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

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

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).


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.


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.


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 .


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.


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?


Two-source internal DLA cluster built from overlappingsingle-source clusters.


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 .


Diaconis-Fulton sum of two squares in Z2 overlapping in asmaller square.


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.


Divisible sandpile sum of two squares in Z2 overlapping in asmaller square.


Diaconis-Fulton sum Divisible sandpile sum


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.


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.


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.


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.


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)


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.


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.


Internal DLA Divisible Sandpile Rotor-Router Model


Steps of the Proof

convergence of densities

⇓convergence of obstacles

⇓convergence of majorants

⇓convergence of domains.


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.


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.


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.


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!


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.


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 .


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.


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.


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.


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.


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 .


ZZD

h(x ,y)dx dy = h(−1,0) +h(1,0)


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.


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.


Internal erosion of a disk of radius 250 in Z2.


Internal erosion of a box of side length 500 in Z2.


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)?


Probability of a given site being eroded from a box in Z2.


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.


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.


Two Mystery Shapes


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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