Lattice paths with catastrophes Gascom 2016 Cyril Banderier and - - PowerPoint PPT Presentation

Lattice paths with catastrophes

Gascom 2016 Cyril Banderier and Michael Wallner

Laboratoire d’Informatique de Paris Nord, Universit´ e Paris Nord, France Institute of Discrete Mathematics and Geometry, TU Wien, Austria

1 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

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),

• r 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!).

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Generating functions in combinatorics

recursive combinatorial structures (lists, words, trees, maps, graphs, permutations, walks, tilings...) recurrences series complex analysis asymptotics typical behavior F(z) =

• anzn

In enumerative and analytic combinatorics, generating functions and their nature play a key rˆ

• le:

rational functions (≈ walks on graphs) algebraic functions (≈ walks on N) . . . ♥♥♥ Full final version of the book for free at algo.inria.fr/flajolet ♥♥♥ http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf

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The analytic combinatorics dogma of Flajolet & Sedgewick

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World #1: Rational functions (≈ walks in finite graphs)

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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/pnZ, 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

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Gaussian limit laws for patterns

Gaussian limit law when R is a word (e.g. R = ababb) Nicod` eme & Salvy & Flajolet (2000) “Motif Statistics”: Gaussian limit law for R regular expression (tac)(c + g)∗(cat) P(z, u) =

• n,k≥0

Pr(Xn = k) ukzn

• L = zT(u)

L + 1 = ⇒ P(z, u) = B(z, u) det(I − zT(u)) Perron–Frobenius: ∃′λ(u) of maximal modulus P(z, u) ∼u→1 c(u) 1 − zλ(u) = ⇒ Pr(Xn = k) ∼ c(u)λ(u)n Hwang: quasi-powers theorem = ⇒ Gaussian limit law E[Xn] = µn + c1 + O(An) Var[Xn] = σ2n + c2 + O(An) Pr

Xn − µn

σ√n ≤ x

• →

1 √ 2π

x

−∞

exp−t2/2 dt

Non Gaussian law = [BaBoPoTa2012] Algebraic case = [BanderierDrmota13] 7 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

Borges’ theorem & Gaussian limit laws

”The Library of Babel”, by the Argentinian writer Jorge Luis Borges (1899-1986) ”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.

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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).

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World #2: Algebraic functions (≈ walks on N)

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From language theory to combinatorics in one theorem

Theorem (Chomsky–Sch¨ utzenberger, 1963) Any context-free language has an algebraic generating function F(z) =

• n≥0

fnzn where fn = # words of length n generated by the grammar.

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From language theory to combinatorics in one theorem

Theorem (Chomsky–Sch¨ utzenberger, 1963) Any context-free language has an algebraic generating function F(z) =

• n≥0

fnzn where fn = # words of length n generated by the grammar. (if ambiguity: fn = # number of derivation trees)

Proof: systems of rewriting rules = polynomial system = algebraic function       

S = 1 + aAbS + bBaS A = 1 + aAbA B = 1 + bBaB

      

S(z) = 1 + zA(z)zS(z) + zB(z)zS(z) A(z) = 1 + zA(z)zA(z) B(z) = 1 + zB(z)zB(z)

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Algebraic functions are everywhere in combinatorics

Survey by Stanley99, Bousquet-M´ elou06, 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 fnnzn, 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).

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Algebraic functions are everywhere in combinatorics

Survey by Stanley99, Bousquet-M´ elou06, 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 fnnzn, 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]...

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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≥k0

ak

• (z − ρ)1/rk

Theorem (Flajolet–Odlyzko, 1990) Asymptotics is given via singularity analysis: F(z) ∼ (1 − z/ρ)α (for z ∼ ρ) ⇐ ⇒ fn ∼ 1 Γ(−α)ρ−nn−α−1

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N-algebraic functions (≈ context-free grammars)

Definition (N-algebraic functions)

      

y1 = P1(z, y1, . . . , yd) . . . yd = Pd(z, y1, . . . , yd) where each polynomial Pi is such that [yj]Pi = 1 and has coefficients in N. The power series yi solutions of this system are called N-algebraic functions.

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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) ⇐ ⇒ fn ∼ 1 2√π

1

ρ

n

n−3/2

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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) ⇐ ⇒ fn ∼ 1 2√π

1

ρ

n

n−3/2 Example: B = z + B2 (& works for any t-ary trees B = z + φ(B)) B(z) =

• n≥0

bnzn = 1 − √1 − 4z 2 bn =

2n

n

• n + 1 ∼

4n 4√π n−3/2

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Asymptotics of N-algebraic functions

Theorem (BanderierDrmota2013) N-algebraic functions have a dyadic critical exponent, i.e. fn ∼ 1 Γ(−α)ρ−nn−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).

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Proof for the dyadic critical exponents

                      

y1 = P1(z, y1, y2, y5) y2 = P2(z, y2, y3, y5) y3 = P3(z, y3, y4) y4 = P4(z, y3, y4) y5 = P5(z, y5, y6) y6 = P6(z, y5, y6).

2 5 6 3 4 1

2 5,6 3,4 1

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.

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Proof for the dyadic critical exponents

                      

y1 = P1(z, y1, y2, y5) y2 = P2(z, y2, y3, y5) y3 = P3(z, y3, y4) y4 = P4(z, y3, y4) y5 = P5(z, y5, y6) y6 = P6(z, y5, y6).

2 5 6 3 4 1

2 5,6 3,4 1

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 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,

• r of polar-algebraic type

y(z) = c (1 − z/ρ)m2−k + c′ (1 − z/ρ)(m−1)2−k + · · ·

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Application to lattice paths

bounded queue: rational world (known behavior, omitted here) unbounded queue: algebraic world (next slides)

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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:

−6 −7 −4 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-M´ elou–Denise–Flajolet–Gardy–Gouyou-Beauchamps, F´ edou, Garcia, Merlini, . . . ).

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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:

−6 −7 −4 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-M´ elou–Denise–Flajolet–Gardy–Gouyou-Beauchamps, F´ edou, 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.

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Generating functions

−6 −7 −4 A A A A A

For walks with catastrophes + jumps encoded by P(u) = d

i=−c piui

(hereafter c = 1 for readability, but similar proofs work for any c > 1) Theorem (Generating functions for lattice paths with catastrophes) fn,k = # catastrophe-walks of length n from altitude 0 to altitude k. F(z, u) =

k≥0 Fk(z)uk = n,k≥0 fn,kukzn is algebraic and satisfies

F(z, u) = zp−1 zp−1 − z2p−1( 1−u1(z)

1−zP(1) − u1(z) − u1(z)(1−zp0)−1 zp−1

) u − u1(z) u(1 − zP(u)) where u1(z) is the root of 1 − zP(u) = 0 such that u1(z) ∼ 0 for z ∼ 0.

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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): F(z, u) = 1

• empty walk

+ zP(u)F(z, u)

• jump from P(u)

− z p−1 u F0(z)

• neg. jump at alt. 0

+ zu0 (F(z, 1) − F0(z) − F1(z)) ,

• catastrophe at altitude > 1

which is convenient to rewrite as (1 − zP(u))F(z, u) = 1 − z p−1 u F0(z) + z (F(z, 1) − F0(z) − F1(z)) . This single equation contains 4 unknowns ⇒ can we create additional equations?

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(1−zP(u))F(z, u) = 1−z p−1 u F0(z)+z (F(z, 1) − F0(z) − F1(z)) . (∗∗∗) First, setting u = 1 leads to (1 − zP(1))F(z, 1) = 1 − zp−1F0(z) + z (F(z, 1) − F0(z) − F1(z)) . (1) Extracting [u1] in the functional equation gives F0(z) − zp−1F1(z) − zp0F0(z) = 1 + z (F(z, 1) − F0(z) − F1(z)) . (2) Finally, if we solve 1 − zP(u) = 0 with respect to u, we get 1 + d roots,

• ne of them is such that u1(z) ≈ 0 for z ≈ 0. Then plugging this root

u1(z) in the functional equation (***) leads to 0 = 1 − z p−1 u1(z)F0(z) + z (F(z, 1) − F0(z) − F1(z)) . (3) We thus got a system of 3 equations involving the 3 unknowns F(z, 1), F0(z), and F1(z). Solving it gives F(z, u).

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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) = 1 1 − z(M(z, 1) − E(z) − M1(z)), Walks ending at altitude 0 and 1: F0(z) = S(z)E(z), and F1(z) = S(z)M1(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) − M1(z)) . E (excursions), M (meanders), and M1 (meanders ending at altitude 1) are classical walks with no catastrophe. M(z, u) =

u−u1(z) u(1−zP(u)) is the GF of meanders [BanderierFlajolet2001]

M1(z) = E(z)p1zE(z)

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Corollary (Generating functions for Dyck paths with catastrophes) mn := # Dyck meanders with catastrophes of length n starting from 0. F(z, 1) =

• n≥0

mnzn = z(u1(z) − 1) z2 + (z2 + z − 1)u1(z) = 1 + z + 2z2 + 4z3 + O(z4), where u1(z) = 1−

√ 1−4z2 2z

. en := # Dyck excursions with catastrophes of length n ending at 0. F0(z) =

• n≥0

enzn = (2z − 1)u1(z) z2 + (z2 + z − 1)u1(z) = 1 + z2 + z3 + 3z4 + O(z5). Moreover, e2n is also the number of Dumont permutations of the first kind

• f length 2n avoiding the patterns 1423 and 4132. [Burstein05]

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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

• ption 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

Dyck arch with catastrophe

1

1-horizontal Dyck arch

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Excursions

Let en be the number of Dyck paths with catastrophes of length n, and let hn 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: en = hn. Proof: A first proof that hn = en 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

hnzn = 1 1 − z2 1 − z − z2C(z) , where C(z) is the generating function of classical Dyck paths, C(z) = 1/(1 − z2C(z)). . . .

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Enumeration of Dyck arches with catastrophes

Proposition The number an Dyck arches with catastrophes is an =

• n − 2

⌊ n−3

2 ⌋

• .

Proof: using M(z), E(z) and M1(z) the generating functions of classical Dyck walks for meanders, excursions, and meanders ending at 1, one has: A(z) =

• n≥0

anzn = z M(z) − E(z) − M1(z) E(z) = 2z2 + z − 1 +

• (1 − 2z) (1 + 2z) (1 − z)2

2(1 − 2z) = z3 + z4 + 3z5 + 4z6 + 10z7 + 15z8 + 35z9 + O(z10). an+2 =

• n

⌊ n

2⌋

• meanders

− 1 n/2 + 1

• n

n 2

• excursions

[ [n even] ]

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Asymptotics and limit laws

Proposition (Asymptotics of Dyck paths with catastrophes) The number of Dyck paths with catastrophes en, and Dyck meanders with catastrophes mn is asymptotically equal to en = Ceρ−n

• 1 + O

1

n

• ,

mn = Cmρ−n

• 1 + O

1

n

• ,

where ρ ≈ 0.46557 is the unique positive root of ρ3 + 2ρ2 + ρ − 1, Ce ≈ 0.10381 is the unique positive root of 31C3

e − 62C2 e + 35Ce − 3,

Cm ≈ 0.32679 is the unique positive root of 31C3

m − 31C2 m + 16Cm − 3.

Proof: via singularity analysis. simple pole at ρ = 1

6

• 116 + 12

√ 93

1/3 + 2

3

• 116 + 12

√ 93

−1/3 − 2

3 ≈ 0.46557 which is

strictly smaller than 1/2 which is the dominant singularity of u1(z): F0(z) = C 1 − z/ρ + O(1), for z → ρ.

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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(ρh) > ρg, that g is analytic in |z| < R for some R > ρg, with a unique dominant singularity at ρg, which is a simple pole, and that h is

• aperiodic. Furthermore, let q(z) be analytic for |z| < ρh. Then the

number χ of H-components in a random Fn-structure, corresponding to the probability distribution [ukzn]F(z, u)/[zn]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)).

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Proposition (Perturbed supercritical sequences) For a schema F = Q × SEQ(uH) such that h(ρh) > 1, (with q(z) analytic for |z| < ρ, where ρ is the positive root of h(ρ) = 1), the number Xn of H-components in a random Fn-structure of large size n is, asymptotically Gaussian with E(Xn) ∼ n ρh′(ρ), V(Xn) ∼ nρh′′(ρ) + h′(ρ) − ρh′(ρ)2 ρ2h′(ρ)3 . Proof: previous Prop with g(z) = (1 − z)−1 and ρg replaced by 1. The second part results from the bivariate generating function F(z, u) = q(z) 1 − (u − 1)hmzm − 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 = 1.

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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) :=

• n,k,ℓ≥0

cnkℓznvkwℓ = 1 1 − vA(z) − w A(z) , where A(z) is the generating function of arches with catastrophes, and

• A(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.

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Average number of catastrophes

C(z, v) := C(z, v, 1) = E(z) 1 − vA(z)E(z), P (Xn = k) = [znvk]C(z, v) [zn]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 Xn, Xn − µn σ√n , µ ≈ 0.0708358118, σ2 ≈ 0.05078979113, where µ is the unique positive real root of 31µ3 + 31µ2 + 40µ − 3, and σ2 is the unique positive real root of 29791σ6 − 59582σ4 + 60579σ2 − 2927, converges in law to a Gaussian variable N(0, 1) : lim

n→∞ P

Xn − µn

σ√n ≤ x

• =

1 √ 2π

x

−∞

e−y2/2 dy.

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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) = u1(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 ρA of A(z). By Stirling’s formula: an ∼ K

2n

√n

• ,

This implies that A(z) ∼ ˜ K(1 − 2z)−1/2 for z → 1/2. As E(1/2) = 1, one has limz→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.

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Average number of returns to zero

From now on, let Xn 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 (Xn = k) = [znvk]C(z, v, v) [zn]C(z, 1, 1) . Theorem The number of returns to zero of a random Dyck path with catastrophes

• f length n is normally distributed. The standardized version of Xn,

Xn − µn σ√n , µ ≈ 0.1038149281, σ2 ≈ 0.1198688826, where µ is the unique positive real root of 31µ3 − 62µ2 + 35µ − 3, and σ2 is the unique positive real root of 29791σ6 + 231σ2 − 79, converges in law to a Gaussian variable N(0, 1).

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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) − ε0 √ 1 − 2z, for z → 1/2, where ε0 is a computable constant. Thus, B(z) ∼ K0 + K1 √1 − 2z for constant K0, K1. So limz→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.

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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 = v1(ρ)−1 ≈ 0.6823278: P (Xn = k) ∼ (1 − λ) λk. Proof: Let pn(u) := [zn]F(z,u)

[zn]F(z,1) be the probability generating function of Xn.

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

Final altitude limit law (proof)

Let us now fix u ∈ (0, 1) and treat is henceforth as a parameter. The probability generating function of Xn is pn(u) = [zn]S(z)M(z, u) [zn]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]: pn(u) ∼ M(ρ, u)[zn]S(z) M(ρ, 1)[zn]S(z) = M(ρ, u) M(ρ, 1). The branches allow us to factor the kernel equation into u(1 − zP(u)) = −zp1(u − u1(z))(u − v1(z)). Thus, M(ρ, u) = 1 ρp−1(v1(ρ) − u), the limit probability generating function of a geometric distribution.

38 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

Final altitude

Figure: The limit law for the final altitude in the case of a jump polynomial P(u) = u40 + 10u3 + 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

Uniform random generation

Generalized Dyck paths (meanders and excursions) can be generated by pushdown-automata/context-free grammars. dynamic programming approach, O(n2) time and O(n3) 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 − ǫ)n, (1 + ǫ)n]. [Banderier-Wallner16] : generating trees+holonomy theory → O(n2) time, O(1) memory.

40 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes

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,kf k+j n−(m+1),0

f 0

n,0

. Then, for each pair (i, k), theory of D-finite functions applied to our algebraic functions gives the recurrence for fm (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 fm’s in floating point arithmetic, instead of rational numbers (although all the fm 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(n2) & O(1) memory is enough

to output the n jumps of the lattice path, step after step, as a stream.

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Conclusion

universality of the Gaussian limit law, and its counter-examples (rational/algebraic world). Universal behaviour: asymptotics fn ∼ KAnnα 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

Conclusion

universality of the Gaussian limit law, and its counter-examples (rational/algebraic world). Universal behaviour: asymptotics fn ∼ KAnnα 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!”

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