GAUSSIAN LAWS MICHÈLE SORIA. 8 october 2015 IHP Séminaire Flajolet IN ANALYTIC COMBINATORICS

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1 GAUSSIAN LAWS IN ANALYTIC COMBINATORICS MICHÈLE SORIA 8 october 205 IHP Séminaire Flajolet / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

2 Analytic Combinatorics Estimate properties of large structured combinatorial objects Symbolic Method Combinatorial Specification Equations on (univariate) Generating Fonctions encoding counting sequences Complex Analysis GF as Analytic Functions extract asymptotic information on counting sequences Properties of Large Structures laws governing parameters in large random objects similar to enumeration deformation (adjunction of auxiliary variable) perturbation (effect of variations of auxiliary variable) Gaussian Laws with Singularity Analysis + Combinatorial applications 2 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

3 Univariate Asymptotics Combinatorial Specification Univariate Generating Functions Asymptotic Enumeration Enumeration of general Catalan trees G(z) = g n z n, g n = # trees of size n G = SEQ(G) G(z) = ( ) G(z) = 2 4z ( ) G(z) = 2 ) 4z = g n+ = 4 n n 3/2 n+ ( 2n n π stirling approx. n! = 2πn(n/e) n ( + 0(/n)) z G(z) G(z) has a singularity of square-root type at ρ = 4 = g n c 4 n n 3/2 Singularity Analysis 3 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

4 Bivariate Asymptotics Combinatorial Specification Bivariate Generating Functions Asymptotic Behavior of Parameters Number of leaves G(z, u) = g n,k z n u k, g n,k = # trees of size n with k leaves G = + SEQ (G ) G(z, u) = zu + zg(z, u) G(z, u) G(z, u) = 2 ( + (u )z 2(u + )z + (u ) 2 z 2 ) = g n,k = n ( n k )( n 2 ) k Mean value µ n = n/2 ; standard deviation σ n = O( n) for k = n 2 + x n, g n,k g n n σ n 2π e x 2 2σn 2 4 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

5 Bivariate Asymptotics Combinatorial Specification Generating Functions Bivariate Asymptotic Behavior of Parameters G(z, u) = 2 ( + (u )z 2(u + )z + (u ) 2 z 2 ) = g(z, u) h(z, u) z ρ(u), ρ(u) = ( + u) 2 Probability generating function p n (u) = k g n,k g n u k = [zn ]G(z,u) [z n ]G(z,) p n (u) = ( ) n ρ(u) ( + o()) ρ() Perturbation of Singularity Analysis + Quasi-Powers approximation Central Limit Theorem 5 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

6 TOOLS Generating Functions and Limit Laws Bivariate Generating Functions Continuous Limit Laws Continuity Theorem for Characteristic functions Quasi-Powers Theorem Central Limit Theorem Quasi-Powers Singularity Analysis Asymptotic enumeration Perturbation of Singularity Analysis 6 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

7 Bivariate Generating Functions and Limit Laws Combinatorial class A Uniform Model Size. : A N Counting Generating Function A(z) = α A z α = n a nz n Parameter χ : A N Bivariate Generating Function A(z, u) = α A uχ(α) z α = n,k a n,ku k z n Random Variables (X n ) n 0 P(X n = k) = a n,k a n Probability Generating Function p n (u) k P(X n = k)u k = E(u X ) = [zn ]A(z,u) [z n ]A(z) Mean value µ n E(X n ) = p n() Variance σ 2 n = p n () µ 2 n + µ n Question : Asymptotic behavior of X n? moments, limit law (density, cumulative), tails of distribution Answer by Analytic Combinatorics : Evaluate [z n ]A(z, u) in different u-domains 7 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

8 Examples of discrete Limit Law µ n = O() and σ 2 n = O() discrete limit law : k, P(X n = k) n P(X = k), X discrete RV Ex. root degree distribution in Catalan trees : Geometric 8 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

9 Examples of continuous Limit Law µ n and σ 2 n Quicksort : number of comparisons 0 < N 50 Normalized variable X n = X n µ n σ n continuous limit law 9 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

10 Continuous Limit Laws Y Continuous Random Variable Distribution Function F Y (x) := P(Y x) Characteristic function (Fourier Transform) ϕ Y (t) = E(e ity ) F Y (x) differentiable : density f Y (x) := F Y (x) (X n ) sequence of (norm. combinatorial) Random Variables X n = Xn µn σ n Weak Convergence X n Y x, lim n F X n (x) = F Y (x) Continuity Theorem (Lévy 922) Local Limit Law X n Y P(X n = µ n + xσ n ) = σ n f (x) X n Y ϕ X n ϕ Y Xn µn µn Xn µn it it it it ϕ X n (t) = E(e σn ) = e σn E(e σn ) = e σn p n (e it σn ) 0 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

11 Gaussian Limit Laws N (0, ) N (0, ) Distribution Function F (x) = 2π x e t2 2 Probability density f (x) = 2π e x2 2 Characteristic Function ϕ(t) = e t2 2 dt Number of a s in words on {a, b} of length n Bivariate GF F(z, u) = z zu ( Binomial Distribution P(X n = k) = n ) 2 n k / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

12 Central Limit Theorem Central Limit Theorem (Demoivre, Laplace, Gauss) Let T,..., T n be i.i.d RV, with finite mean µ and variance σ 2, and let S n = T i. The normalized variable S n weekly converges to a normalized Gaussian distribution : S n S n nµ σ n N (0, ) Proof by Levy s theorem : characteristic function E(e itsn ) = (φ T (t)) n, with φ T (t) = + itµ t 2 2 (µ2 + σ 2 ) + o(t 2 ), t 0 Sn nµ it σ n nµ it φ S n (t) = E(e ) = e σ it n E(e σ it n ) = e σ n (φt ( t ( σ n ))n ) log φ S n (t) = it nµ σ + n log + itµ n σ t 2 n 2σ 2 n (µ2 + σ 2 ) + o( t 2 n ) = t O( Thus φ S n (t) e t2 2 n ) Xn nµ φ S n (t) = p n (u) = p n (u) u = e it σn : u when n 2 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

13 Quasi-Powers Theorem Example Cycles in Permutations P = SET( CYCLE(Z)) F (z, u) = exp ( ) u log z F (z, u) = n,k f n,ku k zn n! = n p n(u)z n with p n (u) = P(X n = k)u k F (z, u) = ( z) u = ( ) n+u n z n p n (u) = u(u + )... (u + n ) n! p n (u) = ( Γ(u) (eu ) log n + O Theorem (Goncharov 944) = Γ(u + n) Γ(u)Γ(n + ) ( )), uniform for u n The number of cycles in a random permutation is asymptotically Gaussian, with µ n = H n and σ n = log n + o(). 3 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

14 Quasi-Powers Theorem Quasi-powers Theorem (H-K Hwang 994) X n non-negative discrete random variables with PGF p n (u). Assume that, uniformly for u (u in a fixed neighbourhood of ) p n (u) = A(u)B(u) λn ( + O( κ n )), with λ n, κ n, n with A(u), B(u) analytic at u =, A() = B() = and "variability conditions" on B(u). Then Mean E(X n ) = λ n B () + A () + O( κ n ), Variance Var(X n ) = λ n (B () + B () B 2 ()) +... Distribution asymptotically Gaussian : P(X n < µ n + xσ n ) x e t2 2 dt 2π ( Speed of Convergence O κ n + βn ) Proof : analyticity + uniformity error terms transfer to differentiation. Like in CLT only local properties near u = are needed since λ n. Normalization, Characteristic function + Levy Theorem 4 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

15 Other Properties Local Limit Theorem (H-K Hwang 994) If the Quasi-Powers approximation holds on the circle u = then σ n P(X n = µ n + xσ n ) = e x2 2 2π Large Deviations Theorem (H-K Hwang 994) If the Quasi-Powers approximation holds on an interval containing then log P(X n < xλ n ) λ n W (x) + O() where W (x) = min log(b(u)/u x ) 5 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

16 Singularity Analysis Theorem : Singularity Analysis (Flajolet-Odlyzko 90) F(z) = f n z n analytic function at 0, unique dominant sigularity at z = ρ, and F(z) analytic in some -domain ( -analytic) and locally, then F(z) ( z/ρ) α α R Z f n ρ n n α Γ(α) Cauchy coefficient integral : [z n ] F(z) = F (z) dz 2iπ γ z n+ F alg-log singularity F(z) ( z/ρ) log β α ( z/ρ) f n ρ n n α Γ(α) log β n Hankel Contour 6 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

17 Perturbation of Singularity Analysis Robust : contour integrals uniform error terms for parameters p n (u) = [zn ]F(z, u) [z n ]F (z, ) u parameter Sing. of F (z, ) Sing. of F(z, u) for u Movable singularities F (z, u) ( z/ρ(u)) α p n(u) ( ) n ρ(u), uniform for u. ρ() = Gaussian limit law, with a mean and variance of order n. Variable exponent F (z, u) ( z/ρ) α(u) p n(u) n α(u) α(), uniform for u = Gaussian limit law, with a mean and variance of order log n. 7 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

18 ANALYTICO-COMBINATORIAL SCHEMES Perturbation of Meromorphic Asymptotics Meromorphic functions Linear systems Perturbation of Singularity Analysis Alg-Log Scheme : movable singularities Algebraic Systems Exp-Log Scheme : variable exponent Differential equations 8 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

19 MEROMORPHIC SCHEMES 9 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

20 Example : Eulerian numbers Rises in permutations F(z, u) = u( u) e (u )z u F(z, ) = z ρ(u) = log u + 2ikπ u u ρ = [z n ]F(z, u) = F(z, u) dz 2iπ z =/2 z n+ uniformly = (ρ(u)) n + O(2 n ) Quasi-Powers Theorem = Number of rises in permutations Gaussian limit law µ n n 2, σ2 n n 2 20 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

21 Meromorphic Scheme : Theorem Meromorphic Theorem (Bender 93) Then F(z, u) analytic at (0, 0), non negative coefficients. F(z, ) single dominant simple pole at z = ρ. B(z, u) F(z, u) =, for z < ρ + ε and u < η C(z, u) B(z, u), C(z, u) analytic for u < η, z < ρ + ε (ρ simple zero of C(z, ) and ρ(u) analytic u, C(u, ρ(u) = 0)) Non degeneracy + variability conditions Gaussian limit law, mean and variance asymptotically linear : N (c n, c 2 n) Speed of Convergence O(n /2 ) Proof : f n (u) = Residue + exponentially small uniform for u 2 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

22 Applications : Supercritical Sequences Supercritical Composition Substitution F = G(H) f (z) = g(h(z)), h(0) = 0 τ h = lim z ρ h(z). The composition is supercritical if τ h > ρ g h Supercritical Sequences F = SEQ(H) f (z) = h(z), τ h > The number of components in a random supercritical sequence of size n is asymptotically Gaussian, with µ n =, where ρ is the positive root of h(ρ) =, and σ 2 (n) = c ρ n n ρh (ρ) blocks in a surjection F(z, u) = u(e z ) parts in an integer composition F(z, u) = uh(z) cycles in an alignment F (z, u) = u log( z), h(z) = i I zi 22 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

23 Other applications Occurrences of a fixed pattern of length k in words F(z, u) = (u )(c(z) ) 2z (u )(z k +( 2z)(c(z) )) Parallelogram polyominoes of area n perimeter width height F (z, u) = J (z,u) J 0 (z,u) Bessel Functions GCD of polynomials over finite field P monic polynomials of d o n in F p (z) : P(z) = pz steps in Euclid s algorithm u 0 = q u + u 2... u h 2 = q h u h + u h u h = q h u h + 0 F (z, u) = ug(z) P(z) = u p(p )z p(z) pz Gaussian limit law with mean and variance asymptotically linear 23 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

24 Linear Systems Theorem Gaussian Limiting Distribution Bender 93 Y(z, u) = V(z, u) + T(z, u).y(z, u) each V i and T i,j polynomial in z, u with non negative coefficients. Technical conditions : irreducibility, aperiodicity, unicity of dominant pole of Y (z, ). Then Limiting distribution of the additional parameter u : N (c n, c 2 n) Applications Irreducible and aperiodic finite Markov chain, after n transitions, number of times a certain state is reached asymptotically Gaussian. Set of patterns in words : number of occurrences asymptotically Gaussian 24 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

25 ALG-LOG SCHEMES 25 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

26 Example : leaves in Catalan Trees G(z, u) = zu + F(z, u) = G(z, u 2 ) satisfies F(z, u) = z(u2 ) 2 F (z, u) = A(z, u) B(z, u)( [z n ]F (z, u) = b u ρ(u) n n 3/2 2 π ( + O ( n )) Singularity Analysis = uniformity for u zg(z, u) G(z, u) z( u) 2 z( + u) 2 2 z ρ(u) )/2 with ρ(u) = ( + u) 2 p n (u) ( ρ(u) ρ() ) n Quasi-Powers Theorem Gaussian limit law mean and variance linear u [ 2, 3 2 ] 26 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

27 Alg-Log Scheme : Theorem Algebraic Singularity Theorem (Flajolet-S. 93, Drmota 97) Then F(z, u) analytic at (0, 0), non negative coefficients. F(z, ) single dominant alg-log singularity at z = ρ For z < ρ + ε and u < η F(z, u) = A(z, u) + B(z, u)c(z, u) α, α R Z A(z, u), B(z, u), C(z, u) analytic for u < η, z < ρ + ε Non degeneracy + variability conditions. Gaussian limit law, mean and variance asymptotically linear : N (c n, c 2 n) Speed of Convergence O(n /2 ) Proof : Uniform lifting of univariate SA for u Implicit Function Theorem :ρ(u) ; C(u, ρ(u) = 0) analytic u, + Quasi-Powers Theorem 27 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

28 Applications Functional equation (Bender, Canfield, Meir-Moon) y(z) = Φ(z, y(z)) Φ(z, y) has a power series expansion at (0, 0) with non-negative coefficients, non linear (Φ yy (z, y) 0), and well-defined (Φ z (z, y) 0). Let ρ > 0, τ > 0 such that τ = Φ(ρ, τ) and = Φ y (ρ, τ). Then g(z), h(z) analytic functions such that locally y(z) = g(z) h(z) z/ρ with g(ρ) = τ and h(ρ) 0. Simple family of trees Y (z) = zφ(z, Y (z)) Catalan, Cayley, Motzkin,... number of leaves number of occurrences of a fixed pattern number of nodes of any fixed degree Gaussian limiting distribution, mean and variance linear. 28 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

29 Random Mappings { M = SET(CYCLE(T )) T = SET(T ) ( ) M(z) = exp log T (z), T (z) = ze T (z) number of leaves T (z) = 2( ez) + sot. number of nodes with a fixed number of predecessors Gaussian limiting distribution, with mean and variance asymptotically linear. 29 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

30 other applications Extends to Pólya operators : Otter Trees : B(z) = z + 2 B2 (z) + 2 B(z2 ), General non plane trees : H(z) = z exp( H(z k ) k ) number of leaves, number of nodes with a fixed number of predecessors asymptotically Gaussian Characteristics of random walks in the discrete plane : number of steps of any fixed kind, number of occurrences of any fixed pattern asymptotically Gaussian Planar maps : number of occurrences of any fixed submap asymptotically Gaussian 30 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

31 Algebraic Systems Theorem Gaussian Limiting Distribution Drmota, Lalley Y(z) = P(z, Y(z), u) positive and well defined polynomial (entire) system of equations, in z, Y, u which has a solution F(z, u). Strongly connected dependency graph (non linear case), locally f j (z, u) = g j (z, u) h j (z, u) z/ρ(u) g j (z, u), h j (z, u), ρ(u) analytic non zero functions. Limiting distribution of the additional parameter u : N (c n, c 2 n) General dependancy graph Drmota, Banderier 204 Gaussian Limiting Distribution when all strongly connected components have different radius of convergence. 3 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

32 Algebraic Systems : Applications number of independent sets in a random tree number of patterns in a context-free language Gaussian limiting distribution, mean and variance asymptotically linear. Non-crossing familiy : trees, forests, connect. or general graphs number of connected components number of edges Forests F 3 (z) + (z 2 z 3)F 2 (z) + (z + 3)F = 0 irreducible aperiodic system F = + zfu U = + UV V = zfu 2 = square-root type singularity 32 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

33 EXP-LOG SCHEMES 33 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

34 Example : cycles in permutations ( ) F (z, u) = exp u log z = ( z) u p n (u) = [z n ( ]F(z, u) = nu Γ(u) + O( n )) Singularity Analysis uniformity for u Stirling numbers of st kind Quasi-Powers approximation with β n = log n p n (u) Γ(u) (exp(u ))log n Number of cycles in a random permutation asymptotically Gaussian, with µ n and σ 2 n equivalent to log n 34 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

35 Exp-log Scheme : Theorem Variable exponent Theorem (Flajolet-S. 93) Then F(z, u) analytic at (0, 0), non negative coefficients. F(z, ) single dominant alg-log singularity at z = ρ For z < ρ + ε and u < η F(z, u) = A(z, u) + B(z, u)c(z) α(u), α() R Z A(z, u), B(z, u) analytic for u < η, z < ρ + ε ; α(u) analytic for u < η ; C(z) analytic for z < ρ + ε and unique root C(ρ) = 0. Non degeneracy + variability conditions. Gaussian limit law, mean and variance asymptotically logaritmic : N (α () log n, (α () + α ()) log n) Speed of Convergence O((log n) /2 ) Proof : Singularity Analysis + Quasi-Powers Theorem 35 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

36 Number of components in a set G of logarithmic type : G(z) singular at ρ, and locally G(z) = k log z/ρ + λ + O(log 2 ( z/ρ)) Labelled exp-log schema F = SET( G) F (z, u) = exp(ug(z)) F (z, u) Number of components N (k log n, k log n) ( z/ρ) uk Unlabelled exp-log schem F = MSET( G) or F = PSET( G) ( ) ( u k ) ( u) k F (z, u) = exp k G(zk ) or F (z, u) = exp G(z k ) k ρ < = F(z, u) = H(z, u) exp(ug(z)) H(z, u) analytic u < η, z < ρ + ε Number of components N (k log n, k log n) 36 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

37 Connected components in Random Mappings Random Mappings (labelled) { M = SET( CYCLE(T )) T = SET(T ) { ( ) M(z) = exp log T (z) T (z) = ze T (z) T (z) = 2( ez) + sot. Connected components M(z, u) = ( T (z)) u (2( ez)) u 2 Gaussian limiting law mean and variance 2 log n. Random Mapping Patterns (unlabelled) MP = MSET( L) L = CYCLE(H) H = MSET(H) ( ) MP(z) = exp L(z k ) k L(z) = φ(k) k log H(z k ) H(z) = z exp( H(z k ) k ) H(z) = γ ( z/η) + sot., η < Connected components MP(z, u) c u ( z/η) u 2 Gaussian limiting law mean and variance 2 log n. 37 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

38 Irreducible factors in a polynomial d o n in F p P monic polynomials of d o n in F p (z) : Polar singularity /p P n = p n, P(z) = Unique factorization property : pz P = MSET(I) P(z) = exp( I(z k ) ) k Mobius inversion : I(z) = k µ(k) k log(p(z k )) Logarithmic type : I(z) = log pz + k 2 µ(k) k log(p(z k )) Thus number of irreducible factors asymptotically Gaussian with mean and variance log n. (Erdös-Kac Gaussian Law for the number of prime divisors of natural numbers.) 38 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

39 Linear Differential Equations node level in BST Binary Search Trees (increasing trees) F = + Z F F F(z) = + Internal nodes z 0 F 2 (t)dt z F (z, u) = + 2u F (t, u) dt 0 t F z(z, u) = 2u F (z, u), with F(0, u) = F(z, u) = z ( z) 2u Singularity analysis ( [z n ]F(z, u) = n2u Γ(2u) + O( n )), uniform for u Distribution of depth of a random node in a random Binary Search Trees asymptotically Gaussian, with mean and variance 2 log n 39 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

40 Linear Differential Equations Linear differential equations (Flajolet-Lafforgue 94) Then F(z, u) analytic at (0, 0), non negative coefficients. F(z, ) single dominant alg-log singularity at ρ : f n Cρ n n σ F(z, u) satisfies a linear differential equation a 0 (z, u)f (r) (z, u) + a (z,u) (ρ z) F (r ) ar (z,u) (z, u) (ρ z) F(z, u) = 0 r a i (z, u) analytic at z = ρ Non degeneracy + variability conditions. Indicial polynomial J(θ) = a 0 (ρ, )(θ) r + a (ρ, )(θ) r a (ρ, ) unique simple root σ > 0. (θ) r = θ(θ )... (θ r + ) Gaussian limit law, mean and variance asymptotically logarithmic : N (c log n, c 2 log n) 40 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

41 Node level in quadtrees F(z, u) = + 4u I. JF(z, u) z I[g](z) = g(t) dt 0 t z dt J[g](z) = g(t) 0 t( t) Indicial polynomial J(θ, u) = θ 2 4u root σ(u) = 2 u Depth of a random external node in a random quadtree asymptotically Gaussian, mean and variance logn 4 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

42 Non Linear Differential Equations? Binary Search Trees, parameter χ, F (z, u) = T λ(t )uχ(t ) z T, where λ(t ) = V T V Paging of BST : index + pages of size b F z(z, u) = uf(z, u) 2 + ( u) d ( ) z b+, F (0, u) = dz z Occurences of a pattern P in BST F z(z, u) = F(z, u) 2 + P λ(p)(u )z P, F(0, u) = Riccati equations : Y = ay 2 + by + c W = AW + BW with Y = W Poles and movable singularity ρ(u) analytic for u Gaussian limit with mean and variance cn Varieties of increasing trees F = Z φ(f) F z(z) = φ(f(z)), F (0) = 0 Nodes fixed degree : Gaussian limit law with mean and variance cn aw 42 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics

43 Conclusion Analytic Combinatorics methods : Combinatorial Decomposability + Strong Analycity + Smooth Singulatity Perturbation = Gaussian Laws + Local limits + Large deviations Also Gaussian limit laws with analytic perturbation of Saddle-point method and Sachkov Quasi-Powers Discontinuity of singularity for u (confluence,...) non Gaussian continuous limit laws : Rayleigh, Airy,... Beyond the scope of Analytic Combinatorics : functional limit theorems (Probabilistic approach ) 43 / 43 Michèle Soria Gaussian Laws in Analytic Combinatorics