Diffusion Limited Aggregation Forest Jacob J. Kagan PIMS-mPrime - - PowerPoint PPT Presentation

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Diffusion Limited Aggregation Forest Jacob J. Kagan PIMS-mPrime - - PowerPoint PPT Presentation

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Introduction Monotonic IDLA forest First Passage percolation Diffusion Limited Aggregation Forest Jacob J. Kagan PIMS-mPrime Probability summer school 2012 Joint work with Noam Berger (HUJI) and Eviatar B. Procaccia (WIZ) June 18, 2012 1 /

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Introduction Monotonic IDLA forest First Passage percolation

Diffusion Limited Aggregation Forest

Jacob J. Kagan

PIMS-mPrime Probability summer school 2012 Joint work with Noam Berger (HUJI) and Eviatar B. Procaccia (WIZ)

June 18, 2012

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Introduction Monotonic IDLA forest First Passage percolation

The internal diffusion limited aggregation (IDLA) was first proposed by Meakin and Deutch (1986) to model industrial chemical processes like electropolishing, corrosion and etching.

Figure: IDLA of n = 4 × 105 points. (Jerison, Levine and Sheffield 2012)

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Introduction Monotonic IDLA forest First Passage percolation

The internal diffusion limited aggregation (IDLA) was first proposed by Meakin and Deutch (1986) to model industrial chemical processes like electropolishing, corrosion and etching.

Figure: IDLA of n = 4 × 105 points. (Jerison, Levine and Sheffield 2012)

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Introduction Monotonic IDLA forest First Passage percolation

We consider a model similar in spirit, for which we can show an interesting behavior. We call our model Monotonic IDLA Forest (MIF).

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Introduction Monotonic IDLA forest First Passage percolation

We consider a model similar in spirit, for which we can show an interesting behavior. We call our model Monotonic IDLA Forest (MIF).

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Introduction Monotonic IDLA forest First Passage percolation

the model

Consider the upper half of the Z2 rotated lattice. With every even vertex on the x axis associate an independent poisson clock.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

the model

Consider the upper half of the Z2 rotated lattice. With every even vertex on the x axis associate an independent poisson clock.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

When a clock of some vertex rings, start a random walk from the vertex until reaching the cluster’s boundary. If the walk tries to move to an adjacent tree or to close a loop, it dies.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

When a clock of some vertex rings, start a random walk from the vertex until reaching the cluster’s boundary. If the walk tries to move to an adjacent tree or to close a loop, it dies.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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SLIDE 10

Introduction Monotonic IDLA forest First Passage percolation

When a clock of some vertex rings, start a random walk from the vertex until reaching the cluster’s boundary. If the walk tries to move to an adjacent tree or to close a loop, it dies.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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SLIDE 11

Introduction Monotonic IDLA forest First Passage percolation

When a clock of some vertex rings, start a random walk from the vertex until reaching the cluster’s boundary. If the walk tries to move to an adjacent tree or to close a loop, it dies.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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SLIDE 12

Introduction Monotonic IDLA forest First Passage percolation

When a clock of some vertex rings, start a random walk from the vertex until reaching the cluster’s boundary. If the walk tries to move to an adjacent tree or to close a loop, it dies.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

This way we get a forest that will eventually reach every vertex.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

Question: Are all the trees finite?

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50

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Introduction Monotonic IDLA forest First Passage percolation

Question: Or do infinite trees exist?

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50

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Introduction Monotonic IDLA forest First Passage percolation

First Passage percolation

First passage percolation (FPP) is a random metric on a graph. Each edge e is given a weight w(e). The distance between two vertices is the minimal sum of weights over paths connecting the vertices. dω(x, y) = min

γ:x→y

  • e∈γ

w(e) note that this definition generalizes in an obvious way to a distance between sets. dω(A, B) = min

γ:x→y

x∈A,y∈B

  • e∈γ

w(e)

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Introduction Monotonic IDLA forest First Passage percolation

First Passage percolation

First passage percolation (FPP) is a random metric on a graph. Each edge e is given a weight w(e). The distance between two vertices is the minimal sum of weights over paths connecting the vertices. dω(x, y) = min

γ:x→y

  • e∈γ

w(e) note that this definition generalizes in an obvious way to a distance between sets. dω(A, B) = min

γ:x→y

x∈A,y∈B

  • e∈γ

w(e)

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SLIDE 18

Introduction Monotonic IDLA forest First Passage percolation

First Passage percolation

First passage percolation (FPP) is a random metric on a graph. Each edge e is given a weight w(e). The distance between two vertices is the minimal sum of weights over paths connecting the vertices. dω(x, y) = min

γ:x→y

  • e∈γ

w(e) note that this definition generalizes in an obvious way to a distance between sets. dω(A, B) = min

γ:x→y

x∈A,y∈B

  • e∈γ

w(e)

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SLIDE 19

Introduction Monotonic IDLA forest First Passage percolation

First Passage percolation

First passage percolation (FPP) is a random metric on a graph. Each edge e is given a weight w(e). The distance between two vertices is the minimal sum of weights over paths connecting the vertices. dω(x, y) = min

γ:x→y

  • e∈γ

w(e) note that this definition generalizes in an obvious way to a distance between sets. dω(A, B) = min

γ:x→y

x∈A,y∈B

  • e∈γ

w(e)

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Introduction Monotonic IDLA forest First Passage percolation

the coupling

We couple the process to a FPP process with weights ∼ exp( 1

2n )

  • n level n, denoting by ˆ

T the measure on trees which are the union of geodesics from all vertices to the x axis.

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

w w w w w w~exp(1/4) w'~exp(1/8) w' w' w' w' w' 10 / 17

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Introduction Monotonic IDLA forest First Passage percolation

Lemma 1 There exists a coupling measure Q such that Q(∀x, T(x) = ˆ T(x)) = 1 Proof sketch. For each monotone path γ ⊆ { ˆ T(x) ∪ ∂ ˆ T(x)}, γ = (e1, . . . , el(γ)) from x we assign rings: if γ ⊂ ˆ T(x) we assign the ring l(γ)

i=1 ω(ei), and the path of

the particle will be γ. if γ ˆ T(x) we assign the ring sequence l(γ)

i=1 ω(ei),

l(γ)

i=1 ω(ei) + Poisson(el(γ)), for each ring in this sequence the

particle will be assigned the path γ. Note that in the second case, all the particles will vanish

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Introduction Monotonic IDLA forest First Passage percolation

Lemma 1 There exists a coupling measure Q such that Q(∀x, T(x) = ˆ T(x)) = 1 Proof sketch. For each monotone path γ ⊆ { ˆ T(x) ∪ ∂ ˆ T(x)}, γ = (e1, . . . , el(γ)) from x we assign rings: if γ ⊂ ˆ T(x) we assign the ring l(γ)

i=1 ω(ei), and the path of

the particle will be γ. if γ ˆ T(x) we assign the ring sequence l(γ)

i=1 ω(ei),

l(γ)

i=1 ω(ei) + Poisson(el(γ)), for each ring in this sequence the

particle will be assigned the path γ. Note that in the second case, all the particles will vanish

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SLIDE 23

Introduction Monotonic IDLA forest First Passage percolation

Lemma 1 There exists a coupling measure Q such that Q(∀x, T(x) = ˆ T(x)) = 1 Proof sketch. For each monotone path γ ⊆ { ˆ T(x) ∪ ∂ ˆ T(x)}, γ = (e1, . . . , el(γ)) from x we assign rings: if γ ⊂ ˆ T(x) we assign the ring l(γ)

i=1 ω(ei), and the path of

the particle will be γ. if γ ˆ T(x) we assign the ring sequence l(γ)

i=1 ω(ei),

l(γ)

i=1 ω(ei) + Poisson(el(γ)), for each ring in this sequence the

particle will be assigned the path γ. Note that in the second case, all the particles will vanish

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SLIDE 24

Introduction Monotonic IDLA forest First Passage percolation

Lemma 1 There exists a coupling measure Q such that Q(∀x, T(x) = ˆ T(x)) = 1 Proof sketch. For each monotone path γ ⊆ { ˆ T(x) ∪ ∂ ˆ T(x)}, γ = (e1, . . . , el(γ)) from x we assign rings: if γ ⊂ ˆ T(x) we assign the ring l(γ)

i=1 ω(ei), and the path of

the particle will be γ. if γ ˆ T(x) we assign the ring sequence l(γ)

i=1 ω(ei),

l(γ)

i=1 ω(ei) + Poisson(el(γ)), for each ring in this sequence the

particle will be assigned the path γ. Note that in the second case, all the particles will vanish

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SLIDE 25

Introduction Monotonic IDLA forest First Passage percolation

Lemma 1 There exists a coupling measure Q such that Q(∀x, T(x) = ˆ T(x)) = 1 Proof sketch. For each monotone path γ ⊆ { ˆ T(x) ∪ ∂ ˆ T(x)}, γ = (e1, . . . , el(γ)) from x we assign rings: if γ ⊂ ˆ T(x) we assign the ring l(γ)

i=1 ω(ei), and the path of

the particle will be γ. if γ ˆ T(x) we assign the ring sequence l(γ)

i=1 ω(ei),

l(γ)

i=1 ω(ei) + Poisson(el(γ)), for each ring in this sequence the

particle will be assigned the path γ. Note that in the second case, all the particles will vanish

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Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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SLIDE 29

Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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SLIDE 30

Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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SLIDE 31

Introduction Monotonic IDLA forest First Passage percolation

Theorem 3.1 All the trees are finite almost surely. Proof sketch. Assume an infinite tree exists. Denote p = P(|T(0)| = ∞) by translational invariance we have E[|T m(0)||T(0) = ∞] ≤ 1 p we try to kill the tree at a sequence of levels where the tree has a fixed diameter. this is a sequence of independent, finite events with positive probability a Borel-Cantelli argument gives a contradiction.

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Introduction Monotonic IDLA forest First Passage percolation

y x

2 8

  • 2

4 6

  • 6
  • 4

2 4 6 8 10 12 14

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Introduction Monotonic IDLA forest First Passage percolation

FPP with decreasing weights

Theorem 3.2 All the trees with edge weights exp(2n) are finite almost surely.

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Introduction Monotonic IDLA forest First Passage percolation

20 40 60 80 100 120 140 160 180 200 10 20 30 40 50 60 70 80

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Introduction Monotonic IDLA forest First Passage percolation 16 / 17

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Introduction Monotonic IDLA forest First Passage percolation

Questions?

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