Table Of ContentLecture Notes in Mathematics 1875
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Subseries:
Ecoled’EtédeProbabilitésdeSaint-Flour
J. Pitman
Combinatorial Stochastic
Processes
Ecole d’Eté de Probabilités
de Saint-Flour XXXII – 2002
Editor:JeanPicard
ABC
Author Editor
JimPitman JeanPicard
DepartmentofStatistics LaboratoiredeMathématiquesAppliquées
UniversityofCalifornia,Berkeley UMRCNRS6620
367EvansHall UniversitéBlaisePascal(Clermont-Ferrand)
Berkeley,CA94720-3860 63177AubièreCedex
USA France
e-mail:[email protected] e-mail:[email protected]
Cover:BlaisePascal(1623–1662)
LibraryofCongressControlNumber:2006921042
MathematicsSubjectClassification(2000):05Axx,60C05,60J65,60G09,60J80
ISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
ISSNEcoled’EtédeProbabilitésdeSt.Flour,printedition:0721-5363
ISBN-10 3-540-30990-XSpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30990-1SpringerBerlinHeidelbergNewYork
DOI10.1007/b11601500
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Foreword
Three series of lectures were given at the 32nd Probability Summer School in
Saint-Flour(July7–24,2002),bytheProfessorsPitman,TsirelsonandWerner.
ThecoursesofProfessorsTsirelson(“Scalinglimit,noise,stability”)andWerner
(“Random planar curves and Schramm-Loewner evolutions”) have been pub-
lishedinapreviousissueofLectures Notes in Mathematics (volume1840).This
volume contains the course “Combinatorial stochastic processes” of Professor
Pitman. We cordially thank the author for his performance in Saint-Flour and
for these notes.
76 participants have attended this school. 33 of them have given a short
lecture. The lists of participants and of short lectures are enclosed at the end of
the volume.
The Saint-Flour Probability Summer School was founded in 1971. Here are
the references of Springer volumes which have been published prior to this one.
AllnumbersrefertotheLectureNotesinMathematics series,exceptS-50which
refers to volume 50 of the Lecture Notes in Statistics series.
1971: vol 307 1980: vol 929 1990: vol 1527 1998: vol 1738
1973: vol 390 1981: vol 976 1991: vol 1541 1999: vol 1781
1974: vol 480 1982: vol 1097 1992: vol 1581 2000: vol 1816
1975: vol 539 1983: vol 1117 1993: vol 1608 2001: vol 1837 & 1851
1976: vol 598 1984: vol 1180 1994: vol 1648 2002: vol 1840
1977: vol 678 1985/86/87: vol 1362 & S-50 1995: vol 1690 2003: vol 1869
1978: vol 774 1988: vol 1427 1996: vol 1665
1979: vol 876 1989: vol 1464 1997: vol 1717
Further details can be found on the summer school web site
http://math.univ-bpclermont.fr/stflour/
Universit´e Blaise Pascal Jean Picard
September 2005
Contents
0 Preliminaries 1
0.0 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Brownian motion and related processes. . . . . . . . . . . . . . . 3
0.3 Subordinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Bell polynomials and Gibbs partitions 13
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Partitions and compositions . . . . . . . . . . . . . . . . . . . . . 14
1.3 Moments and cumulants . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Random sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Gibbs partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Exchangeable random partitions 37
2.1 Finite partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Infinite partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Structural distributions . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Limits of Gibbs partitions . . . . . . . . . . . . . . . . . . . . . . 50
3 Sequential constructions of random partitions 55
3.1 The Chinese restaurant process . . . . . . . . . . . . . . . . . . . 56
3.2 The two-parameter model . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 A branching process construction . . . . . . . . . . . . . . . . . . 72
4 Poisson constructions of random partitions 77
4.1 Size-biased sampling . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Poisson representation of the two-parameter model . . . . . . . . 81
4.3 Representation of infinite Gibbs partitions . . . . . . . . . . . . . 85
4.4 Lengths of stable excursions . . . . . . . . . . . . . . . . . . . . . 87
4.5 Brownian excursions . . . . . . . . . . . . . . . . . . . . . . . . . 90
VIII Contents
5 Coagulation and fragmentation processes 97
5.1 Coalescents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Fragmentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Representations of infinite partitions . . . . . . . . . . . . . . . . 109
5.4 Coagulation and subordination . . . . . . . . . . . . . . . . . . . 112
5.5 Coagulation – fragmentation duality . . . . . . . . . . . . . . . . 116
6 Random walks and random forests 121
6.1 Cyclic shifts and Lagrange inversion . . . . . . . . . . . . . . . . 122
6.2 Galton-Watson forests . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Brownian asymptotics for conditioned Galton-Watson trees . . . 129
6.4 Critical random graphs. . . . . . . . . . . . . . . . . . . . . . . . 135
7 The Brownian forest 143
7.1 Plane trees with edge-lengths . . . . . . . . . . . . . . . . . . . . 144
7.2 Binary Galton-Watson trees . . . . . . . . . . . . . . . . . . . . . 146
7.3 Trees in continuous paths . . . . . . . . . . . . . . . . . . . . . . 149
7.4 Brownian trees and excursions . . . . . . . . . . . . . . . . . . . 152
7.5 Plane forests with edge-lengths . . . . . . . . . . . . . . . . . . . 162
7.6 Sampling at downcrossing times. . . . . . . . . . . . . . . . . . . 165
7.7 Sampling at Poisson times . . . . . . . . . . . . . . . . . . . . . . 167
7.8 Path decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.9 Further developments . . . . . . . . . . . . . . . . . . . . . . . . 174
8 Brownian local times 177
8.1 Stopping at an inverse local time . . . . . . . . . . . . . . . . . . 177
8.2 Squares of Bessel processes . . . . . . . . . . . . . . . . . . . . . 179
8.3 Stopping at fixed times . . . . . . . . . . . . . . . . . . . . . . . 182
8.4 Time-changed local time processes . . . . . . . . . . . . . . . . . 185
8.5 Branching process approximations . . . . . . . . . . . . . . . . . 187
9 Brownian bridge asymptotics 193
9.1 Basins and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.2 Mapping walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.3 Brownian asymptotics . . . . . . . . . . . . . . . . . . . . . . . . 199
9.4 The diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.5 The height profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.6 Non-uniform random mappings . . . . . . . . . . . . . . . . . . . 206
10 Random forests and the additive coalescent 207
10.1 Random p-forests and Cayley’s multinomial expansion . . . . . . 208
10.2 The additive coalescent . . . . . . . . . . . . . . . . . . . . . . . 210
10.3 The standard additive coalescent . . . . . . . . . . . . . . . . . . 213
10.4 Poisson cutting of the Brownian tree . . . . . . . . . . . . . . . . 215
Contents IX
Bibliography 223
Index 249
List of participants 253
List of short lectures 255
Preliminaries
0.0. Preface
This is a collection of expository articles about various topics at the interface
between enumerative combinatorics and stochastic processes. These articles ex-
pand on a course of lectures given at the Ecole d’Et´e de Probabilit´es de St.
Flour in July 2002. The articles are also called ‘chapters’. Each chapter is fairly
self-contained,soreaderswithadequatebackgroundcanstartreadinganychap-
ter, with occasional consultation of earlier chapters as necessary. Following this
Chapter 0, there are 10 chapters, each divided into sections. Most sections con-
clude with some Exercises. Those for which I don’t know solutions are called
Problems.
Acknowledgments Muchoftheresearchreviewedherewasdonejointlywith
DavidAldous.Muchcreditisduetohim,especiallyforthebigpictureofcontin-
uumapproximationstolargecombinatorialstructures.Thanksalsotomyother
collaborators in this work, especially Jean Bertoin, Michael Camarri, Steven
Evans,SashaGnedin,BenHansen,JacquesNeveu,MihaelPerman,RaviSheth,
Marc Yor and Jim Young. A preliminary version of these notes was developed
in Spring 2002 with the help of a dedicated class of ten graduate students in
Berkeley: Noam Berger, David Blei, Rahul Jain, S¸erban Nacu, Gabor Pete,
Lea Popovic, Alan Hammond, Antar Bandyopadhyay, Manjunath Krishnapur
and Gr´egory Miermont. The last four deserve special thanks for their contri-
butions as research assistants. Thanks to the many people who have read ver-
sions of these notes and made suggestions and corrections, especially David
Aldous, Jean Bertoin, Aubrey Clayton, Shankar Bhamidi, Rui Dong, Steven
Evans, Sasha Gnedin, B´en´edicte Haas, Jean-Fran¸cois Le Gall, Neil O’Connell,
Mihael Perman, Lea Popovic, Jason Schweinsberg. Special thanks to Marc Yor
and Matthias Winkel for their great help in preparing the final version of these
notes for publication. Thanks also to Jean Picard for his organizational efforts
in making arrangements for the St. Flour Summer School. This work was sup-
ported in part by NSF Grants DMS-0071448 and DMS-0405779.
2 Preliminaries
0.1. Introduction
The main theme of this course is the study of various combinatorial models of
random partitions and random trees, and the asymptotics of these models re-
latedtocontinuousparameterstochasticprocesses.Abasicfeatureofmodelsfor
random partitions is that the sum of the parts is usually constant. So the sizes
of the parts cannot be independent. But the structure of many natural mod-
els for random partitions can be reduced by suitable conditioning or scaling to
classical probabilistic results involving sums of independent random variables.
Limit models for combinatorially defined random partitions are consequently
related to the two fundamental limit processes of classical probability theory:
Brownian motion and Poisson processes. The theory of Brownian motion and
related stochastic processes has been greatly enriched by the recognition that
some fundamental properties of these processes are best understood in terms of
how various random partitions and random trees are embedded in their paths.
Thishasledtorapiddevelopments,particularlyinthetheoryofcontinuumran-
domtrees,continuousstatebranchingprocesses,andMarkoviansuperprocesses,
whichgofarbeyondthescopeofthiscourse.Followingisalistofthemaintopics
to be treated:
• modelsforrandomcombinatorialstructures,suchastrees,forests,permu-
tations, mappings, and partitions;
• probabilisticinterpretationsofvariouscombinatorialnotionse.g.Bellpoly-
nomials, Stirling numbers, polynomials of binomial type, Lagrange inver-
sion;
• Kingman’stheoryofexchangeablerandompartitionsandrandomdiscrete
distributions;
• connections between random combinatorial structures and processes with
independent increments: Poisson-Dirichlet limits;
• random partitions derived from subordinators;
• asymptotics of random trees, graphs and mappings related to excursions
of Brownian motion;
• continuum random trees embedded in Brownian motion;
• Brownian local times and squares of Bessel processes;
• various processes of fragmentation and coagulation, including Kingman’s
coalescent, the additive and multiplicative coalescents
Next,anincompletelistandtopicsofcurrentinterest,withinadequaterefer-
ences.Thesetopicsareclosetothosejustlisted,andcertainlypartoftherealm
of combinatorial stochastic processes, but not treated here:
• probability on trees and networks, as presented in [292];
• random integer partitions [159, 104], random Young tableaux, growth of
Young diagrams, connections with representation theory and symmetric
functions [245, 420, 421, 239];
• longestincreasingsubsequenceofapermutation,connectionswithrandom
matrices [28];
0.2 Brownian motion and related processes 3
• random partitions related to uniformly chosen invertible matrices over a
finite field, as studied by Fulman [160];
• random maps, coalescing saddles, singularity analysis, and Airy phenom-
ena, [81];
• random planar lattices and integrated superbrownian excursion [94].
The reader of these notes is assumed to be familiar with the basic theory
of probability and stochastic processes, at the level of Billingsley [64] or Dur-
rett [122], including continuous time stochastic processes, especially Brownian
motion and Poisson processes. For background on some more specialized top-
ics (local times, Bessel processes, excursions, SDE’s) the reader is referred to
Revuz-Yor [384]. The rest of this Chapter 0 reviews some basic facts from this
probabilistic background for ease of later reference. This material is organized
as follows:
0.2. Brownian motion and related processes Thissectionprovidessome
minimal description of the background expected of the reader to follow
some of the more advanced sections of the text. This includes the defin-
ition and basic properties of Brownian motion B := (B ,t ≥ 0), and of
t
some important processes derived from B by operations of scaling and
conditioning.TheseprocessesincludetheBrownianbridge,Brownianme-
ander and Brownian excursion. The basic facts of Itoˆ’s excursion theory
for Brownian motion are also recorded.
0.3. Subordinators This section reviews a few basic facts about increasing
L´evy processes in general, and some important facts about gamma and
stable processes in particular.
0.2. Brownian motion and related processes
Let S :=X +···+X where the X are independent random variables with
n 1 n i
mean 0 and variance 1, and let S for real t be defined by linear interpolation
t
between integer values. According to Donsker’s theorem [64, 65, 122, 384]
√
(S / n,0≤t≤1) →d (B ,0≤t≤1) (0.1)
nt t
in the usual sense of convergence in distribution of random elements of C[0,1],
where (B ,t ≥ 0) is a standard Brownian motion meaning that B is a process
t
with continuous paths and stationary independent Gaussian increments, with
√
d
B = tB where B is standard Gaussian.
t 1 1
Brownian bridge Assuming now that the Xi are integer valued, some c√on-
ditioned forms of Donsker’s theorem can be present√ed as√follows. Let o( n)
denote any sequence of possible values of S with o( n)/ n → 0 as n → ∞.
n
Then [128]
√ √
(S / n,0≤t≤1|S =o( n)) →d (Bbr,0≤t≤1) (0.2)
nt n t
