Limit law for number of components of fixed sizes of graphs with degree one or two
LLimit law for the numbers of components of fixed sizesof graphs with degree one or two ´Elie de Panafieu ∗ Nicolas Broutin † April 25, 2019
In this note, we answer a question of Giardin`a, Giberti, van der Hofstad and Prioriello [3].We consider labelled simple graphs with all vertices of degree 1 or 2. We denote by G n ,n theset of such graphs with n vertices of degree 1, and n vertices of degree 2. Throughout thedocument, the size refers to the number of vertices. The random variable U j counts the numberof connected components of size j in a graph drawn uniformly at random from G n ,n . The mainresult is that for any fixed integer q ≥ α >
0, for n = 2 k is a large evennumber and n = (cid:98) αn / (cid:99) , the random vector ( U , . . . , U q ) has a Gaussian limit distributionas k → ∞ . In Section 4, we extend those results to multigraphs, with the distribution inducedby the configuration model. The vector ( U , . . . , U q ) has the same Gaussian limit distribution,while U follows a Poisson law. Theorem 1.
Let α > . For n an even let n = (cid:98) αn / (cid:99) . For j ≥ , let U j denote the numberof connected components of size j in a uniformly random graph from G n ,n . Then, for every j ,as n → ∞ along the even integers, E [ U j ] ∼ α j − (1 + α ) j − n and V ar( U j ) = O (cid:16) n (cid:17) . Furthermore, for any integer q ≥ , as n → ∞ along the even integers, the vector (cid:112) n / (cid:18) U j − α j − (1 + α ) j − n (cid:19) ≤ j ≤ q converges in distribution to a multivariate Gaussian N ( , H ( α )) , where the positive semi-definitematrix H ( α ) = ( H i,j ( α ) , ≤ i, j ≤ q ) is given by H i,j ( α ) = − α i + j − (1 + α ) i + j − (cid:18) i − − α )( j − − α ) α (1 + α ) (cid:19) + i = j α (cid:18) α α (cid:19) i − . Specifically, in order to prove the theorem, we show that in the neighborhood of the vector , the multivariate Laplace transform of the rescaled random variables converges point-wise tothe Laplace transform of a multivariate Gaussian distribution.In order to simplify the formulation of the lemmas, we introduce the uniform Landau notation f ( z, u ) = z → z O V ( g ( z ))which means that there exists two constants K and δ > u such that for all u in V and | z − z | ≤ δ , | f ( z, u ) | ≤ Kg ( z ) . As usual with the Landau notation, the limit z is often implicit when the context leaves noambiguity. This definition extends naturally to the “ small o ” Landau notation. ∗ RISC – Johannes Kepler University, Linz – Austria.
Email: depanafi[email protected] † Inria Paris–Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay – France.
Email: [email protected] a r X i v : . [ m a t h . C O ] D ec Expression of the generating function
To derive the limit distribution of the sizes of the components in G n ,n , we analyze its generatingfunction, defined below. Definition 1.
Let q ≥ be a natural number. Let u denote the vector ( u , . . . , u q ) , and G n ,n ( u ) denote the ordinary multivariate generating function of graphs in G n ,n where for j = 2 , , . . . , q ,the variable u j marks the number of connected components of size j . Therefore, the number ofgraphs in G n ,n with m j components of size j for all ≤ j ≤ q is [ u m . . . u m q q ] G n ,n ( u ) . The following notations will also prove useful throughout the paper.
Definition 2.
The sequence ( v n ,n ) and the multivariate generating functions Path( z, u ) and Cycle( z, u ) are defined by v n ,n = ( n + n )!2 n / ( n / , Path( z, u ) = 11 − z + q (cid:88) j =2 ( u j − z j − , Cycle( z, u ) = 12 log 11 − z − z − z q (cid:88) j =3 ( u j − z j j . In the following lemma, we derive a simple exact formula for the generating function of thegraphs in G n ,n with variables marking the components of sizes from 2 to q . Lemma 1.
The generating function G n ,n ( u ) is zero when n is odd, otherwise, it is given by G n ,n ( u ) = v n ,n [ z n ] e Cycle( z, u ) Path( z, u ) n / . (1) Proof.
A component of a graph in G n ,n is either a non-oriented path of size at least 2, or anon-oriented cycle of size at least 3. For simplicity, in the following we refer to those connectedgraphs as paths and cycles .Let us first consider vertices of degree 1 as unlabelled (we will label them later on). Thenumber of oriented paths with n vertices of degree 2 is then n ! and the number of non-orientedcycles n ! / (2 n ). Let the variable z mark the vertices of degree 2, then the exponential generatingfunctions of oriented paths and non-oriented cycles are respectively (cid:88) n ≥ n ! z n n ! = 11 − z , (cid:88) n ≥ n !2 n z n n ! = 12 log (cid:18) − z (cid:19) − z − z . Now for all 2 ≤ j ≤ q , we introduce the variable u j to mark the components of size j . Sincea path with n vertices of degree 2 is a connected component of size n + 2, in the generatingfunction of oriented paths, for j from 2 to q , the j th coefficient is multiplied by u j +2 . Similarly,the j th coefficient of the generating function of cycles is multiplied by u j . Finally, the generatingfunctions of oriented paths and non-oriented cycles, exponential with respect to z and ordinary2ith respect to all u j , arePath( z, u ) = 11 − z + q (cid:88) j =2 ( u j − z j − , Cycle( z, u ) = 12 log 11 − z − z − z q (cid:88) j =3 ( u j − z j j . Note that each path contains exactly two vertices of degree 1, while all the vertices of a cyclehave degree 2. Therefore, G n ,n is empty when n is odd and a graph in G n ,n is a set of n / ( n + n )! n ! n ! subsetsof size n of { , , . . . , n + n } . We then need to choose a permutation of size n to associate toeach vertex of degree 1 its label. (Note here that the generating function Path( z, u ) above counts oriented paths, so that the vertices of degree one are distinguished.) Furthermore, each non-oriented path matches exactly two oriented paths, so we replace Path( z, u ) with Path( z, u ) / G n ,n ( u ) is G n ,n ( u ) = ( n + n )! n ! n ! n ! n ![ z n ] e Cycle( z, u ) (Path( z, u ) / n / ( n / , which reduces to the result of the lemma.We will obtain in Lemma 2 a uniform asymptotic estimate of G n ,n ( u ) using the Fourier–Laplace method. In the next corollary, we reformulate the exact expression derived in Lemma 1to adopt a form that is a more adapted to this method. In particular, the coefficient extraction isreplaced by an integral, and the variables u , u , . . . , u q are considered as positive real numbers. Corollary 1.
For any ζ ∈ (0 , , and u in a neighborhood of , G n ,n ( u ) = v n ,n π Path( ζ, u ) n / ζ n (cid:90) π − π A ( θ, u ) e − φ ( θ, u ) n / dθ, where φ and A are defined by φ ( θ, u ) = log(Path( ζ, u )) − log(Path( ζe iθ , u )) + iαθ,A ( θ, u ) = exp (cid:16) Cycle( ζe iθ , u ) (cid:17) . Proof.
For every u , the generating function G ( z, u ) has radius of convergence 1. We rewrite thecoefficient extraction of Equation (1) as a Cauchy integral on a circle of radius ζ ∈ (0 , G n ,n ( u ) = v n ,n π (cid:90) π − π exp (cid:16) Cycle( ζe iθ , u ) (cid:17) Path( ζe iθ , u ) n / ( ζe iθ ) n dθ, = v n ,n π Path( ζ, u ) n / ζ n (cid:90) π − π A ( θ, u ) e − φ ( θ, u ) n / dθ, with φ and A defined as in the lemma.The Laplace method requires to locate the minimum of the function θ (cid:55)→ φ ( θ, u ) and thebehavior of A and φ in its vicinity. This information is derived in the following lemma.3 emma 2. Let α > , n be an even number and n the closest integer to αn / . Let ζ denotethe unique solution in (0 , of the equation ζ∂ z log(Path( ζ, u )) = α. (2) There exists a neighborhood V ⊂ R q> of , such that the functions φ and A satisfy the followingproperties for θ in a complex neighborhood of :1. uniformly for u ∈ V , it holds that φ ( θ, u ) = ∂ θ φ (0 , u ) θ O V ( θ ) ,A ( θ, u ) = A (0 , u ) + O V ( θ ) ,
2. for all u ∈ V , we have ∂ θ φ (0 , u ) > , and A (0 , u ) (cid:54) = 0 ,3. for all u ∈ V , the real part of φ ( θ, u ) is non-negative, Re( φ ( θ, u )) ≥ , with equality onlyat θ = 0 . Before proceeding to the proof, observe that the value of ζ in the statement actually dependson α and u . When needed, we shall write ζ u instead of ζ to avoid any ambiguity. Proof.
By definition, φ (0 , u ) = 0. We choose ζ such that ∂ θ φ ( ζ, u ) = 0 , which is equivalent with Equation (2). A simple computation reduces this last expression to ζ − ζ ) (cid:80) qj =3 ( u j − j − ζ j − − ζ + (1 − ζ ) (cid:80) qj =2 ( u j − ζ j − = α. (3)In particular, when u = , we have ζ = α α . When the components u , u , . . . , u q of u are positive numbers, the analytic function z (cid:55)→ z ∂ z Path( z, u )Path( z, u )has positive coefficients, so it is strictly increasing. Furthermore, using Expression (3), wesee that this function tends to 0 (resp. infinity) when z goes to 0 (resp. 1) along the real axis.Therefore, for any positive real numbers α and u , u , . . . , u q , Equation (2) has a unique solution ζ in (0 , u = , Equation (3) becomes ζ / (1 − ζ ) = α, which implies that ζ = α α . Equation (2) defines ζ implicitly as a function of α and u , and has a solution ζ for u = .Furthermore, the derivative with respect to ζ of the left-hand side of Equation (2) does notvanish at u = , since ∂ ζ (cid:18) ζ ∂ z Path( ζ, u )Path( ζ, u ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) u = = ∂ (cid:32) ζ ∂ (cid:0) (1 − ζ ) − (cid:1) (1 − ζ ) − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ = ζ = (1 + α ) . Therefore, according to the Implicit Function Theorem, for all positive α , there exists a neigh-borhood of on which the function u (cid:55)→ ζ is continuous and thus close to ζ .4y continuity, for any α >
0, there exists a neighborhood of (0 , ) where the function( θ, u ) (cid:55)→ | ∂ θ φ ( θ, u ) | is bounded. Using Taylor’s Theorem, we conclude that for every α >
0, there exists a neighbor-hood V of such that φ ( θ, u ) = ∂ θ φ (0 , u ) θ O V ( θ ) ,A ( θ, u ) = A (0 , u ) + O V ( θ ) . Since ∂ θ φ (0 , ) and A (0 , ) are non-zero (the first one is equal to α (1 + α ), the second anexponential), by a continuity argument, we can choose V small enough to ensure that ∂ θ φ (0 , u )is positive and A (0 , u ) does not cancel. In this section, we obtain the asymptotics for the coefficients of the generating function G n ,n ( u ).The result relies on the following technical lemma. The idea is very classical, but we coud not finda reference to this multidimensional version and we include a proof for the sake of completeness. Lemma 3.
Let us consider a real vector u in R d , a real neighborhood V ⊂ R q of u and (cid:15) > . Let A ( θ, u ) and φ ( θ, u ) denote two continuous complex functions satisfying the followingproperties:1. for all u ∈ V , the functions θ (cid:55)→ A ( θ, u ) and θ (cid:55)→ φ ( θ, u ) are analytic at the origin andhave a radius of convergence greater than (cid:15) ,2. uniformly for u ∈ V , φ ( θ, u ) = ∂ θ φ (0 , u ) θ O V ( θ ) ,A ( θ, u ) = A (0 , u ) + O V ( θ ) ,
3. for all u ∈ V , we have ∂ θ φ (0 , u ) > and A (0 , u ) (cid:54) = 0 ,4. for all u ∈ V , and θ ∈ [ − (cid:15), (cid:15) ] , the real part of φ ( θ, u ) is non-negative, Re( φ ( θ, u )) ≥ ,with equality only at θ = 0 .Then there exists a neighborhood W ⊂ V of u where (cid:90) (cid:15) − (cid:15) A ( θ, u ) e − nφ ( θ, u ) dθ = √ πA (0 , u ) (cid:113) n∂ θ φ (0 , u ) (1 + O W ( n − / )) . Proof.
We follow the same steps as in the proof of the
Large Power Theorem [2, Theorem VIII.8page 587], but all the intermediate results need to be uniform with respect to u . First, wereduce the domain of integration, then the integrand is approximated by a Gaussian integrandand finally the domain of integration is extended to R , and we replace the classic Gaussianintegral by its value. Reduction of the domain of integration.
Let C ⊂ V denote a compact set that containsa neighborhood of u . We first show that the main contribution of the integral I n ( u ) = (cid:90) (cid:15) − (cid:15) A ( θ, u ) e − nφ ( θ, u ) dθ θ = 0. Specifically, we introduce the integral˜ I n ( u ) = (cid:90) n − / − n − / A ( θ, u ) e − nφ ( θ, u ) dθ and prove that there exists a constant K independent of u such that I n ( u ) = ˜ I n ( u ) + O C (cid:16) e − Kn / (cid:17) . (4)We start with the inequality (cid:12)(cid:12)(cid:12) I n ( u ) − ˜ I n ( u ) (cid:12)(cid:12)(cid:12) ≤ (cid:90) n − / ≤| θ |≤ (cid:15) (cid:12)(cid:12)(cid:12) A ( θ, u ) e − nφ ( θ, u ) (cid:12)(cid:12)(cid:12) dθ (5)where the right-hand side is at most2 (cid:15) sup | θ |≤ (cid:15) u ∈ C | A ( θ, u ) | exp − n inf n − / ≤| θ |≤ (cid:15) u ∈ C Re ( φ ( θ, u )) . Since the function ( θ, u ) (cid:55)→ | A ( θ, u ) | is continuous, it reaches a finite maximum in the compactset [ − (cid:15), (cid:15) ] × C , so 2 (cid:15) sup | θ |≤ (cid:15) u ∈ C | A ( θ, u ) | = O C (1) . According to Assumptions 2 and 3,Re( φ ( θ, u )) = ∂ θ φ (0 , u ) θ O C ( θ ) , so n inf n − / ≤| θ |≤ (cid:15) u ∈ C Re ( φ ( θ, u )) = (cid:18)
12 inf u ∈ C ∂ θ φ (0 , u ) (cid:19) n / + O C ( n − / ) . We conclude that (cid:90) n − / ≤| θ |≤ (cid:15) (cid:12)(cid:12)(cid:12) A ( θ, u ) e − nφ ( θ, u ) (cid:12)(cid:12)(cid:12) dθ = O C (cid:16) e − Kn / (cid:17) where K = inf u ∈ C ∂ θ φ (0 , u ) >
0. Combined with Equation (5), this last result proves Equal-ity (4).
Approximation of the integrand.
We inject the expressions of A ( θ, u ) and φ ( θ, u ) fromAssumption 2 in the definition of ˜ I n ( u )˜ I n ( u ) = (cid:90) n − / − n − / ( A (0 , u ) + O C ( θ )) e − n∂ θ φ (0 , u ) θ / n O C ( θ ) dθ. Uniformly with respect to u ∈ C and θ ∈ [ − n − / , n − / ], we have e n O C ( θ ) = 1 + O u ∈ C, | θ |≤ n − / ( n − / ) ,A (0 , u ) + O C ( θ ) = A (0 , u ) (cid:16) O u ∈ C, | θ |≤ n − / ( n − / ) (cid:17) . Remark that this property holds because we reduced the domain of integration. We then obtain˜ I n ( u ) = (cid:90) n − / − n − / A (0 , u ) e − n∂ θ φ (0 , u ) θ / dθ (cid:16) O C ( n − / ) (cid:17) , I n ( u ) = A (0 , u ) (cid:113) n∂ θ φ (0 , u ) (cid:90) n / − n / e − t / dt (cid:16) O C ( n − / ) (cid:17) (6)after the linear change of variable t = θ (cid:113) n∂ θ φ (0 , u ) . Gaussian integral.
To conclude the proof, we quickly prove the classic fact (cid:90) n / − n / e − t / dt = √ π (cid:16) O ( e − n / / ) (cid:17) . Indeed, the complete Gaussian integral is (cid:82) ∞−∞ e − t / dt = √ π while (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ e − t / dt − (cid:90) n / − n / e − t / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ∞ n / te − t / dt = 2 e − n / / as soon as n / is greater than 1. Injecting this relation in Equations (6) and (4), we obtain I n ( u ) = √ πA (0) (cid:113) n∂ θ φ (0 , u ) (cid:16) O C ( n − / ) (cid:17) (cid:16) O C ( e − n / / ) (cid:17) + O C ( e − Kn / )which concludes the proof.The error term of the previous lemma could be improved up to O W ( n − ), but this wouldrequire more work. Actually, in the following we will simply use a o W (1) error term, which issufficient for our purpose and reduces the notations.Combining Corollary 1 and Lemmas 2 and 3, we obtain the asymptotics of G n ,n ( u ) for u , u , . . . , u q in small but fixed real neighborhood of 1, n even and 2 n /n close to a fixedpositive constant α . Corollary 2.
With the notations of Corollary 1 and Lemma 2, for all α > , there is a neigh-borhood W of such that, when n is even and n = (cid:98) αn / (cid:99) , G n ,n ( u ) = v n ,n √ π A (0 , u ) (cid:113) ∂ θ φ (0 , u ) n / ζ, u ) n / ζ n (1 + o W (1)) . We now exploit the expression in Corollary 2 to obtain the limit distribution of the vector( U , U , . . . , U q ) of counts of connected components of sizes 2 , , . . . , q in a graph drawn uniformlyat random from G n ,n . Lemma 4.
Let α > , t , . . . , t q > , n an even integer and n = (cid:98) αn / (cid:99) . For all ≤ j ≤ q ,let U j be the random variable counting the number of connected components of size j in a graphdrawn uniformly from G n ,n , and let V j denote the rescaled random variable V j = 1 (cid:112) n / (cid:18) U j − α j − (1 + α ) j − n (cid:19) . hen the limit when n goes to infinity of the multivariate Laplace transform of V , . . . , V q is lim n →∞ L ( V ) n ,n ( t , . . . , t q ) = e t ·H ( α ) · t where H ( α ) is a positive semi-definite symmetric matrix with rows and columns indexed from to q (i.e. its upper-left coefficient is H , ( α ) ) and defined by H i,j ( α ) = − α i + j − (1 + α ) i + j − (cid:18) i − − α )( j − − α ) α (1 + α ) (cid:19) + i = j α (cid:18) α α (cid:19) i − . As a consequence ( V , . . . , V q ) converges in distribution to the multivariate Gaussian N ( , H ( α )) .Proof. This lemma is a multivariate version of the
Quasi-Power Theorem of Hwang [4], alsoavailable in [2, Lemma IX.1 page 646], applied to a particular case. (Note however that themain point of Hwang’s theorem is the improvement on the rate of convergence; here, we onlyuse the same approach but do not try to obtain the best possible rate.) When t = ( t , . . . , t q ) isa vector, the notation e t denotes the vector ( e t , . . . , e t q ).Note first that the asymptotics of the Laplace transform L ( U ) n ,n ( t ) about t = 0 yield asymp-totics for the moments of the vector U = ( U , U , . . . , U q ). We have L ( U ) n ,n ( t ) = B ( t ) e n χ ( t ) / (1 + o W (1)) (7)where B ( t ) and χ ( t ) are defined by B ( t ) = A (0 , e t ) A (0 , ) (cid:115) ∂ θ φ (0 , ) ∂ θ φ (0 , e t ) χ ( t ) = log (cid:18) Path( ζ e t , e t )Path( ζ , ) (cid:19) − α log (cid:18) ζ e t ζ (cid:19) and are such that B ( ) = 1 and χ ( ) = 0. So, for 2 ≤ j ≤ q , E [ U j ] = ∂ t j L ( U ) n ,n ( t ) (cid:12)(cid:12)(cid:12) t =0 ∼ n ∂ t i χ ( ) V ar( U j ) = ∂ t j L ( U ) n ,n ( t ) − ( ∂ t j L ( U ) n ,n ( t )) (cid:12)(cid:12)(cid:12) t =0 ∼ n ∂ t j B ( ) ∂ t j χ ( ) + n ∂ t j χ ( ) . So E [ U j ] ∼ n ∂ t j χ ( ) / V ar( U j ) = O ( n ). So for the limit distribution of U =( U , U , . . . , U q ), the natural rescaling involves V = ( V j ) ≤ j ≤ q = (cid:32) U j − n ∂ t j χ ( ) / (cid:112) n / (cid:33) ≤ j ≤ q . It now suffices to obtain pointwise convergence of the corresponding Laplace transform L ( V ) n ,n ( t ) = E [ e tV ]. so that, for any t , L ( V ) n ,n ( t ) = E exp q (cid:88) j =2 U j t j (cid:112) n / e − √ n / ∇ χ ( ) · t = L ( U ) n ,n (cid:32) t (cid:112) n / (cid:33) e − √ n / ∇ χ ( ) · t .
8n the following, we write n = 2 k , for an integer k , and we want asymptotics as k → ∞ .However, by definition of G n ,n ( u ), L ( U ) n ,n ( t / √ k ) = G n ,n ( e t / √ k ) G n ,n ( ) . Now, for any fixed t , for all n large enough t / √ k is in a neighborhood of , and u = e t / √ k isin a neighborhood of . Thus, we can apply Lemma 2 to obtain the asymptotics for the Laplacetransform L ( V ) n ,n ( t ), as the number of nodes tend to infinity, and this uniformly with respectto t in a fixed neighborhood W of : L ( U ) n ,n ( t / √ k ) = B ( t / √ k ) e kχ ( t / √ k ) (1 + o W (1)) . (8)The multivariate Taylor expansion of χ ( t ) near t = is χ ( t ) = ∇ χ ( ) · t + 12 t · H · t + O ( (cid:107) t (cid:107) )where ∇ χ ( ) and H denote respectively the gradient and Hessian matrix of χ at . Following(7), we are interested in asymptotics of kχ ( t /k ), as k → ∞ : kχ (cid:18) t √ k (cid:19) = √ k ∇ χ ( ) · t + 12 t · H · t + O ( k − / (cid:107) t (cid:107) ) . The uniform convergence in Equation (7) allows us to apply this rescaling to the Laplace trans-form of U , . . . , U q L ( U ) n ,n (cid:18) t √ k (cid:19) = B (cid:18) t √ k (cid:19) e kχ ( t / √ k ) (1 + o W (1))Since B is continuous and its value at is 1, we can rewrite this expression as L ( U ) n ,n (cid:18) t √ k (cid:19) = e √ k ∇ χ ( ) · t e t ·H· t (1 + o W (1)) . (9)It follows that, in a fixed neighborhood of , we have L ( V ) n ,n ( t ) = e t ·H· t (1 + o W (1)) . (10)From (10) above, in order to complete the proof of the convergence in distribution of V =( V , V , . . . , V q ), it suffices to verify that the right-hand side above is the Laplace transform ofa multivariate Gaussian, which reduces to checking that the Hessian matrix H is positive semi-definite. To see that this is the case, it suffices to consider the cumulant generating functionlog L ( V ) n ,n ( t ), which is a convex function for every n and n [see, e.g., 1]. It follows that thequadratic form t H t is convex, so that the matrix H is positive semi-definite.Finally, we turn to the evaluation of ∇ χ ( ) and H . Remark that, although this is not explicitin the notation, both ∇ χ ( ) and H depend on α . By definition of the Gradient and the Hessianmatrix, with the convention that rows and columns are indexed from 2 to q , the j th componentof ∇ χ ( t ) is ∂ t j χ ( ) and the coefficient ( i, j ) of H is ∂ t i ∂ t j χ ( ). Since χ ( t ) = (cid:18) log (cid:18) Path( ζ u , u )Path( ζ , ) (cid:19) − α log (cid:18) ζ u ζ (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) u = e t , we have ∂ t j χ ( t ) = e t j ∂ u j (log(Path( ζ u , u )) − α log( ζ u )) | u = e t = e t j (cid:18) ∂ u j log(Path( z, u )) | z = ζ u + ( ∂ u j ζ u ) ∂ z log(Path( ζ u , u )) − αζ u ∂ u j ζ u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) u = e t .
9y the definition (2) of ζ u , ∂ z log (Path( ζ u , u )) − αζ u = 0 , so ∂ t j χ ( t ) = e t j ∂ u j log(Path( z, u )) | z = ζ e t , u = e t . (11)According to the value of ζ derived in Lemma 2 and the expression of Path( z, u ), we have ζ = α α , Path( ζ , ) = 1 + α,∂ u j Path( z, u ) = z j − . It follows that the j th component of the Gradient ∇ χ ( ) is ∂ t j χ ( ) = ∂ u j Path( ζ , u ) | u = Path( ζ , ) = α j − (1 + α ) j − . Now we compute the coefficient ( i, j ) of the Hessian matrix H . Let f ( z, u ) denote the function f ( z, u ) = log(Path( z, u )) = log (cid:32) − z + q (cid:88) j =2 ( u j − z j − (cid:33) , then by derivation of Equation (11), ∂ t i ∂ t j χ ( ) = ( ∂ u i ζ ) ∂ z ∂ u j f ( ζ , ) + ∂ u i ∂ u j f ( ζ , ) + i = j ∂ u i f ( ζ , ) . Deriving Equation (2) with respect to u i and rearranging the terms leads to ∂ u i ζ = − ζ ∂ u i ∂ z f ( ζ , ) ∂ z f ( ζ , ) + ζ ∂ z f ( ζ , ) , so ∂ t i ∂ t j χ ( ) = − ζ ∂ u i ∂ z f ( ζ , ) ∂ z f ( ζ , ) + ζ ∂ z f ( ζ , ) ∂ z ∂ u j f ( ζ , ) + ∂ u i ∂ u j f ( ζ , ) + i = j ∂ u i f ( ζ , ) , (12)where i = j denotes the indicator that i = j . Simple computations on the expression of f ( z, u )yield ∂ z f ( ζ , ) = 1 + α,∂ z f ( ζ , ) = (1 + α ) ,∂ u i f ( ζ , ) = α i − (1 + α ) i − ∂ u i ∂ z f ( ζ , ) = α i − (1 + α ) i − ( i − − α ) ,∂ u i ∂ u j f ( ζ , ) = − α i + j − (1 + α ) i + j − . Injecting those relations in Equation (12) leads to H i,j = − α i + j − (1 + α ) i + j − (cid:18) i − − α )( j − − α ) α (1 + α ) (cid:19) + i = j α (cid:18) α α (cid:19) i − , which completes the proof. 10 Configuration model
The multigraph process , also known as the uniform graph model , produces a random vertex-labelled multigraph with n vertices and m edges by drawing 2 m labelled vertices v w . . . v m w m uniformly and independently in [1 , n ], and adding to the multigraph the edges v i w i for i from 1to m : edge( G ) = { v i w i | ≤ i ≤ m } . When the set of degrees of the output is constrained, the multigraph process and the configura-tion model generate the same distribution on multigraphs. The number of sequences of vertices v w . . . v m w m that correspond to a given multigraph G is denoted by seqv( G )seqv( G ) = |{ v w . . . v m w m | { v i w i | ≤ i ≤ m } = edge( G ) }| . Observe that a multigraph G with m edges contains neither loops nor multiple edges (in whichcase it is simple ) if and only if its number of sequences of vertices seqv( G ) is equal to 2 m m !.For this reason, [5] introduced the compensation factor κ ( G ) = seqv( G )2 m m ! . The probability forthe multigraph process to produce a multigraph G in a family F is then proportional to κ ( G ).Therefore, we associate to F the generating function F ( z ) = (cid:88) G ∈F κ ( G ) z n ( G ) n ( G )! , where n ( G ) denotes the number of vertices of G .With this convention, the generating function of paths is the same in the multigraph processas before, because paths are simple multigraphs and their compensation factors are equal to one.The generating function of cycles becomesCycle( z, u ) = 12 log 11 − z + q (cid:88) j =1 ( u j − z j j . because in the multigraph process, a cycle of size one is a loop with compensation factor 1 / /
2. There is now one morevariable u that marks the components of size 1, which correspond to the loops.The set MG n ,n contains the multigraphs with n vertices of degree 1 and n of degree 2.It is equipped with the distribution induced by the compensation factors or, equivalently, by theconfiguration model. We redefine the generating function G n ,n ( u ) such that the sum of thecompensation factors of multigraphs in MG n ,n with m i components of size i for all 1 ≤ i ≤ q is [ u m . . . u m q q ] G n ,n ( u ) . With those new definitions, Lemma 1, Corollary 1, Lemma 2 and Corollary 2 hold true. Observethat ζ u has the same value as for simple graphs, because its implicit characterization (2) onlydepends on the generating function Path( z, u ), which is unchanged.For an integer q ≥
2, we define the new random variables V , . . . , V q as in Lemma 4, and set V = U . They are gathered into a vector V = ( V , . . . , V q ). Following the proof of the lemma,the multivariate Laplace transform of the variables V is L ( V ) n ,n ( t ) = A (0 , ( e t , , . . . , A (0 , ) e ( t ··· t q ) ·H· ( t ··· t q ) (1 + o W (1)) . By definition, A ( θ, u ) = exp (cid:0) Cycle( ζ u e iθ , u ) (cid:1) , so A (0 , u ) = exp
12 log (cid:18) − ζ u (cid:19) + q (cid:88) j =1 ( u j − ζ j u j ζ = α α . Therefore, injecting the relation A (0 , ( e t , , . . . , A (0 , ) = e α α ( t − in the expression of the Laplace transform, we obtain L ( V ) n ,n ( t ) = e α α ( t − e t ·H· t (1 + o W (1)) . It follows that the limit law of V is Poisson with parameter α α , while the limit law of V , . . . , V q is Gaussian. Theorem 2.
Let α > . For n an even let n = (cid:98) αn / (cid:99) . For j ≥ , let U j denote the numberof connected components of size j in a multigraph produced by the configuration model in G n ,n .Then, for every j ≥ , as n → ∞ along the even integers, E [ U j ] ∼ α j − (1 + α ) j − n and V ar( U j ) = O (cid:16) n (cid:17) , and E [ U ] ∼ α α and V ar( U ) ∼ α α . Furthermore, for any integer q ≥ , as n → ∞ along the even integers, the vector (cid:112) n / (cid:18) U j − α j − (1 + α ) j − n (cid:19) ≤ j ≤ q converges in distribution to a multivariate Gaussian N ( , H ( α )) , where the positive semi-definitematrix H is defined in Theorem 1, and U converges in distribution to a Poisson random variableof parameter α α . References [1] A. Dembo and O. Zeitouni.
Large Deviation Techniques and Applications . Springer, secondedition, 1998.[2] P. Flajolet and R. Sedgewick.
Analytic Combinatorics . Cambridge University Press, Cam-bridge, UK, 2009.[3] C. Giardin`a, C. Giberti, R. van der Hofstad, and M.L. Prioriello. Quenched central limittheorems for the Ising model on random graphs.
Submitted , 2014.[4] H.K. Hwang. On the convergence rates in the central limit theorems for combinatorialstructures.
European Journal of Combinatorics , 19:329—343, 1998.[5] S. Janson, D.E. Knuth, T. (cid:32)Luczak, and B. Pittel. The birth of the giant component.