Lattice paths with catastrophes

Transcription

1 Lattice paths with catastrophes Gascom 2016 Cyril Banderier and Michael Wallner Laboratoire d Informatique de Paris Nord, Université Paris Nord, France Institute of Discrete Mathematics and Geometry, TU Wien, Austria 1 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

2 Question from colleagues from queueing theory: You guys in combinatorics, can you do exact enumeration for the Bernoulli walk, for which one also allows at any time some catastrophe (=unbounded jump from anywhere directly to 0). Typical properties of such walks, distribution of patterns? How to generate them? Caveat: The limiting object is not Brownian motion at all (infinite negative drift!). [blackboard drawing with (un)bounded queues] Motivation: financial mathematics (catastrophe = bankrupt), or evolution of the queue of printer (catastrophe = reset of the printer), population genetics (species extinctions by pandemic),... Our answer: cute model, we have nice tools for this (and these tools offer for free a generalization to any set of jumps!). 2 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

3 Generating functions in combinatorics Full final version of the book for free at algo.inria.fr/flajolet 3 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes recursive combinatorial structures (lists, words, trees, maps, graphs, permutations, walks, tilings...) recurrences series complex analysis asymptotics typical behavior F (z) = a n z n In enumerative and analytic combinatorics, generating functions and their nature play a key rôle: rational functions ( walks on graphs) algebraic functions ( walks on N)...

4 The analytic combinatorics dogma of Flajolet & Sedgewick 4 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

5 World #1: Rational functions ( walks in finite graphs) 5 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

6 Rational functions Rational generating functions appear very often: Markov chains = automata theory = random walks in graphs = regular expressions (=closed by +, x, ) = system of linear equations = N-rational functions number of integer points of a curve in Z/p n Z, in polytopes. P-partitions Universal asymptotic behavior is known polar singularities, Perron Frobenius Gaussian Limit law (if technical conditions, strongly connected) if transitions Q then probabilities of patterns Q 6 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

7 Gaussian limit laws for patterns Gaussian limit law when R is a word (e.g. R = ababb) Nicodème & Salvy & Flajolet (2000) Motif Statistics : Gaussian limit law for R regular expression (tac)(c + g) (cat) P(z, u) = n,k 0 L = zt (u) L + 1 = P(z, u) = Pr(X n = k) u k z n Perron Frobenius: λ(u) of maximal modulus P(z, u) u 1 B(z, u) det(i zt (u)) c(u) 1 zλ(u) = Pr(X n = k) c(u)λ(u) n Hwang: quasi-powers theorem = Gaussian limit law E[X n ] = µn + c 1 + O(A n ) Var[X n ] = σ 2 n + c 2 + O(A n ) ( ) Xn µn Pr σ x 1 x n 2π exp t2 /2 dt 7 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes Non Gaussian law = [BaBoPoTa2012] Algebraic case = [BanderierDrmota13]

8 Borges theorem & Gaussian limit laws The Library of Babel, by the Argentinian writer Jorge Luis Borges ( ) A half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum. Theorem (Borges s theorem (a Flajolet meta-theorem )) In any large enough structure, any possible pattern will appear with non-zero probability, and its number of occurrences will follow a Gaussian limit law. 8 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

9 Borges theorem & Gaussian limit laws & Bible code This Borges theorem applies e.g. to patterns in words, trees, maps, graphs... This allows to refute the revelations of the stupid Bible Code another avatar: The book Moby Dick predicted in full details the accidental death of Lady Di (nice example due to Brendan McKay). 9 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

10 World #2: Algebraic functions ( walks on N) 10 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

11 From language theory to combinatorics in one theorem Theorem (Chomsky Schützenberger, 1963) Any context-free language has an algebraic generating function F (z) = f n z n n 0 where f n = # words of length n generated by the grammar. 11 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

12 From language theory to combinatorics in one theorem Theorem (Chomsky Schützenberger, 1963) Any context-free language has an algebraic generating function F (z) = f n z n n 0 where f n = # words of length n generated by the grammar. (if ambiguity: f n = # number of derivation trees) Proof: systems of rewriting rules = polynomial system = algebraic function S = 1 + aabs + bbas S(z) = 1 + za(z)zs(z) + zb(z)zs(z) A = 1 + aaba A(z) = 1 + za(z)za(z) B = 1 + bbab B(z) = 1 + zb(z)zb(z) 11 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

13 Algebraic functions are everywhere in combinatorics Survey by Stanley99, Bousquet-Mélou06, FlajoletSedgewick09. Quite often, algebraicity comes from: a tree-like structure (dissections of polygons: a result going back to Euler in 1751, one of the founding problems of analytic combinatorics!), a grammar description (polyominoes, directed animals, tiling problems, lattice paths, RNA in bioinformatics), the diagonal of a bivariate rational function f nn z n, solution of functional equations solvable by the kernel method: K(z, u)f (z, u) = sum of unknowns, e.g. for avoiding-pattern permutations (Knuth), p-automatic sequences, more mysterious reasons (Gessel walks, urn models). 12 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

14 Algebraic functions are everywhere in combinatorics Survey by Stanley99, Bousquet-Mélou06, FlajoletSedgewick09. Quite often, algebraicity comes from: a tree-like structure (dissections of polygons: a result going back to Euler in 1751, one of the founding problems of analytic combinatorics!), a grammar description (polyominoes, directed animals, tiling problems, lattice paths, RNA in bioinformatics), the diagonal of a bivariate rational function f nn z n, solution of functional equations solvable by the kernel method: K(z, u)f (z, u) = sum of unknowns, e.g. for avoiding-pattern permutations (Knuth), p-automatic sequences, more mysterious reasons (Gessel walks, urn models). Their asymptotics are crucial for establishing (inherent) ambiguity of context-free languages Flajolet87, for the analysis of lattice paths [BanderierFlajolet02], or planar maps [BaFlScSo01] / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

15 Coefficients of algebraic functions F (z) is algebraic= P Q[z, x] such that P(z, F (z)) = 0. Theorem (Newton 1676, Puiseux 1850) Any algebraic function has a Puiseux series expansion F (z) = k k 0 a k Theorem (Flajolet Odlyzko, 1990) Asymptotics is given via singularity analysis: ((z ρ) 1/r ) k F (z) (1 z/ρ) α (for z ρ) f n 1 Γ( α) ρ n n α 1 13 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

16 N-algebraic functions ( context-free grammars) Definition (N-algebraic functions) y 1 = P 1 (z, y 1,..., y d ). y d = P d (z, y 1,..., y d ) where each polynomial P i is such that [y j ]P i 1 and has coefficients in N. The power series y i solutions of this system are called N-algebraic functions. 14 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

17 Universality of square-root behavior Theorem (Drmota Lalley Woods, 1997) Any positive, strongly connected, algebraic system of equations has a critical exponent -3/2 (i.e., a Puiseux exponent 1/2). F (z) (1 z/ρ) 1/2 (for z ρ) f n 1 ( ) 1 n 2 n 3/2 π ρ 15 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

18 Universality of square-root behavior Theorem (Drmota Lalley Woods, 1997) Any positive, strongly connected, algebraic system of equations has a critical exponent -3/2 (i.e., a Puiseux exponent 1/2). F (z) (1 z/ρ) 1/2 (for z ρ) f n 1 ( ) 1 n 2 n 3/2 π ρ Example: B = z + B 2 (& works for any t-ary trees B = z + φ(b)) B(z) = b n z n = 1 ( 1 4z 2n ) n b n = 2 n + 1 4n 4 π n 3/2 n 0 15 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

19 Asymptotics of N-algebraic functions Theorem (BanderierDrmota2013) N-algebraic functions have a dyadic critical exponent, i.e. f n 1 Γ( α) ρ n n α 1 with α = 2 k for some k 1 ( or α = m2 k for some m 1 and some k 0). Theorem (Neat corollary) Planar maps and several families of lattice paths (like Gessel walks) are not N-algebraic (i.e., not generated by an unambiguous context-free grammar). 16 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

20 Proof for the dyadic critical exponents y 1 = P 1 (z, y 1, y 2, y 5 ) y 2 = P 2 (z, y 2, y 3, y 5 ) y 3 = P 3 (z, y 3, y 4 ) y 4 = P 4 (z, y 3, y 4 ) y 5 = P 5 (z, y 5, y 6 ) y 6 = P 6 (z, y 5, y 6 ) ,6 3,4 A positive system, its dependency graph, its reduced dependency graph. None of these graphs are here strongly connected: e.g. the state 1 is a sink; it is thus a typical example of system not covered by the Drmota Lalley Woods theorem, but covered by our new result. 17 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

21 Proof for the dyadic critical exponents y 1 = P 1 (z, y 1, y 2, y 5 ) y 2 = P 2 (z, y 2, y 3, y 5 ) y 3 = P 3 (z, y 3, y 4 ) y 4 = P 4 (z, y 3, y 4 ) y 5 = P 5 (z, y 5, y 6 ) y 6 = P 6 (z, y 5, y 6 ) ,6 Then, follow inductively the Puiseux exponents on the reduced dependency graph (it s a DAG, directed acyclic graph!): the singular behavior of any y(z) around ρ is either of algebraic type 1 3,4 y(z) = y(ρ) + c(1 z/ρ) 2 k + c (1 z/ρ) 2 2 k +, where c 0 and where k is a positive integer, or of polar-algebraic type y(z) = c (1 z/ρ) + c m2 k (1 z/ρ) + (m 1)2 k 18 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

22 Application to lattice paths bounded queue: rational world (known behavior, omitted here) unbounded queue: algebraic world (next slides) 19 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

23 Lattice paths with Catastrophes [Chang & Krinik & Swift: Birth-multiple catastrophe processes, 2007] [Krinik & Rubino: The Single Server Restart Queueing Model, 2013] Decomposition of a Dyck path with 3 catastrophes into 5 arches: A A A A A Reminiscent of rewriting rules from the ECO methodology of the Florentine school (Pinzani Pergola Barcucci et al., West, Banderier Bousquet-Mélou Denise Flajolet Gardy Gouyou-Beauchamps, Fédou, Garcia, Merlini,... ). 20 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

24 Lattice paths with Catastrophes [Chang & Krinik & Swift: Birth-multiple catastrophe processes, 2007] [Krinik & Rubino: The Single Server Restart Queueing Model, 2013] Decomposition of a Dyck path with 3 catastrophes into 5 arches: A A A A A Reminiscent of rewriting rules from the ECO methodology of the Florentine school (Pinzani Pergola Barcucci et al., West, Banderier Bousquet-Mélou Denise Flajolet Gardy Gouyou-Beauchamps, Fédou, Garcia, Merlini,... ). Caveat: Dyck + catastrophes = many direct decompositions! But for more general jumps the corresponding grammars are then too huge (billions of rules). So we present hereafter an approach which offers the advantage to solve also *Generalized* Dyck paths + catastrophes, in a very efficient and compact way, shortcutting the too huge grammar problem, and leading to asymptotics. 20 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

25 Generating functions A A A A A For walks with catastrophes + jumps encoded by P(u) = d i= c p i u i (hereafter c = 1 for readability, but similar proofs work for any c > 1) Theorem (Generating functions for lattice paths with catastrophes) f n,k = # catastrophe-walks of length n from altitude 0 to altitude k. F (z, u) = k 0 F k(z)u k = n,k 0 f n,ku k z n is algebraic and satisfies F (z, u) = zp 1 zp 1 z 2 p 1 ( 1 u 1(z) 1 zp(1) u 1(z) u 1(z)(1 zp 0 ) 1 zp 1 ) u u 1 (z) u(1 zp(u)) where u 1 (z) is the root of 1 zp(u) = 0 such that u 1 (z) 0 for z / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

26 Proof via variant of the kernel method Proof: either make all jumps encoded by P(u) (not jumping below 0) or a catastrophe (i.e., a direct jump to 0, when we are at altitude > 1): 1 }{{} empty walk + zp(u)f (z, u) }{{} jump from P(u) which is convenient to rewrite as F (z, u) = z p 1 u F 0(z) }{{} neg. jump at alt. 0 + zu 0 (F (z, 1) F 0 (z) F 1 (z)), }{{} catastrophe at altitude > 1 (1 zp(u))f (z, u) = 1 z p 1 u F 0(z) + z (F (z, 1) F 0 (z) F 1 (z)). This single equation contains 4 unknowns can we create additional equations? 22 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

27 (1 zp(u))f (z, u) = 1 z p 1 u F 0(z)+z (F (z, 1) F 0 (z) F 1 (z)). First, setting u = 1 leads to ( ) (1 zp(1))f (z, 1) = 1 zp 1 F 0 (z) + z (F (z, 1) F 0 (z) F 1 (z)). (1) Extracting [u 1 ] in the functional equation gives F 0 (z) zp 1 F 1 (z) zp 0 F 0 (z) = 1 + z (F (z, 1) F 0 (z) F 1 (z)). (2) Finally, if we solve 1 zp(u) = 0 with respect to u, we get 1 + d roots, one of them is such that u 1 (z) 0 for z 0. Then plugging this root u 1 (z) in the functional equation (***) leads to 0 = 1 z p 1 u 1 (z) F 0(z) + z (F (z, 1) F 0 (z) F 1 (z)). (3) We thus got a system of 3 equations involving the 3 unknowns F (z, 1), F 0 (z), and F 1 (z). Solving it gives F (z, u). 23 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

28 A bijective interpretation for the generating function Theorem (Generating functions for lattice paths with catastrophes) F (z, u) = S(z)M(z, u), with S(z) = Walks ending at altitude 0 and 1: 1 1 z(m(z, 1) E(z) M 1 (z)), F 0 (z) = S(z)E(z), and F 1 (z) = S(z)M 1 (z). Proof: Walk= Sequence(Arches ending with a catastrophe) Meander. Arches ending with cat = meander ending at > 1, followed by a catastrophe: A = z(m(z, 1) E(z) M 1 (z)). E (excursions), M (meanders), and M 1 (meanders ending at altitude 1) are classical walks with no catastrophe. u u 1(z) u(1 zp(u)) M(z, u) = is the GF of meanders [BanderierFlajolet2001] M 1 (z) = E(z)p 1 ze(z) 24 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

29 Corollary (Generating functions for Dyck paths with catastrophes) m n := # Dyck meanders with catastrophes of length n starting from 0. F (z, 1) = n 0 m n z n = where u 1 (z) = 1 1 4z 2 2z. z(u 1 (z) 1) z 2 + (z 2 + z 1)u 1 (z) = 1 + z + 2z2 + 4z 3 + O(z 4 ), e n := # Dyck excursions with catastrophes of length n ending at 0. F 0 (z) = n 0 e n z n = (2z 1)u 1 (z) z 2 + (z 2 + z 1)u 1 (z) = 1 + z2 + z 3 + 3z 4 + O(z 5 ). Moreover, e 2n is also the number of Dumont permutations of the first kind of length 2n avoiding the patterns 1423 and [Burstein05] 25 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

30 Bijection with Motzkin paths (solving conjectures by Alois P. Heinz, R. J. Mathar, and other contributors in the On-Line-Encyclopedia of Integer Sequences) 1 Dyck paths with catastrophes are Dyck paths with the additional option of jumping to the x-axis from any altitude h > 0; and 2 1-horizontal Dyck paths are Dyck paths with the additional allowed horizontal step (1, 0) at altitude 1. Bijection between Dyck arches with catastrophes and 1-horizontal Dyck arches: 6 1 Dyck arch with catastrophe 1-horizontal Dyck arch 26 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

31 Excursions Let e n be the number of Dyck paths with catastrophes of length n, and let h n be the number of 1-horizontal Dyck paths of length n. Then, one has Theorem (Bijection for Dyck paths with catastrophes) The number of Dyck paths with catastrophes of length n is equal to the number of 1-horizontal Dyck paths of length n: e n = h n. Proof: A first proof that h n = e n consists in using the continued fraction point of view (each level k + 1 of the continued fraction encodes the jumps allowed at altitude k): H(z) = n 0 h n z n = 1 1 z 2, 1 z z 2 C(z) where C(z) is the generating function of classical Dyck paths, C(z) = 1/(1 z 2 C(z)) / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

32 Enumeration of Dyck arches with catastrophes Proposition The number a n Dyck arches with catastrophes is a n = ( ) n 2 n 3 2. Proof: using M(z), E(z) and M 1 (z) the generating functions of classical Dyck walks for meanders, excursions, and meanders ending at 1, one has: A(z) = a n z n = z M(z) E(z) M 1(z) E(z) n 0 = 2z2 + z 1 + (1 2z) (1 + 2z) (1 z) 2 2(1 2z) = z 3 + z 4 + 3z 5 + 4z z z z 9 + O(z 10 ). ( ) ) n 1 n a n+2 = n 2 }{{ n [[n even]] n/2 + 1( 2 }}{{} meanders excursions 28 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

33 Asymptotics and limit laws Proposition (Asymptotics of Dyck paths with catastrophes) The number of Dyck paths with catastrophes e n, and Dyck meanders with catastrophes m n is asymptotically equal to ( )) ( )) 1 1 e n = C e ρ (1 n + O, m n = C m ρ (1 n + O, n n where ρ is the unique positive root of ρ 3 + 2ρ 2 + ρ 1, C e is the unique positive root of 31C 3 e 62C 2 e + 35C e 3, C m is the unique positive root of 31C 3 m 31C 2 m + 16C m 3. Proof: via singularity analysis. simple pole at ( ρ = /3 ( 93) /3 93) which is strictly smaller than 1/2 which is the dominant singularity of u 1 (z): F 0 (z) = C + O(1), for z ρ. 1 z/ρ 29 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

34 Supercritical composition Variant of the supercritical composition scheme [Proposition IX.6 Flajolet Sedgewick09], where a perturbation function q(z) is added. Proposition (Perturbed supercritical composition) If F (z, u) = q(z)g(uh(z)) where g(z) and h(z) satisfy the supercriticality condition h(ρ h ) > ρ g, that g is analytic in z < R for some R > ρ g, with a unique dominant singularity at ρ g, which is a simple pole, and that h is aperiodic. Furthermore, let q(z) be analytic for z < ρ h. Then the number χ of H-components in a random F n -structure, corresponding to the probability distribution [u k z n ]F (z, u)/[z n ]F (z, 1) has a mean and variance that are asymptotically proportional to n; after standardization, the parameter χ satisfies a limiting Gaussian distribution, with speed of convergence O(1/ n). Proof: As q(z) is analytic at the dominant singularity, it contributes only a constant factor. +Hwang s quasi-powers theorem on F (z, u) = g(uh(z)). 30 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

35 Proposition (Perturbed supercritical sequences) For a schema F = Q SEQ(uH) such that h(ρ h ) > 1, (with q(z) analytic for z < ρ, where ρ is the positive root of h(ρ) = 1), the number X n of H-components in a random F n -structure of large size n is, asymptotically Gaussian with E(X n ) n ρh (ρ), V(X n) n ρh (ρ) + h (ρ) ρh (ρ) 2 ρ 2 h (ρ) 3. Proof: previous Prop with g(z) = (1 z) 1 and ρ g replaced by 1. The second part results from the bivariate generating function F (z, u) = q(z) 1 (u 1)h m z m h(z), and from the fact, that u close to 1 induces a smooth perturbation of the pole of F (z, 1) at ρ, corresponding to u = / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

36 Average number of catastrophes Proposition The trivariate generating function of Dyck paths with catastrophes where z marks the length, v the number of catastrophes, and w the number of returns to zero with 1 jumps is given by C(z, v, w) := c nkl z n v k w l = n,k,l va(z) wã(z), where A(z) is the generating function of arches with catastrophes, and Ã(z) is the generating function of arches without catastrophes. Proof: Observe that counting catastrophes is equivalent with counting arches with catastrophes, whereas counting returns to zero with 1 jumps is equivalent to counting arches. Every Dyck path with catastrophes can be uniquely decomposed into a sequence of arches with catastrophes and arches without catastrophes. 32 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

37 Average number of catastrophes C(z, v) := C(z, v, 1) = E(z) 1 va(z)e(z), P (X n = k) = [zn v k ]C(z, v) [z n ]C(z, 1) Theorem The number of catastrophes of a random Dyck path with catastrophes of length n is normally distributed. The standardized version of X n, X n µn σ n, µ , σ , where µ is the unique positive real root of 31µ µ µ 3, and σ 2 is the unique positive real root of 29791σ σ σ , converges in law to a Gaussian variable N (0, 1) : ( ) lim P Xn µn n σ x = 1 x e y 2 /2 dy. n 2π 33 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

38 Average number of catastrophes (proof) Proof: The structure of the generating function C(z, v) is exactly the one needed in the perturbed supercritical sequence scheme + check the analytical conditions. The perturbing factor q(z) is in this case E(z) = u 1 (z)/z, while the inner function h(z) is equal to A(z)E(z). On the one hand, the radius of convergence of E(z) is equal to 1/2. On the other hand, this is also the radius of convergence ρ A of A(z). By Stirling s formula: a n K ( 2 n n ), This implies that A(z) K(1 2z) 1/2 for z 1/2. As E(1/2) = 1, one has lim z 1/2 A(z)E(z) =, this shows the supercriticality condition. Note that due to C(z, 1) = C(z), ρ is such that A(ρ)E(ρ) = 1. Finally, some resultant computations give the constants. 34 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

39 Average number of returns to zero From now on, let X n be the random variable, representing that Dyck path with catastrophes of length n consists of k returns to zero. In other words the probability is defined as P (X n = k) = [zn v k ]C(z, v, v) [z n ]C(z, 1, 1). Theorem The number of returns to zero of a random Dyck path with catastrophes of length n is normally distributed. The standardized version of X n, X n µn σ n, µ , σ , where µ is the unique positive real root of 31µ 3 62µ µ 3, and σ 2 is the unique positive real root of 29791σ σ 2 79, converges in law to a Gaussian variable N (0, 1). 35 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

40 Average number of returns to zero (proof) Proof: This theorem is a direct consequence of the supercritical composition scheme. In [BanderierFlajolet02] it was shown that E(z) is singular at 1/2, with the Puiseux expansion E(z) E(1/2) ε 0 1 2z, for z 1/2, where ε 0 is a computable constant. Thus, B(z) K 0 + K 1 1 2z for constant K0, K 1. So lim z 1/2 A(z) + B(z) =, which shows the supercriticality condition. The radius of convergence ρ is such that A(ρ) + B(ρ) = 1. Finally, some resultant computations give the constants. 36 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

41 Final altitude limit law Theorem The final altitude of a random Dyck path with catastrophes of length n admits a geometric limit distribution with parameter λ = v 1 (ρ) : P (X n = k) (1 λ) λ k. Proof: Let p n (u) := [zn ]F (z,u) [z n ]F (z,1) be the probability generating function of X n. Then, one directly shows that it converges pointwise for u (0, 1) to the probability generating function of the geometric type. By the fundamental continuity theorem [Feller68] for probability generating functions, this yields convergence in law of the corresponding discrete distribution. 37 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

42 Final altitude limit law (proof) Let us now fix u (0, 1) and treat is henceforth as a parameter. The probability generating function of X n is p n (u) = [zn ]S(z)M(z, u) [z n ]S(z)M(z, 1). By [Banderier Flajolet02], M(z, u) and M(z, 1) are singular at z = 1/2 > ρ (ρ is the singularity of S(z)). By [Flajolet Sedgewick09]: p n (u) M(ρ, u)[zn ]S(z) M(ρ, 1)[z n ]S(z) = M(ρ, u) M(ρ, 1). The branches allow us to factor the kernel equation into u(1 zp(u)) = zp 1 (u u 1 (z))(u v 1 (z)). Thus, M(ρ, u) = 1 ρp 1 (v 1 (ρ) u), the limit probability generating function of a geometric distribution. 38 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

43 Final altitude Figure: The limit law for the final altitude in the case of a jump polynomial P(u) = u u 3 + 2u 1. The picture shows a period 40 behavior, which is explained by a sum of 40 geometric-like basic limit laws. 39 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

44 Uniform random generation Generalized Dyck paths (meanders and excursions) can be generated by pushdown-automata/context-free grammars. dynamic programming approach, O(n 2 ) time and O(n 3 ) bits in memory. [Hickey and Cohen83]: context-free grammars. [Flajolet Zimmermann Van Cutsem 94]: the recursive method, a wide generalization to combinatorial structures, so such paths of length n can be generated in O(n ln n) average-time. [Goldwurm95] proved that this can be done with the same time-complexity, with only O(n) memory. [Duchon Flajolet Louchard Schaeffer04] : Boltzmann method. Linear average-time random generator for paths of length [(1 ɛ)n, (1 + ɛ)n]. [Banderier-Wallner16] : generating trees+holonomy theory O(n 2 ) time, O(1) memory. 40 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

45 Uniform random generation (generating tree+holonomy) Each transition is computed via P(jump j at altitude k, and length m, ending at 0 at length n) = f 0 m,k f k+j n (m+1),0 fn,0 0 Then, for each pair (i, k), theory of D-finite functions applied to our algebraic functions gives the recurrence for f m (computable in O( m) via an algorithm of [Chudnovsky & Chudnovsky 86] for P-recursive sequence). Possible win on the space complexity and bit complexity: computing the f m s in floating point arithmetic, instead of rational numbers (although all the f m are integers, it is often the case that the leading term of the P-recursive recurrence is not 1, and thus it then implies rational number computations, and time loss in gcd computations). Global cost n m=1 O( m)o( n m) = O(n 2 ) & O(1) memory is enough to output the n jumps of the lattice path, step after step, as a stream.. 41 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

46 Conclusion universality of the Gaussian limit law, and its counter-examples (rational/algebraic world). Universal behaviour: asymptotics f n KA n n α involving dyadic critical exponents α (coherent with [Banderier Drmota15]) Generalized Dyck paths with unbounded jumps can be exactly enumerated and asymptotically analyzed. Not Brownian limit objects: some more tricky fractal periodic geometrically amortized limit laws (and also Gaussian laws). We gave a uniform random generation algorithm. 42 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

47 Conclusion universality of the Gaussian limit law, and its counter-examples (rational/algebraic world). Universal behaviour: asymptotics f n KA n n α involving dyadic critical exponents α (coherent with [Banderier Drmota15]) Generalized Dyck paths with unbounded jumps can be exactly enumerated and asymptotically analyzed. Not Brownian limit objects: some more tricky fractal periodic geometrically amortized limit laws (and also Gaussian laws). We gave a uniform random generation algorithm. Old dream of Flajolet & Steyaert (70 s): input=description of an algorithm, output = cost of the algorithm on entries of size n. This dream only partially became true (ΛυΩ system + Maple [Salvy & Zimmermann]), but still remains the main challenge of the field, forcing us to dig further in effective multivariate analysis... The Flajolet Sedgewick Gascom motto: If you can specify it, you can analyze it, you can generate it! 42 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes