Algorithmic Probability and Combinatorics: Ams Special Sessions on Algorithmic Probability and Combinatorics, October 5-6, 2007, Depaul University, ... 208, University of

251 Pages·2010·1.89 MB·English

by  Manuel E. Lladser

C
ONTEMPORARY
M
ATHEMATICS
520
Algorithmic Probability and
Combinatorics
AMS Special Sessions on
Algorithmic Probability and Combinatorics
October 5–6, 2007
DePaul University
Chicago, Illinois
October 4–5, 2008
University of British Columbia
Vancouver, BC, Canada
Manuel E. Lladser
Robert S. Maier
Marni Mishna
Andrew Rechnitzer
Editors
American Mathematical Society
Algorithmic Probability and
Combinatorics
This page intentionally left blank
C
ONTEMPORARY
M
ATHEMATICS
520
Algorithmic Probability and
Combinatorics
AMS Special Sessions on
Algorithmic Probability and Combinatorics
October 5–6, 2007
DePaul University
Chicago, Illinois
October 4–5, 2008
University of British Columbia
Vancouver, BC, Canada
Manuel E. Lladser
Robert S. Maier
Marni Mishna
Andrew Rechnitzer
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary 05–06,60–06, 41–06, 82–06;
Secondary 05A15, 05A16, 60C05, 41A60.
Library of Congress Cataloging-in-Publication Data
AMSSpecialSessiononAlgorithmicProbabilityandCombinatorics(2007: DePaulUniversity)
Algorithmicprobabilityandcombinatorics: AMSSpecialSession,October5-6,2007,DePaul
University, Chicago, Illinois : AMS Special Session, October 4-5, 2008, University of British
Columbia,Vancouver,BC,Canada/ManuelE.Lladser... [etal.],editors.
p. cm. –(Contemporarymathematics;v. 520)
Includesbibliographicalreferences.
ISBN978-0-8218-4783-1(alk. paper)
1. Combinatorial analysis—Congresses. 2. Approximation theory—Congresses. 3. Mathe-
maticalstatistics—Congresses. I.Lladser,Manuel,1970- II.AMSSpecialSessiononAlgorith-
micProbabilityandCombinatorics(2008: UniversityofBritishColumbia) III.Title.
QA164.A474 2007
511(cid:2).6—dc22
2010011434
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Contents
Preface vii
Walks with small steps in the quarter plane
Mireille Bousquet-M´elou and Marni Mishna 1
Quantum random walk on the integer lattice: Examples and phenomena
Andrew Bressler, Torin Greenwood, Robin Pemantle,
and Marko Petkovˇsek 41
A case study in bivariate singularity analysis
Timothy DeVries 61
Asymptotic normality of statistics on permutation tableaux
Pawe(cid:2)l Hitczenko and Svante Janson 83
Rotor walks and Markov chains
Alexander E. Holroyd and James Propp 105
Approximate enumeration of self-avoiding walks
E. J. Janse van Rensburg 127
Fuchsian differential equations from modular arithmetic
Iwan Jensen 153
Random pattern-avoiding permutations
Neal Madras and Hailong Liu 173
Analytic combinatorics in d variables: An overview
Robin Pemantle 195
Asymptotic expansions of oscillatory integrals with complex phase
Robin Pemantle and Mark C. Wilson 221
v
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Preface
This volume contains refereed articles by speakers in the AMS Special Ses-
sions on Algorithmic Probability and Combinatorics, held on October 5–6, 2007
at DePaul University in Chicago, IL, and on October 4–5, 2008 at the University
of British Columbia in Vancouver, BC. The articles cover a wide range of topics
in analytic combinatorics and in the study, both analytic and computational, of
combinatorial probabilistic models. The authors include pure mathematicians, ap-
plied mathematicians, and computational physicists. A few of the articles have an
expositoryflavor,withextensivebibliographies,butoriginalresearchpredominates.
This is the first volume that the AMS has published in this interdisciplinary
area. Our hope is that these articles give an accurate picture of its variety and
vitality, anditstiestootherareasof mathematics. These areasinclude asymptotic
analysis,algebraicgeometry,specialfunctions,theanalysisofalgorithms,statistical
mechanics, stochastic simulation, and importance sampling.
As co-organizers and co-editors, we thank all participants, contributors, and
referees. We are grateful to the American Mathematical Society for assistance in
organizingthespecialsessions,andinthepublicationofthisvolume. Weespecially
thank Christine Thivierge of the AMS staff, for her efficient support in the latter.
Manuel E. Lladser
Robert S. Maier
Marni Mishna
Andrew Rechnitzer
vii
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CCoonntteemmppoorraarryyMMaatthheemmaattiiccss
Volume520,2010
Walks with small steps in the quarter plane
Mireille Bousquet-M´elou and Marni Mishna
Abstract. Let S ⊂ {−1,0,1}2 \{(0,0)}. We address the enumeration of
plane lattice walks with steps in S, that start from (0,0) and remain in the
firstquadrant{(i,j):i(cid:2)0,j(cid:2)0}. Apriori,thereare28 modelsofthistype,
but someare trivial. Some others are equivalent to models ofwalks confined
to a half-plane, and can therefore be treated systematically using the kernel
method,whichleadstoageneratingfunctionthatisalgebraic.
Wefocusontheremainingmodels,andshowthatthereare79inherently
different ones. To eachof the 79, we associatea groupG ofbirational trans-
formations. We show that this group is finite (in fact dihedral, and of order
at most 8) in 23 cases, and is infinite in the other 56 cases. We present a
unifiedwayofdealingwith22ofthe23modelsassociatedwithafinitegroup.
For each, we find the generating function to be D-finite; and in some cases,
algebraic. The23rdmodel,knownasGessel’swalks,hasrecentlybeenproved
byBostanetal.tohaveanalgebraic(andhenceD-finite)generatingfunction.
Weconjecturethattheremaining56models,eachassociatedwithaninfinite
group,havegeneratingfunctionsthatarenon-D-finite.
Our approach allows us to recover and refine some known results, and
also to obtain new ones. For instance, we prove that walks with N, E, W, S,
SWandNEstepsyieldanalgebraicgeneratingfunction.
1. Introduction
The enumeration of lattice walks is a classic topic in combinatorics. Many
combinatorialobjects(trees,maps,permutations,latticepolygons,Youngtableaux,
queues...) can be encoded as lattice walks, so that lattice path enumeration has
many applications. Given a lattice, for instance the hypercubic lattice Zd, and
a finite set of steps S ⊂ Zd, a typical problem is to determine how many n-step
walks with steps taken from S, starting from the origin, are confined to a certain
region A of the space. If A is the whole space, then the length generating function
ofthesewalksisasimplerationalseries. IfAisahalf-space,boundedbyarational
hyperplane,thentheassociatedgeneratingfunctionisanalgebraicseries. Instances
of the latter problem have been studied in many articles since at least the end of
the 19th century [1, 6]. It is now understood that the kernel method provides a
2000 MathematicsSubjectClassification. Primary05A15.
MBM was supported by the French “Agence Nationale de la Recherche,” project SADA
ANR-05-BLAN-0372.
MMwassupportedbyaCanadianNSERCDiscoverygrant.
(cid:2)c2010AmericanMathematicalSociety
1