Limit Theorems for Internal Aggregation Models by Lionel Timothy Levine AB Harvard University

143 Pages·2007·2.6 MB·English

Limit Theorems for Internal Aggregation Models
by
Lionel Timothy Levine
A.B. Harvard University, 2002
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Mathematics
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge:
Professor Yuval Peres, Chair
Professor Lawrence C. Evans
Professor Elchanan Mossel
Fall 2007
The dissertation of Lionel Timothy Levine is approved:
Chair Date
Date
Date
University of California, Berkeley
Fall 2007
Limit Theorems for Internal Aggregation Models
Copyright 2007
by
Lionel Timothy Levine
1
Abstract
Limit Theorems for Internal Aggregation Models
by
Lionel Timothy Levine
Doctor of Philosophy in Mathematics
University of California, Berkeley
Professor Yuval Peres, Chair
We study the scaling limits of three different aggregation models on Zd: internal DLA, in
which particles perform random walks until reaching an unoccupied site; the rotor-router
model,inwhichparticlesperformdeterministicanaloguesofrandomwalks;andthedivisible
sandpile, in which each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same scaling limit,
which we describe as the solution to a certain PDE free boundary problem in Rd. In
particular, internal DLA has a deterministic scaling limit. We find that the scaling limits
are quadrature domains, which have arisen independently in many fields such as potential
theory and fluid dynamics. Our results apply both to the case of multiple point sources and
to the Diaconis-Fulton smash sum of domains.
In the special case when all particles start at a single site, we show that the scaling
limit is a Euclidean ball in Rd and give quantitative bounds on the rate of convergence to a
ball. Forthedivisiblesandpile,theerrorintheradiusisboundedbyaconstantindependent
of the total starting mass. For the rotor-router model in Zd, the inner error grows at most
logarithmically in the radius r, while the outer error is at most order r1−1/dlogr. We also
improve on the previously best known bounds of Le Borgne and Rossin in Z2 and Fey and
Redig in higher dimensions for the shape of the classical abelian sandpile model.
Lastly, we study the sandpile group of a regular tree whose leaves are collapsed
to a single sink vertex, and determine the decomposition of the full sandpile group as a
product of cyclic groups. For the regular ternary tree of height n, for example, the sandpile
2
group is isomorphic to (Z )2n−3 ⊕(Z )2n−4 ⊕...⊕Z ⊕Z . We use this result to
3 7 2n−1−1 2n−1
prove that rotor-router aggregation on the regular tree yields a perfect ball.
i
To my parents, Lance and Terri.
ii
Acknowledgments
This work would not have been possible without the incredible support and insight of my
advisor, Yuval Peres. Yuval taught me not only a lot of mathematics, but also the tools,
techniques, instincts and heuristics essential to the working mathematician. Somehow, he
evenmanagedtoteachmealittlebitaboutlifeaswell. Myentirephilosophyandapproach,
and my sense of what is important in mathematics, are colored by his ideas.
I would like to thank Jim Propp for introducing me to this beautiful area of
mathematics, and for many inspiring conversations over the years. Thanks also to Scott
Armstrong, Henry Cohn, Darren Crowdy, Craig Evans, Anne Fey, Chris Hillar, Wilfried
Huss, Itamar Landau, Karola M´esz´aros, Chandra Nair, David Pokorny, Oded Schramm,
Scott Sheffield, Misha Sodin, Kate Stange, Richard Stanley, Parran Vanniasegaram, Grace
Wang, and David Wilson for many helpful conversations. Richard Liang taught me how to
write image files using C, so that I could write programs to generate many of the figures.
In addition, Itamar Landau and Yelena Shvets helped create several of the figures.
I also thank the NSF for supporting me with a Graduate Research Fellowship
during much of the time period when this work was carried out.
Finally,Iwouldliketothankmyfamilyandfriendsforsupportingmeandbelieving
in me through the best and worst of times. Your love and support mean more to me than
I can possibly express in words.
iii
Contents
1 Introduction 1
1.1 Three Models with the Same Scaling Limit . . . . . . . . . . . . . . . . . . 1
1.2 Single Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Multiple Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Quadrature Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Aggregation on Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Spherical Asymptotics for Point Sources 18
2.1 Basic Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Divisible Sandpile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Abelian Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Proof of Theorem 1.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Classical Sandpile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Rotor-Router Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Inner Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Outer Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Scaling Limits for General Sources 42
3.1 Potential Theory Background . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Least Superharmonic Majorant . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Superharmonic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.3 Boundary Regularity for the Obstacle Problem . . . . . . . . . . . . 49
3.1.4 Convergence of Obstacles, Majorants and Domains . . . . . . . . . . 56
3.1.5 Discrete Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Divisible Sandpile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Convergence of Odometers . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.2 Convergence of Domains . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Rotor-Router Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.1 Convergence of Odometers . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Convergence of Domains . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Internal DLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Inner Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.2 Outer Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
iv
3.5 Multiple Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5.1 Associativity and Hausdorff Continuity of the Smash Sum . . . . . . 93
3.5.2 Smash Sums of Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5.3 Algebraic Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Aggregation on Trees 99
4.1 The Sandpile Group of a Tree . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.1 General Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 Regular Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.3 Proof of Toumpakari’s Conjecture . . . . . . . . . . . . . . . . . . . 109
4.2 The Rotor-Router Model on Trees . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.1 The Rotor-Router Group . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.2 Aggregation on Regular Trees . . . . . . . . . . . . . . . . . . . . . . 114
4.2.3 Recurrence and Transience . . . . . . . . . . . . . . . . . . . . . . . 118
5 Conjectures and Open Problems 125
5.1 Rotor-Router Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Sandpile Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Bibliography 129
v
Index of Notation
o, origin in Zd, 1 A¯, closure, 47
(cid:3)
∆, discrete Laplacian, 3 x , box of side δ centered at x, 60
n
|x| = (x2+...+x2)1/2, 4 x::, closest lattice point, 60
1 d
(cid:3)
A⊕B, smash sum, 5 A , union of boxes, 60
δ , lattice spacing, 5 A:: = A∩δ Zd, 60
n n
(cid:3)
D , inner (cid:15)-neighborhood, 6 f , step function, 60
(cid:15)
D(cid:15), outer (cid:15)-neighborhood, 6 f::, restriction to δ Zd, 60
n
B , discrete ball, 9 g (x,y), Green’s function on δ Zd, 60
r n n
X , simple random walk, 18 M, uniform bound on σ, 61
k
g (x,y), Green’s function on Zd, 18 DC(σ), discontinuities of σ, 62
1
γ , obstacle for a single point source, 19 G σ , discrete potential, 62
d n n
∂A, boundary of A ⊂ Zd, 19 σ , mass density on δ Zd, 62
n n
L, Lebesgue measure on Rd, 24 γ , obstacle on δ Zd, 63
n n
H, hole depth, 25 s , superharmonic majorant in δ Zd, 63
n n
S , sandpile with hole depth H, 25 u , divisible sandpile odometer, 64
n,H n
Q(x,k), discrete cube, 26 λ, bounds σ away from 1, 69
∇, discrete gradient, 28 D(cid:101), enlarged noncoincidence set, 69
div, discrete divergence, 29 S f, binomial smoothing, 71
k
A u, average in a ball, 42 α(n), scale of smoothing, 72
r
Gσ, superharmonic potential, 44 M , 81
(cid:15)
g(x,y), Green’s function on Rd, 45 L , 81
(cid:15)
σ, mass density on Rd, 46 L(cid:101)(cid:15), 82
γ, obstacle on Rd, 46 f , 82
n,(cid:15)
s, superharmonic majorant in Rd, 46 g , Green’s function on D::, 82
n,(cid:15) (cid:15)
D, noncoincidence set, 46 τy, first exit time of D::, 82
(cid:15) (cid:15)
Ao, interior, 47 τy, first hitting time of z, 82
z