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), 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
Generating functions in combinatorics recursive combinatorial structures (lists, words, trees, maps, graphs, permutations, walks, tilings...) � recurrences � series � complex analysis � asymptotics � typical behavior � a n z n F ( z ) = In enumerative and analytic combinatorics, generating functions and their nature play a key rˆ ole: rational functions ( ≈ walks on graphs) algebraic functions ( ≈ walks on N ) . . . ♥♥♥ Full final version of the book for free at algo.inria.fr/flajolet ♥♥♥ 3 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf
The analytic combinatorics dogma of Flajolet & Sedgewick 4 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
World #1: Rational functions ( ≈ walks in finite graphs) 5 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
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
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 ) � Pr( X n = k ) u k z n P ( z , u ) = n , k ≥ 0 B ( z , u ) L = zT ( u ) � � L + � 1 = ⇒ P ( z , u ) = det( I − zT ( u )) Perron–Frobenius: ∃ ′ λ ( u ) of maximal modulus c ( u ) ⇒ Pr( X n = k ) ∼ c ( u ) λ ( u ) n P ( z , u ) ∼ u → 1 1 − z λ ( u ) = Hwang: quasi-powers theorem = ⇒ Gaussian limit law E [ X n ] = µ n + c 1 + O ( A n ) Var[ X n ] = σ 2 n + c 2 + O ( A n ) � X n − µ n � � x 1 exp − t 2 / 2 dt Non Gaussian law = [BaBoPoTa2012] Pr σ √ n ≤ x → √ 2 π 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. 8 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
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
World #2: Algebraic functions ( ≈ walks on N ) 10 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
From language theory to combinatorics in one theorem Theorem (Chomsky–Sch¨ utzenberger, 1963) Any context-free language has an algebraic generating function � f n z n F ( z ) = n ≥ 0 where f n = # words of length n generated by the grammar. 11 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
From language theory to combinatorics in one theorem Theorem (Chomsky–Sch¨ utzenberger, 1963) Any context-free language has an algebraic generating function � f n z n F ( z ) = 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
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 � 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
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 � 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]... 12 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
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 � ( z − ρ ) 1 / r � k � F ( z ) = a k k ≥ k 0 Theorem (Flajolet–Odlyzko, 1990) Asymptotics is given via singularity analysis: 1 F ( z ) ∼ (1 − z /ρ ) α Γ( − α ) ρ − n n − α − 1 (for z ∼ ρ ) ⇐ ⇒ f n ∼ 13 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
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
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). � 1 � n 1 F ( z ) ∼ − (1 − z /ρ ) 1 / 2 n − 3 / 2 (for z ∼ ρ ) ⇐ ⇒ f n ∼ 2 √ π ρ 15 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
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). � 1 � n 1 F ( z ) ∼ − (1 − z /ρ ) 1 / 2 n − 3 / 2 (for z ∼ ρ ) ⇐ ⇒ f n ∼ 2 √ π ρ Example: B = z + B 2 (& works for any t-ary trees B = z + φ ( B )) b n z n = 1 − √ 1 − 4 z � 2 n � � 4 n 4 √ π n − 3 / 2 n B ( z ) = b n = n + 1 ∼ 2 n ≥ 0 15 / 42 Cyril Banderier, Michael Wallner Lattice paths with catastrophes
Recommend
More recommend
