, if k → ∞ as n → ∞ then Pr h X ( L ) n, M = k i = O ( k − k ) . roof. It suffices to consider graphs whose connected components are trees and unicyclic graphs.Using the symbolic method we obtain that the probability that G ( n, M ) contains exactly k unicycliccomponents containing an L -cycle is equal toPr h X ( L ) n, M = k i = n ! (cid:0) ( n ) M (cid:1) [ x n ] W − ( x ) n − M ( n − M )! λ L ( T ( x )) k k ! e W ( x ) − λ L ( T ( x )) . (9)The term λ L ( T ( x )) k k ! encodes the components containing an L -cycle, while the term e W ( x ) − λ L ( T ( x )) encodes the rest of unicyclic components (whose lengths do not belong to L ). Using Cauchy integral’sformula we get [ x n ] W − ( x ) n − M λ L ( T ( x )) k e W ( x ) − λ L ( T ( x )) = (10) M − n πi H (2 W − ( x )) n − M λ L ( T ( x )) k e W ( x ) − λ L ( T ( x )) dxx n +1 . After the change of variables z = T ( x ), it becomes[ x n ] W − ( x ) n − M λ L ( T ( x )) k e W ( x ) − λ L ( T ( x )) = 2 M − n πi I g ( z ) λ L ( z ) k e nh ( z ) dzz , (11)where g ( z ) = (1 − z ) e λ ( z ) − λ L ( z ) , (12) h ( z ) = z − log z + (cid:0) − Mn (cid:1) log (cid:0) z − z (cid:1) . (13)Note that the function h ( z ) given by (13) is exactly the same as [2, Equation (30)], which satisfiesthe conditions h ′ (2 c ) = h ′ (1) = 0. In the range M = cn with 0 < c < , we can apply saddle-pointmethods by choosing a circular path { ce iθ , θ ∈ [ − π, π ) } as the contour of integration. As shownin [4], we split the integral in (11) into three parts, namely R − θ − π + R θ − θ + R πθ . It suffices to integratefrom − θ to θ , for a convenient value of θ , because the remaining integrals can be bounded by themagnitude of the central integrand. Following the proof of [2, Theorem 3.2] and choosing θ = n − / (so that nθ → ∞ but nθ → n → ∞ ) we haveexp (cid:0) nh (2 ce iθ ) (cid:1) = exp (cid:16) nh (2 c ) − nc (1 − c )2(1 − c ) θ (cid:17) (cid:0) iO ( nθ ) + O ( nθ ) (cid:1) , (14)and for all choices of θ in [ − π, − θ ] ∪ [ θ , π ) we have (cid:12)(cid:12) exp (cid:0) nh (2 ce iθ ) − nh (2 c ) (cid:1)(cid:12)(cid:12) = exp (cid:16) − O ( n / ) (cid:17) . (15)As 2 c < θ , we have g (cid:0) ce iθ (cid:1) = g (2 c ) (cid:0) iO ( θ ) + O ( θ ) (cid:1) , (16)and λ L (2 ce iθ ) k = λ L (2 c ) k (cid:0) iO ( θ ) + O ( θ ) (cid:1) (17)for fixed k >
0. Using expansions (14), (16), (17) and the bound (15) we have I g ( z ) λ L ( z ) k e nh ( z ) dzz = i Z θ − θ g (2 ce iθ ) λ L (2 ce iθ ) k e nh (2 ce iθ ) dθ (cid:16) e − O ( n / ) (cid:17) = ig (2 c ) λ L (2 c ) k e nh (2 c ) Z + θ − θ e − nσ θ · (1 + iO ( θ ) + O ( θ ) + iO ( nθ ) + O ( nθ )) dθ (cid:16) e − O ( n / ) (cid:17) , I g ( z ) λ L ( z ) k e nh ( z ) dzz = i Z θ − θ g (2 ce iθ ) λ L (2 ce iθ ) k e nh (2 ce iθ ) dθ (cid:16) e − O ( n / ) (cid:17) = ig (2 c ) λ L (2 c ) k e nh (2 c ) Z + θ − θ e − nσ θ · (1 + iO ( θ ) + O ( θ ) + iO ( nθ ) + O ( nθ )) dθ (cid:16) e − O ( n / ) (cid:17) , where σ = c (1 − c )1 − c . If we set x = √ nσθ the integral in the above equation becomes1 √ nσ Z σ / n / − σ / n / e − x (cid:16) iO (cid:16) x √ nσ (cid:17) + O (cid:16) x nσ (cid:17) + iO (cid:16) n x √ nσ (cid:17) + O (cid:16) x nσ (cid:17)(cid:17) dx. (18) bserve that σ = O (1) and the estimate (18) is a real number (because (10) is a real number). Hence(18) is equal to 1 √ σn Z σ / n / − σ / n / e − x (cid:18) O (cid:18) x n (cid:19) + O (cid:18) x n (cid:19)(cid:19) dx. It follows that Z θ − θ g (2 ce iθ ) λ L (2 ce iθ ) k e nh (2 ce iθ ) dθ = r πσn g (2 c ) λ L (2 c ) k e nh (2 c ) (cid:16) O (cid:0) n − (cid:1) + e − O ( n / ) (cid:17) . That is[ x n ] W − ( x ) n − M λ L ( T ( x )) k e W ( x ) − λ L ( T ( x )) = 2 M − n √ πσn g (2 c ) λ L (2 c ) k e nh (2 c ) (cid:0) O (cid:0) n − (cid:1)(cid:1) (19)Using Stirling’s formula for the corresponding range of M , we have1 (cid:0) ( n ) M (cid:1) n !( n − M )! k ! = 1 k ! r πnMn − M M n n M M n M ( n − M ) n − M exp (cid:18) − M + Mn + M n (cid:19) (cid:0) O (cid:0) n − (cid:1)(cid:1) . (20)Multiplying (19) and (20), after cancellations we obtainPr h X ( L ) n, M = k i = e − λ L (2 c ) λ L (2 c ) k k ! (cid:0) O (cid:0) n − (cid:1)(cid:1) . This proves the first part of the theorem.Now, suppose that k → ∞ as n → ∞ . The previous arguments work in a similar way. Instead ofusing the estimate (17), which is only valid when θ is small enough, we exploit the fact that λ L hasnon-negative Taylor coefficients. Hence, Equation (17) can be replaced by the relation (cid:12)(cid:12) λ L (2 ce iθ ) k (cid:12)(cid:12) λ L (2 c ) k , which is valid for each choice of θ ∈ [ − π, π ). Applying the same arguments as before and that k ! < e k k k for large k , we conclude thatPr h X ( L ) n, M = k i e − λ L (2 c ) λ L (2 c ) k k ! (cid:0) O (cid:0) n − (cid:1)(cid:1) < Ce − λ L (2 c ) ( eλ L (2 c )) k k k , for a suitable constant C . The second result in the theorem follows from the fact that c < , andhence λ L (2 c ) is bounded. (cid:3) Barely subcritical regime.
In the barely subcritical regime the asymptotic structure of G ( n, M )is the same as in the subcritical regime. However, the integration countour we use is slightly morecomplicated in order to encode cycles of arbitrary length. Theorem 3.2.
Let M = n (1 − µn − / ) with µ tending to infinity with µ = o (cid:0) n / (cid:1) . Let L ⊆ N > and λ L ( z ) = P ℓ ∈ L z ℓ ℓ . Then the random variable X ( L ) n, M equal to the number of L -cycles satisfies Pr h X ( L ) n,M = k i = e − λ L ( Mn ) λ L (cid:0) Mn (cid:1) k k ! (cid:0) O (cid:0) µ − (cid:1)(cid:1) . (21) Assume moreover that lim n λ L (cid:0) Mn (cid:1) = ∞ . Then for fixed real numbers y < y Pr y X n, M − λ L (cid:0) Mn (cid:1)q λ L (cid:0) Mn (cid:1) y → √ π Z y y e − u / du, as n → ∞ . (22) Proof.
The arguments and notation are similar to the ones in the proof of Theorem 3.1. As mentionedin the proof of Theorem 3.1, a.a.s. in this regime G ( n, M ) contains only trees and unicyclic graphs ascomponents. We need estimates for (9) in this new range of M . We use again the same methods as inthe proof of [2, Theorem 3.2]. Let ω ( n ) = ( n − M ) / n / , τ = n ( n − M ) M ( n − M ) , θ = r τn ω ( n ) . Then nθ → ∞ and nθ → n → ∞ . The expansion of h in the vicinity of θ is h (cid:18) Mn e iθ (cid:19) = h (cid:18) Mn (cid:19) − M ( n − M )2 n ( n − M ) θ − i ( n − nM + 2 M ) M n − M ) θ + O ( θ ) . (23) or θ ∈ [ − θ , + θ ], k = Θ (cid:0) λ L ( Mn ) (cid:1) , the expansion of λ L in the vicinity of θ is λ L (cid:0) Mn e iθ (cid:1) k λ L (cid:0) Mn (cid:1) k = 1 + iO kλ L (cid:0) Mn (cid:1) n ( n − M ) θ ! + O k λ L (cid:0) Mn (cid:1) n ( n − M ) θ ! = 1 + iO (cid:18) n ( n − M ) θ (cid:19) + O (cid:18) n ( n − M ) θ (cid:19) . (24)The integrand can be bounded on [ − π, − θ ) ∪ ( θ , π ) because (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) nh (cid:18) Mn e iθ (cid:19) − nh (cid:18) Mn (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = O ( e − ω ( n ) / ) . (25)Combining (23), (24) and (25), we havePr h X ( L ) n,M = k i = n ! (( n ) M ) ( n − M )! 2 M − n π g (cid:0) Mn (cid:1) exp (cid:0) nh (cid:0) Mn (cid:1)(cid:1) λ L ( Mn ) k k ! × R θ − θ e − nτ θ (cid:16) iO (cid:16) n ( n − M ) θ (cid:17) + O (cid:16) n ( n − M ) θ (cid:17)(cid:17) × (cid:16) in ( n − nM +2 M ) M n − M ) θ + O ( nθ ) (cid:17) dθ (cid:16) O ( e − ω ( n ) / ) (cid:17) . We set θ = p τ /nx and the integral becomes r τn Z ω ( n ) − ω ( n ) e − x (cid:18) iO (cid:18) n ( n − M ) / x (cid:19) + O (cid:18) n ( n − M ) x (cid:19)(cid:19) . (cid:18) iO (cid:18) n ( n − M ) / x (cid:19) + O (cid:18) n ( n − M ) x (cid:19)(cid:19) dx = r τn Z ω ( n ) − ω ( n ) e − x (cid:18) O (cid:18) n ( n − M ) x (cid:19)(cid:19) dx = r πτn (cid:18) O (cid:18) n ( n − M ) (cid:19)(cid:19) . After simple algebraic manipulations as in the proof of Theorem 3.1 we obtainPr h X ( L ) n,M = k i = e − λ L ( Mn ) λ L (cid:0) Mn (cid:1) k k ! (cid:0) O (cid:0) µ − (cid:1)(cid:1) . This proves the first part of the theorem.We assume now that lim n λ L (cid:0) Mn (cid:1) = ∞ . Set k = λ L (cid:0) Mn (cid:1) + ρ n q λ L (cid:0) Mn (cid:1) with | ρ n | = o (cid:0) λ L (cid:0) Mn (cid:1)(cid:1) / .We can apply Theorem 2.1 and obtainPr h X ( L ) n, M = k i = 1 q πλ L (cid:0) Mn (cid:1) e − ρ n / ρ n − ρ n q λ L (cid:0) Mn (cid:1) + O ρ n q λ L (cid:0) Mn (cid:1) = 1 q πλ L (cid:0) Mn (cid:1) e − ρ n / (1 + o (1)) . The central limit theorem for X ( L ) n,M follows, that is, for fixed real y < y we havePr y X ( L ) n,M − λ L (cid:0) Mn (cid:1)q λ L (cid:0) Mn (cid:1) y → √ π Z y y e − u / du, as n → ∞ . (cid:3) Critical regime.
In this regime we have to take into account the appearance of complex compo-nents. Let p k ( n, M ; L, r ) be the probability that G ( n, M ) has a total excess r with exactly k unicycliccomponents containing an L -cycle. The following lemma gives an estimate for p k ( n, M ; L, r ). emma 3.3. Let M = n (1 + µn − / ) with µ = O (1) . Let α be the positive solution to µ = α − α . Let k = λ L (cid:16) e − αn − / (cid:17) + ρ q λ L (cid:0) e − αn − / (cid:1) , which satisfies ρ = ω (cid:18) λ L (cid:16) e − αn − / (cid:17) / (cid:19) .Then for fixed r we have p k ( n, M ; L, r ) = e − λ L (cid:18) e − αn − / (cid:19) λ L (cid:16) e − αn − / (cid:17) k k ! √ πe r A (3 r + 1 / , µ ) · (cid:16) O (cid:16) n − / (cid:17)(cid:17) , where e r = (6 r )!2 r r (3 r )! (2 r )! , A ( y, µ ) = e − µ / ( y +1) / X k > ( / µ ) k k !Γ(( y + 1 − k ) / . Moreover, for r large enough there exist absolute constants C > and ε > such that p k ( n, M ; L, r ) e − λ L (cid:18) e − αn − / (cid:19) λ L (cid:16) e − αn − / (cid:17) k k ! Ce − εr . (26) Proof.
The proof is based on analytic techniques introduced in [4] and [6]; see also [9]. The probability p k ( n, M ; L, r ) is given by p k ( n, M ; L, r ) = n ! (cid:0) ( n ) M (cid:1) [ x n ] W − ( x ) n − M + r ( n − M + r )! E r ( x ) λ L ( T ( x )) k k ! e W ( x ) − λ L ( T ( x )) , (27)where E r ( x ) is the EGF of complex components with total excess r given by [6, Equation (6.8)]. Asshown in [6], when r = o ( n / ), the series E r ( x ) can be approximated [6, Equation (6.8)] by e r (1 − T ( x )) r ,where e r = (6 r )!2 r r (3 r )! (2 r )! , and the error term is of order O (cid:16) r / n / (cid:17) . In order to evaluate (27) we have to compute the expression St ( n, M, r )2 πi I (1 − z ) − r e nh ( z ) λ L ( z ) k k ! e W ( z ) − λ L ( z ) E r ( z ) dzz , (28)where St ( n, M, r ) = n ! (cid:0) ( n ) M (cid:1) − n + M − r e n e r ( n − M + r )! , (29) h ( z ) = z − − log z + (cid:18) − Mn (cid:19) log(2 z − z ) . (30)We remark the difference between h ( z ) and the function h ( z ) defined in Equation (13). Note alsothat h ( z ) is exactly the same as in [6, Equation (10.12)], which satisfies h (1) = h ′ (1) = 0 and also h ′′ (1) = 0 if M = n/
2. We now follow the method of the proof of [6, Lemma 3] in order to computeour integral by choosing as path of integration z = z ( t ) = e − αn − / − itn − / , (31)where α is the unique positive solution of µ = α − α , and t belongs to the interval (cid:16) − πn / λ ′ L ( e − αn − / ) − / , πn / λ ′ L ( e − αn − / ) − / (cid:17) . Given that λ L ( e − αn − / − itn − / ) k k ! e − λ L ( e − αn − / − itn − / ) = λ L ( e − αn − / ) k k ! e − λ L ( e − αn − / ) × iO kλ L ( e − αn − / ) λ ′ L (cid:16) e − αn − / (cid:17) n / t + O k λ L (cid:0) e − αn − / (cid:1) λ ′ L (cid:16) e − αn − / (cid:17) n t (32) s long as k = O (cid:16) λ L (cid:16) e − αn − / (cid:17)(cid:17) , our choice ensures that the O terms in (32) can be moved out ofthe integral. By following the proof of [6, Equation (10.1) of Lemma 3] we obtain that, for fixed valuesof rp k ( n, M ; L, r ) = e − λ L ( e − αn − / ) λ L ( e − αn − / ) k k ! √ πe r A (3 r + 1 / , µ ) (cid:18) O (cid:16) n − / (cid:17) + O (cid:18) µ n / (cid:19)(cid:19) . (33)Since k = O (cid:16) λ L (cid:16) e − αn − / (cid:17)(cid:17) in the O terms above we have λ ′ L (cid:16) e − αn − / (cid:17) n / t = O λ ′ L (cid:16) e − αn − / (cid:17) / n / = O (cid:18) − e − αn − / ) / n / (cid:19) = O (cid:16) n − / (cid:17) . This proves the first statement of the theorem.Next let us assume that r → ∞ . We know that E r ( z ) (cid:22) e r (1 − T ( z )) r (see for instance [6, Lemma 4]).From (27) we have p k ( n, M ; L, r ) n ! (cid:0) ( n ) M (cid:1) [ z n ] W − ( x ) n − M + r ( n − M + r )! λ L ( T ( z )) k k ! e W ( z ) − λ L ( T ( z )) e r (1 − T ( z )) r . Then, we obtain p k ( n, M ; L, r ) St ( n, M, r )2 πi I z r (2 − z ) r (1 − z ) r e nh ( z ) λ L ( z ) k k ! e λ ( z ) − λ L ( z ) dzz , (34)with h is defined by (30). In this case we take as contour of integration the circle { δe iθ : θ ∈ [ − π, π ) } with δ = 1 − r / n / <
1. On this circle, since r >
1, for some constant C and function f ( n ) withlim n f ( n ) = + ∞ , we have λ L ( δ ) k k ! e − λ L ( δ ) δ π (cid:18) δ (2 − δ )(1 − δ ) (cid:19) r e nh ( δ ) (1 − δ ) / Z π − π e − f ( n ) θ dθ < C √ n δ r (cid:18) δ (2 − δ )(1 − δ ) (cid:19) r e nh ( δ ) . Note that r M = n (1 + µn − / ). Then for n large enough δ r (2 − δ ) r (1 − δ ) r < n r r r , nh ( δ ) < r + 116 µr / . (35)Using Stirling’s formula we find that n ! (cid:0) ( n ) M (cid:1) ( n − M + r )! e n − n + M − r < n / n r e − µ / / − r , and for r → ∞ , we have e r = (6 r )!2 r r (3 r )! (2 r )! r / (cid:18) r e (cid:19) r . (36)Combining (35) and (36) in (34), we deduce that p k ( n, M ; L, r ) < c r / exp (cid:18) − µ µr / + (cid:18) − (cid:19) r (cid:19) , for some constant c >
0. Since + log − <
0, when r → ∞ we deduce p k ( n, M ; L, r ) e − O ( r ) . (cid:3) We can now proof the main result in the critical regime.
Theorem 3.4.
Let M = n (1 + µn − / ) where µ = O (1) . Let α be the positive solution to µ = α − α .Let L ⊆ N > and let λ L ( z ) = P k ∈ L z k k . Then the random variable X ( L ) n, M equal L -cycles in G ( n, M ) satisfies Pr h X ( L ) n,M = k i = e − λ L (cid:18) e − αn − / (cid:19) λ L (cid:16) e − αn − / (cid:17) k k ! (cid:16) O (cid:16) n − / (cid:17)(cid:17) . (37) oreover, assume that lim n →∞ λ L (cid:16) e − αn − / (cid:17) = + ∞ . Then, for each choice of real y < y Pr y X ( L ) n,M − λ L (cid:16) e − αn − / (cid:17)q λ L (cid:0) e − αn − / (cid:1) y → √ π Z y y e − u / du, when n → ∞ . (38) Proof.
By Lemma 3.3 and the dominated convergence theorem, Pr h X ( L ) n, M = k i is equal to X r > p k ( n, M ; L, r ) = X r > e − λ L (cid:18) e − αn − / (cid:19) λ L (cid:16) e − αn − / (cid:17) k k ! √ πe r A (3 r + 1 / , µ ) · (cid:16) O (cid:16) n − / (cid:17)(cid:17) . For µ = O (1) Janson, Knuth, Luczak and Pittel [6, Equation (13.17) and Corollary p. 61] have shownthat the probability that G ( n, M ) has total excess r is asymptotically √ πe r A (3 r + 1 / , µ ), and thatthe s -th moment of the excess r satisfies P r > √ πe r r s A (3 r + 1 / , µ ) = O ( µ s ) = O (1). Hence P r > √ πe r A (3 r + 1 / , µ ) = 1. This shows relation (37).In order to prove (38), we apply Theorem 2.1 by choosing k = λ L (cid:16) e − αn − / (cid:17) + ρ n q λ L (cid:0) e − αn − / (cid:1) , with λ L (cid:16) e − αn − / (cid:17) / = o ( ρ n ) . (cid:3) Acknowledgments
Part of this research was done when the J.R. was visiting IRIF – Universit´e Paris Denis Diderotin July 2018. J.R. thanks the hospitality of the institution during his research stay. V.R. was par-tially supported by grants ANR 2010 BLAN 0204 (Magnum). V.R. and J.R. were supported by theProject PHC Procope ID-57134837 ‘Analytic, probabilistic and geometric Methods for Random Con-strained graphs’. J.R. was partially supported by the FP7-PEOPLE-2013-CIG project CountGraph(ref. 630749). Finally, M.N. and J.R. were supported by the Spanish project MTM2017-82166-P andby the H2020-MSCA-RISE-2020 project RandNET (ref. 101007705).
References [1] N. Alon and J. H. Spencer.
The probabilistic method . Wiley-Interscience Series in Discrete Mathematics and Opti-mization. Third edition.[2] H. Daud´e and V. Ravelomanana. Random phase transition.
Algorithmica , Special issue of
LATIN
Publ. Math. Inst. Hung. Acad. Sci , 5:17–61, 1960.[4] P. Flajolet, D. E. Knuth, and B. Pittel. The first cycles in an evolving graph.
Discrete Mathematics , 75(1-3):167–215,1989.[5] P. Flajolet and B. Sedgewick.
Analytic Combinatorics . Cambridge University Press, 2009.[6] S. Janson, D. E. Knuth, T. Luczak, and B. Pittel. The Birth of the Giant Component.
Random Structures andAlgorithms , 4(3):233–358, 1993.[7] S. Janson, T. Luczak, and A. Ruci´nski.
Random Graphs . Wiley-Interscience, 2000.[8] V. F. Kolchin.
Random Graphs . Cambridge University Press, 1998.[9] Marc Noy, Vlady Ravelomanana, and Juanjo Ru´e. On the probability of planarity of a random graph near the criticalpoint.
Proc. Amer. Math. Soc. , 143(3):925–936, 2015.
Department of Mathematics, Universitat Polit`ecnica de Catalunya, and Institut de Matem`atiques de laUPC-BarcelonaTech (IMTech), Spain.
Email address , M. Noy: [email protected]
Ecole Normale Sup´erieure, Universit´e d’Antananarivo, Madagascar.
Email address , V. Rasendrahasina: [email protected]
IRIF, Universit´e de Paris, France.
Email address , V. Ravelomanana: [email protected]
Department of Mathematics, Universitat Polit`ecnica de Catalunya, and Barcelona Graduate School ofMathematics (BgsMath), Spain.
Email address , J. Ru´e: [email protected]@upc.edu