Diamond Aggregation Lionel Levine Toronto Probability Seminar - - PowerPoint PPT Presentation

Diamond Aggregation

Lionel Levine Toronto Probability Seminar March 30, 2009 Joint work with Wouter Kager

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Talk Outline

◮

Joint work with Wouter Kager:

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Internal DLA: from random walk to growth model

◮

Uniformly layered walks

◮

Limiting shape and fluctuations ❘

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Talk Outline

◮

Joint work with Wouter Kager:

◮

Internal DLA: from random walk to growth model

◮

Uniformly layered walks

◮

Limiting shape and fluctuations

◮

Joint work with Yuval Peres:

◮

Multiple point sources

◮

Smash sum of two domains in ❘d

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From random walk to growth model

Internal DLA

◮

Given a Markov chain on state space ❩2.

◮

Start with n particles at the origin.

◮

Each particle walks until it finds an unoccupied site, stays there. ❩

Internal DLA

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From random walk to growth model

Internal DLA

◮

Given a Markov chain on state space ❩2.

◮

Start with n particles at the origin.

◮

Each particle walks until it finds an unoccupied site, stays there.

◮

A(n): the resulting random set of n sites in ❩2.

Growth rule:

◮

Let A(1) = {o}, and A(n + 1) = A(n) ∪ {Xn(τn)}

Internal DLA

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From random walk to growth model

Internal DLA

◮

Given a Markov chain on state space ❩2.

◮

Start with n particles at the origin.

◮

Each particle walks until it finds an unoccupied site, stays there.

◮

A(n): the resulting random set of n sites in ❩2.

Growth rule:

◮

Let A(1) = {o}, and A(n + 1) = A(n) ∪ {Xn(τn)} where X1, X2, . . . are independent random walks, and τn = min t | Xn(t) A(n) .

Internal DLA

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The growth rule illustrated

Internal DLA

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The growth rule illustrated

Internal DLA

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The growth rule illustrated

Internal DLA

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Example: simple random walk

Main questions

1. Limiting shape?

Internal DLA

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Example: simple random walk

Main questions

1. Limiting shape? 2. Fluctuation size?

Internal DLA

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Simple random walk

Lawler-Bramson-Griffeath ’92

The limiting shape is a disk: ∀ǫ > 0, with probability 1 B(1−ǫ)n ⊂ A(πn2) ⊂ B(1+ǫ)n eventually.

Internal DLA

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Simple random walk

Lawler-Bramson-Griffeath ’92

The limiting shape is a disk: ∀ǫ > 0, with probability 1 B(1−ǫ)n ⊂ A(πn2) ⊂ B(1+ǫ)n eventually.

Lawler ’95

Strengthened this to show Bn−f(n) ⊂ A(πn2) ⊂ Bn+f(n) eventually for f(n) = n1/3 log4 n.

Internal DLA

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Simple random walk

Lawler-Bramson-Griffeath ’92

The limiting shape is a disk: ∀ǫ > 0, with probability 1 B(1−ǫ)n ⊂ A(πn2) ⊂ B(1+ǫ)n eventually.

Lawler ’95

Strengthened this to show Bn−f(n) ⊂ A(πn2) ⊂ Bn+f(n) eventually for f(n) = n1/3 log4 n.

Someone in the audience ’09 (?)

The true order of fluctuations f(n) is only logarithmic in n.

Internal DLA

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What about other walks?

Modify transition probabilities on the axes:

◮

Steps toward the origin along the x- and y-axes are reflected away from the origin instead. So for x > 0, P((x, 0), (x + 1, 0)) = 1 2 P((x, 0), (x, ±1)) = 1 4.

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What about other walks?

Modify transition probabilities on the axes:

◮

Steps toward the origin along the x- and y-axes are reflected away from the origin instead. So for x > 0, P((x, 0), (x + 1, 0)) = 1 2 P((x, 0), (x, ±1)) = 1 4.

◮

Off the axes, same as simple random walk.

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What about other walks?

Modify transition probabilities on the axes:

◮

Steps toward the origin along the x- and y-axes are reflected away from the origin instead. So for x > 0, P((x, 0), (x + 1, 0)) = 1 2 P((x, 0), (x, ±1)) = 1 4.

◮

Off the axes, same as simple random walk.

◮

Instead of a disk, limiting shape is now a diamond!

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Diamond Aggregation

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Diamond Layers

Notation

◮

• (x, y)
• = |x| + |y|.

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Ln = the diamond layer of radius n =

• z ∈ ❩2 : z = n
• ❩

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Diamond Layers

Notation

◮

• (x, y)
• = |x| + |y|.

◮

Ln = the diamond layer of radius n =

• z ∈ ❩2 : z = n
• ◮

Dn = the diamond of radius n =

• z ∈ ❩2 : z ≤ n
• ◮

dn = #Dn = 2n(n + 1) + 1

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Diamond Layers

Notation

◮

• (x, y)
• = |x| + |y|.

◮

Ln = the diamond layer of radius n =

• z ∈ ❩2 : z = n
• ◮

Dn = the diamond of radius n =

• z ∈ ❩2 : z ≤ n
• ◮

dn = #Dn = 2n(n + 1) + 1

Uniformly layered walk:

Distribution of X(t) is a mixture of uniform distributions on layers Ln.

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Uniformly Layered Walk

Discrete time Markov chain X(t) on state space ❩2 satisfying (U1)

• X(t + 1)
• −
• X(t)
• ≤ 1

(U2) For all n ≥ 1, Po(X(t) ∈ Ln for some t < ∞) = 1. P P

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Uniformly Layered Walk

Discrete time Markov chain X(t) on state space ❩2 satisfying (U1)

• X(t + 1)
• −
• X(t)
• ≤ 1

(U2) For all n ≥ 1, Po(X(t) ∈ Ln for some t < ∞) = 1. (U3) For all k ≥ 0, n ≥ 1 and all x ∈ Ln Pk

• X(t) = x
• X(t) ∈ Ln
• = 1

4n where Pk is the law of the walk started from uniform on layer Lk.

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Shape Theorem

Theorem (Kager-L.)

For any uniformly layered walk, with probability 1 Dn−4√

n log n ⊂ A(dn) ⊂ Dn+20√ n log n

eventually.

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Shape Theorem

Theorem (Kager-L.)

For any uniformly layered walk, with probability 1 Dn−4√

n log n ⊂ A(dn) ⊂ Dn+20√ n log n

eventually.

◮

So all uniformly layered walks have the diamond as their limiting shape.

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Shape Theorem

Theorem (Kager-L.)

For any uniformly layered walk, with probability 1 Dn−4√

n log n ⊂ A(dn) ⊂ Dn+20√ n log n

eventually.

◮

So all uniformly layered walks have the diamond as their limiting shape.

◮

Is

• n log n the right order of fluctuations?

◮

Or do the fluctuations depend on the particular u.l. walk?

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Proof sketch: Containing a large diamond

Fix a site z ∈ Ln−ρ. Want an upper bound on P(z A(dn)). Dn z ρ

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Proof sketch: Containing a large diamond

Fix a site z ∈ Ln−ρ. Want an upper bound on P(z A(dn)).

◮

Among the first dn − 1 walks, let M = # that first hit Ln−ρ at z. L = # that first hit Ln−ρ at z after dropping their particle. Dn z ρ

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Proof sketch: Containing a large diamond

Fix a site z ∈ Ln−ρ. Want an upper bound on P(z A(dn)).

◮

Among the first dn − 1 walks, let M = # that first hit Ln−ρ at z. L = # that first hit Ln−ρ at z after dropping their particle.

◮

Then

• z A(dn)
• ⊂ {L = M}.

Dn z ρ

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Proof sketch: Containing a large diamond

Fix a site z ∈ Ln−ρ. Want an upper bound on P(z A(dn)).

◮

Among the first dn − 1 walks, let M = # that first hit Ln−ρ at z. L = # that first hit Ln−ρ at z after dropping their particle.

◮

Then

• z A(dn)
• ⊂ {L = M}.

Dn z ρ

◮

Both L and M are sums of indicator RV’s.

◮

Main difficulty: The summands of L are dependent.

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Finding independence

Estimating L

Start one new walk from every site of Dn−ρ−1, and let L′ = # of new walks that first hit Ln−ρ at z. Since at most one particle can attach to the cluster at a given site, L ≤ L′. z ρ ❊ ❊ P

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Finding independence

Estimating L

Start one new walk from every site of Dn−ρ−1, and let L′ = # of new walks that first hit Ln−ρ at z. Since at most one particle can attach to the cluster at a given site, L ≤ L′. z ρ

Strategy

Since both L′ and M are sums of independent indicators, show ❊L′ < ❊M and use large deviations to bound P(L′ ≥ M).

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Separating ❊M and ❊L′

Writing ℓ = n − ρ, we have ❊M = (dn − 1)Po(X(τℓ) = x) = 2n(n + 1) 4ℓ > n + ρ 2 . ❊ P P

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Separating ❊M and ❊L′

Writing ℓ = n − ρ, we have ❊M = (dn − 1)Po(X(τℓ) = x) = 2n(n + 1) 4ℓ > n + ρ 2 . ❊L′ =

• y∈Dℓ−1−{o}

Py(X(τℓ) = x) =

ℓ−1

• k=1

4kPk(X(τℓ) = x)

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Separating ❊M and ❊L′

Writing ℓ = n − ρ, we have ❊M = (dn − 1)Po(X(τℓ) = x) = 2n(n + 1) 4ℓ > n + ρ 2 . ❊L′ =

• y∈Dℓ−1−{o}

Py(X(τℓ) = x) =

ℓ−1

• k=1

4kPk(X(τℓ) = x) =

ℓ−1

• k=1

4k 4ℓ = ℓ − 1 2 < n − ρ 2 .

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Final step: Concentration

n/2 M L′ ρ √n √n

Conclusion

By large deviations for sums of independent indicators, P(L = M) ≤ P

• M ≤ n

2

• + P
• L′ ≥ n

2

• has power-law decay for ρ ∼
• n log n.

P

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Final step: Concentration

n/2 M L′ ρ √n √n

Conclusion

By large deviations for sums of independent indicators, P(L = M) ≤ P

• M ≤ n

2

• + P
• L′ ≥ n

2

• has power-law decay for ρ ∼
• n log n.

Done by Borel-Cantelli:

• n≥1
• z∈Dn−ρ

P(z A(dn)) < ∞.

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The order of fluctuations

Dn P

Order of fluctuations

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The order of fluctuations

Dn Dn−f(n) Dn+f(n)

Order of fluctuations:

The slowest rate at which we can let f(n) tend to ∞ so that P

• Dn−f(n) ⊂ A(dn) ⊂ Dn+f(n) eventually
• = 1

Order of fluctuations

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The outward directed layered walk

1/4 1/4 1/4 1/4

Order of fluctuations

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The outward directed layered walk

Order of fluctuations

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The outward directed layered walk

3/4

Order of fluctuations

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The outward directed layered walk

1/8 1/8 3/4

Order of fluctuations

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The outward directed layered walk

1/8 1/8 5/8 3/8 3/4

Order of fluctuations

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The outward directed layered walk

1/8 1/8 5/8 3/8 3/8 5/8 1/8 3/4 1/8 3/4

Order of fluctuations

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The inward directed layered walk

1/6 1/6 5/6 1/2 1/2 5/6 1/6 1 1/6 1

Order of fluctuations

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A natural one-parameter family of walks

Mixed transition matrix:

Q(x, y) = p Q in(x, y) + q Q out(x, y) where p ∈ [0, 1) and p + q = 1.

Transitions between layers:

p p p p q q q q p 1 2 3 4

Order of fluctuations

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Two diamonds from this family

p = 0 walks directed outward p = 3/4 walks biased inward

Order of fluctuations

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Logarithmic Fluctuations

Theorem (Kager-L.)

If p > 1/2, then with probability 1, Dn−6 logr n ⊂ A(dn) ⊂ Dn+6 logr n eventually where r = p/q. P

Order of fluctuations

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Logarithmic Fluctuations

Theorem (Kager-L.)

If p > 1/2, then with probability 1, Dn−6 logr n ⊂ A(dn) ⊂ Dn+6 logr n eventually where r = p/q.

Proof sketch

Fix z ∈ Ln−ρ. If we stop a walk when it hits Ln, then P(z not visited) ≤ 4(n − ρ) q p ρ which has power-law decay for ρ ∼ logr n. Dn z ρ

Order of fluctuations

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The must-drop-somewhere argument

Conclusion (inner fluctuations)

P

• all dn walks visit all sites in Dn−ρ before hitting Ln
• ≥ 1 − n−2.

P P P

Order of fluctuations

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The must-drop-somewhere argument

Conclusion (inner fluctuations)

P

• all dn walks visit all sites in Dn−ρ before hitting Ln
• ≥ 1 − n−2.

Since the walks must drop their particles somewhere, this implies P

• Dn−ρ ⊂ A(dn)
• ≥ 1 − n−2.

P P

Order of fluctuations

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The must-drop-somewhere argument

Conclusion (inner fluctuations)

P

• all dn walks visit all sites in Dn−ρ before hitting Ln
• ≥ 1 − n−2.

Since the walks must drop their particles somewhere, this implies P

• Dn−ρ ⊂ A(dn)
• ≥ 1 − n−2.

The outer fluctuations

P

• all dn walks visit all sites in Dn before hitting Ln+ρ
• ≥ 1 − n−2

implies P

• A(dn) ⊂ Dn+ρ
• ≥ 1 − n−2

Order of fluctuations

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What about a lower bound?

For the outward directed walk (p = 0):

◮

Set Tk = time when (0, k) joins the cluster

Order of fluctuations

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What about a lower bound?

For the outward directed walk (p = 0):

◮

Set Tk = time when (0, k) joins the cluster

◮

Set X1 = T1 and Xk+1 = Tk+1 − Tk.

◮

The Xi are independent geometric RV’s.

Order of fluctuations

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What about a lower bound?

For the outward directed walk (p = 0):

◮

Set Tk = time when (0, k) joins the cluster

◮

Set X1 = T1 and Xk+1 = Tk+1 − Tk.

◮

The Xi are independent geometric RV’s. ⇒ Law of the iterated logarithm for Tk. ∴ Fluctuations are at least order

• n log log n.

Order of fluctuations

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Phase diagram: Order of the fluctuations

p 0 1/2 1 Lower bound Upper bound

• n log n
• n log log n

log n

Order of fluctuations

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Phase diagram: Order of the fluctuations

p 0 1/2 1 Lower bound Upper bound

• n log n
• n log log n

log n

• n log n

Order of fluctuations

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Phase diagram: Order of the fluctuations

p 0 1/2 1 Lower bound Upper bound

• n log n
• n log log n

log n

• n log n
• n log log n?

?

Order of fluctuations

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Phase diagram: Order of the fluctuations

p 0 1/2 1 Lower bound Upper bound

• n log n
• n log log n

log n

• n log n
• n log log n?

? ? ?

Order of fluctuations

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Questions

◮

Are there any uniformly layered walks in ❩d for d > 2? ❩

Further questions

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Questions

◮

Are there any uniformly layered walks in ❩d for d > 2?

◮

In ❩2, what if walks start somewhere other than the origin?

Further questions

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Questions

◮

Are there any uniformly layered walks in ❩d for d > 2?

◮

In ❩2, what if walks start somewhere other than the origin?

◮

The outward directed layered walk, started from (1, 1):

Further questions

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Mystery shapes: Off-center starting point

(3,3) (2,2) (3,2) (1,1) (2,1) (3,1)

Further questions

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