free boundary problems arising from combinatorial and...
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Free Boundary Problems Arising fromCombinatorial and Probabilistic Growth Models
Lionel Levine
February 15, 2008
Joint work with Yuval Peres
Lionel Levine Free Boundary Problems and Growth Models
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Internal DLA with Multiple Sources
I Finite set of points x1, . . . ,xk ∈ Zd .
I Start with m particles at each site xi .
I Each particle performs simple random walk in Zd untilreaching an unoccupied site.
I Get a random set of km occupied sites in Zd .
I The distribution of this random set does not depend on theorder of the walks (Diaconis-Fulton ’91).
Lionel Levine Free Boundary Problems and Growth Models
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100 point sources arranged on a 10×10 grid in Z2.
Sources are at the points (50i ,50j) for 0≤ i , j ≤ 9.Each source started with 2200 particles.
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50 point sources arranged at random in a box in Z2.
The sources are iid uniform in the box [0,500]2.Each source started with 3000 particles.
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Questions
I Fix sources x1, . . . ,xk ∈ Rd .
I Run internal DLA on 1nZd with nd particles per source.
I As the lattice spacing goes to zero, is there a scaling limit?
I If so, can we describe the limiting shape?
I Lawler-Bramson-Griffeath ’92 studied the case k = 1: For asingle source, the limiting shape is a ball in Rd .
I Not clear how to define dynamics in Rd .
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Limiting Shape
I Fix points x1, . . . ,xk ∈ Rd and λ1, . . . ,λk > 0.
I Let In be the random set of occupied sites for internal DLA inthe lattice 1
nZd , if bλindc particles start at each site x ::
i
(closest lattice site to xi ).
I Theorem (L.-Peres) There exists a deterministic domainD ⊂ Rd such that
In→ D as n→ ∞
in the following sense: for any ε > 0, with probability one
D ::ε ⊂ In ⊂ Dε:: for all sufficiently large n,
where Dε,Dε are the inner and outer ε-neighborhoods of D.
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Overlapping Internal DLA Clusters
I Idea: First let the particles at each source xi perform internalDLA ignoring the particles from the other sources.
I Get k overlapping internal DLA clusters, each of which isclose to a ball.
I Hard part: How does the shape change when the particles inthe overlaps continue walking until they reach unoccupiedsites?
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Two-source internal DLA cluster built from overlappingsingle-source clusters.
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Diaconis-Fulton Addition
I Finite sets A,B ⊂ Zd .
I In our application, A and B will be overlapping internal DLAclusters from two different sources.
I Write A∩B = {y1, . . . ,yk}.I To form A+B, let C0 = A∪B and
Cj = Cj−1∪{zj}
where zj is the endpoint of a simple random walk started at yj
and stopped on exiting Cj−1.
I Define A+B = Ck .
I Abeilan property: the law of A+B does not depend on theordering of y1, . . . ,yk .
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Diaconis-Fulton sum of two squares in Z2 overlapping in asmaller square.
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Divisible Sandpile
I Given A,B ⊂ Zd , start withI mass 2 on each site in A∩B; andI mass 1 on each site in A∪B−A∩B.
I At each time step, choose x ∈ Zd with mass m(x) > 1, anddistribute the excess mass m(x)−1 equally among the 2dneighbors of x .
I As t→ ∞, get a limiting region A⊕B ⊂ Zd of sites withmass 1.
I Sites in ∂(A⊕B) have fractional mass.I Sites outside have zero mass.
I Abelian property: A⊕B does not depend on the choices.
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Divisible sandpile sum of two squares in Z2 overlapping in asmaller square.
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Diaconis-Fulton sum Divisible sandpile sum
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Odometer Function
I u(x) = total mass emitted from x .
I Discrete Laplacian:
∆u(x) =1
2d ∑y∼x
u(y)−u(x)
= mass received−mass emitted
= 1−1A(x)−1B(x), x ∈ A⊕B.
I Boundary condition: u = 0 on ∂(A⊕B).
I Need additional information to determine the domain A⊕B.
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Free Boundary Problem
I Unknown function u, unknown domain D = {u > 0}.
u ≥ 0
∆u ≤ 1−1A−1B
u(∆u−1 + 1A + 1B) = 0.
I Alternative formulation:
∆u = 1−1A−1B on D;
u = ∇u = 0 on ∂D.
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Least Superharmonic Majorant
I Given A,B ⊂ Zd , let
γ(x) =−|x |2− ∑y∈A
g(x ,y)− ∑y∈B
g(x ,y),
where g is the Green’s function for simple random walk
g(x ,y) = Ex#{k|Xk = y}.
I Let s(x) = inf{φ(x) | φ is superharmonic on Zd and φ≥ γ}.
I Then odometer = s− γ.
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Defining the Scaling Limit
I A,B ⊂ Rd bounded open sets such that ∂A,∂B have measurezero
I We define the smash sum of A and B as
A⊕B = A∪B ∪{s > γ}
where
γ(x) =−|x |2−ZA
g(x ,y)dy −ZB
g(x ,y)dy
and
s(x) = inf{φ(x)|φ is continuous, superharmonic, and φ≥ γ}
is the least superharmonic majorant of γ.
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I Obstacle for two overlapping disks A and B:
γ(x) =−|x |2−ZA
g(x ,y)dy −ZB
g(x ,y)dy
I Obstacle for two point sources x1 and x2:
γ(x) =−|x |2−g(x ,x1)−g(x ,x2)
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The domain D = {s > γ} for two overlapping disks in R2.
The boundary ∂D is given by the algebraic curve(x2 + y2
)2−2r2(x2 + y2
)−2(x2−y2) = 0.
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Main Result
I Let A,B ⊂ Rd be bounded open sets such that ∂A, ∂B havemeasure zero.
I Theorem (L.-Peres) With probability one
Dn,Rn, In→ A⊕B as n→ ∞,
whereI Dn, Rn, In are the smash sums of A∩ 1
nZd and B ∩ 1nZd ,
computed using divisible sandpile, rotor-router, andDiaconis-Fulton dynamics, respectively.
I A⊕B = A∪B ∪{s > γ}.I Convergence is in the sense of ε-neighborhoods: for all ε > 0
(A⊕B)::ε ⊂Dn,Rn, In⊂ (A⊕B)ε:: for all sufficiently large n.
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Internal DLA Divisible Sandpile Rotor-Router Model
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Steps of the Proof
convergence of densities
⇓convergence of obstacles
⇓convergence of majorants
⇓convergence of domains.
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Convergence of Obstacles
I Green’s function estimates: The Green’s functionsgn on 1
nZd and g on Rd are related by:
gn(x ,y) = nd−2g1(nx ,ny)
= g(x ,y) +O(n−2|x−y |−d).
I Get γn→ γ uniformly on compact sets in Rd .
I Next, want to show sn→ s uniformly on compact sets in Rd ,where s,sn are the least superharmonic majorants of γ,γn.
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Convergence of Majorants
I Taylor’s theorem:
∆ 1n Zd s = 2d∆Rd s +O(n−1||s ′′′||∞).
I Mollify s and restrict to 1nZd to get a function fn with
|s− fn|< ε and ∆fn < δ.
I Since γn→ γ≤ s, the function
φn = fn−δ|x |2 + 2ε
is superharmonic and ≥ γn, hence ≥ sn.
I Hence s ≥ fn− ε≥ φn−3ε≥ sn−3ε.
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Convergence of Majorants
I Other direction: given sn superharmonic on 1nZd , how to
produce a superharmonic function on Rd that is “close” to sn?
I Step function s�n (constant on cubes) is too crude.
I Instead, construct a function whose Laplacian is (∆sn)�:
φn(x) =−Z
Rdg(x ,y)(∆sn)�(y)dy .
I Need to truncate to get the integral to converge.
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Convergence of Domains
I Let un = sn− γn (odometer function), and let
Dn = {un > 0}
be the set of fully occupied sites for the divisible sandpile.
I Want to show
(A⊕B)::ε ⊂ Dn ⊂ (A⊕B)ε::
for sufficiently large n, where
A⊕B = A∪B ∪{s > γ}.
I But converging functions 6⇒ converging supports!
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A Lemma of Caffarelli
I Quadratic Growth Lemma: If x /∈ (A⊕B)ε and x ∈ Dn,there exists y ∈ 1
nZd such that
|x−y | ≤ ε +1
nand un(y)≥ ε
2.
I Proof (Caffarelli): Let B = B(x ,ε). Since
∆un = 1−1A−1B = 1 in B ∩Dn
the functionw(y) = un(y)−|x−y |2
is harmonic in B ∩Dn, so attains its max on the boundary.I Since w(x)≥ 0 and w < 0 on ∂Dn, the max is not attained
on ∂Dn, so it must be attained on ∂B.I Get that if x ∈ Dn, then s(y) > γ(y) for some y ∈ B(x ,2ε),
hence y ∈ A⊕B and x ∈ (A⊕B)2ε.
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A Converse to Quadratic Growth
I Lemma: u : Rd → R, u ≥ 0, u(0) = 0. If |∆u| ≤ λ, then
u(x)≤ cdλ|x |2
for a constant cd depending only on d .
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Adapting the Proof for Rotors
I Rotor-router odometer:
u(x) = total number of particles emitted from x .
I Can try to control ∆u, using the fact that each site emitsapproximately equally many particles to each of its 2dneighbors.
I Instead of ∆u = 1−1A−1B , we only know (in Z2)
−2≤∆u + 1A + 1B ≤ 4.
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Smoothing
I To do better, let
Sku(x) =1
(2k + 1)2 ∑y∈Bk (x)
u(y)
where Bk(x) is a box of side length 2k + 1 centered at x .
I Using ∆ = div grad, we get
∆Sku(x) =1
(2k + 1)2 ∑(y ,z)∈∂Bk (x)
u(z)−u(y)
4
= Sk(1−1A−1B) +O
(1
k
)if all sites in Sk(x) are occupied.
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Multiple Point Sources
I Fix centers x1, . . . ,xk ∈ Rd and λ1, . . . ,λk > 0.
I Theorem (L.-Peres) With probability one
Dn,Rn, In→ D as n→ ∞,
whereI Dn, Rn, In are the domains of occupied sites δnZd , if bλiδ
−dn c
particles start at each site x ::i and perform divisible sandpile,
rotor-router, and Diaconis-Fulton dynamics, respectively.I D is the smash sum of the balls B(xi , ri ), where λi = ωd rd
i .
I Follows from the main result and the case of a single pointsource.
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A Quadrature Identity
I Given A,B ⊂ Rd , if h ∈ C 1(A⊕B
)is harmonic in A⊕B,
then by Green’s theorem,ZA⊕B
h∆u =ZA⊕B
u∆h +Z
∂(A⊕B)
(h
∂u
∂n− ∂h
∂nu
)= 0.
since u and ∇u vanish on ∂(A⊕B).
I Recalling that ∆u = 1−1A−1B , we obtain the identityZA⊕B
hdx =ZA
hdx +ZB
hdx .
Here dx is Lebesgue measure in Rd .
I Holds with inequality for all integrable superharmonicfunctions h on A⊕B.
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Quadrature Domains
I Given x1, . . .xk ∈ Rd and λ1, . . . ,λk > 0.
I D ⊂ Rd is called a quadrature domain for the data (xi ,λi ) if
ZD
h(x)dx ≤k
∑i=1
λih(xi ).
for all integrable superharmonic functions h on D.(Aharonov-Shapiro ’76, Gustafsson, Sakai, ...)
I The smash sum B1⊕ . . .⊕Bk is such a domain, where Bi isthe ball of volume λi centered at xi .
I Generalizes the mean value property of superharmonicfunctions, which corresponds to the case k = 1.
I In two dimensions, the boundary of B1⊕ . . .⊕Bk lies on analgebraic curve of degree 2k .
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ZZD
h(x ,y)dx dy = h(−1,0) +h(1,0)
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Hele-Shaw Flow
I Blob of incompressible fluid sandwiched in the narrow gapbetween two plates.
I Given x1, . . .xk ∈ R2 and λ1, . . . ,λk > 0.
I Identifying one of the plates with R2, inject volume λi of fluidat xi for each i .
I The resulting shape of the blob is the smash sum of disksB1⊕ . . .⊕Bk of area λi centered at xi .
I Stamping problem: Starting with blob A∪B, “stamp” thetwo plates together on A∩B.
I The resulting shape of the blob will be the smash sum A⊕B.
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Internal Erosion, or “Negative Sources”
I Given a finite set A⊂ Zd containing the origin.
I Start a simple random walk at the origin.
I Stop the walk when it reaches a site x ∈ A adjacent to thecomplement of A.
I Lete(A) = A−{x}.
We say that x is eroded from A.
I Iterate this operation until the origin is eroded.
I The resulting random set is called the internal erosion of A.
Lionel Levine Free Boundary Problems and Growth Models
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Internal erosion of a disk of radius 250 in Z2.
Lionel Levine Free Boundary Problems and Growth Models
![Page 38: Free Boundary Problems Arising from Combinatorial and ...pi.math.cornell.edu/~levine/scalinglimitslides0208.pdf100 point sources arranged on a 10 10 grid in Z2. Sources are at the](https://reader035.vdocuments.us/reader035/viewer/2022071409/6101be037401600e623cf34f/html5/thumbnails/38.jpg)
Internal erosion of a box of side length 500 in Z2.
Lionel Levine Free Boundary Problems and Growth Models
![Page 39: Free Boundary Problems Arising from Combinatorial and ...pi.math.cornell.edu/~levine/scalinglimitslides0208.pdf100 point sources arranged on a 10 10 grid in Z2. Sources are at the](https://reader035.vdocuments.us/reader035/viewer/2022071409/6101be037401600e623cf34f/html5/thumbnails/39.jpg)
Questions
I How many sites are eroded?I If A is (say) a square of side length n in Z2, we would guess
thatE#eroded sites = Θ(nα)
for some 1 < α < 2.
I Analogy with diffusion-limited aggregation.I What is the probability that a given site x is eroded?
I For some sites, is this probability o(nα−2)?
Lionel Levine Free Boundary Problems and Growth Models
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Probability of a given site being eroded from a box in Z2.
Lionel Levine Free Boundary Problems and Growth Models
![Page 41: Free Boundary Problems Arising from Combinatorial and ...pi.math.cornell.edu/~levine/scalinglimitslides0208.pdf100 point sources arranged on a 10 10 grid in Z2. Sources are at the](https://reader035.vdocuments.us/reader035/viewer/2022071409/6101be037401600e623cf34f/html5/thumbnails/41.jpg)
Internal Erosion in One Dimension
I Interval A = [−m,n]⊂ Z with −m ≤ 0≤ n.
I Transition probabilities are given by gambler’s ruin:
P([−m,n], [−m,n−1]) =m
m +n;
P([−m,n], [1−m,n]) =n
m +n.
I How large is the remaining interval when the origin getseroded?
I “OK Corral” Process: Gunfight with m fighters on one sideand n on the other. Williams-McIlroy ’98, Kingman ’99,Kingman-Volkov ’03.
Lionel Levine Free Boundary Problems and Growth Models
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The Number of Surviving Gunners
I Let m = n (an equal gunfight).
I Theorem (Kingman-Volkov ’03) Starting from the interval[−n,n], let R(n) be the number of sites remaining when theorigin is eroded. Then as n→ ∞
R(n)
n3/4=⇒
(8
3
)1/4√|Z | (1)
where Z is a standard Gaussian.
Lionel Levine Free Boundary Problems and Growth Models
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Two Mystery Shapes
Lionel Levine Free Boundary Problems and Growth Models