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History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Percolation Theory
Benjamin Hansen
Western Washington University
May 21, 2014
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Differences Among These Graphs?
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
With Largest Cluster Highlighted
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Outline
1 History and Overview
2 Unique Boundary of a Finite Open Cluster
3 Bounding the Critical Probability: pHProof of Lower BoundProof of Upper Bound
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
History and Overview
What is Percolation?Broadbent (1954): Gas MasksFluid vs MediumDiffusion vs PercolationBroadbent and Hammersley: Monte Carlo SimulationsMurray Hill (1961) Program’s Running Time: 39 Hours
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Definitions
Square Lattice, Z2
Sites and BondsOpen and Closed
Dual Lattice, Z2∗
Bond Configuration, p
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Definitions
Square Lattice, Z2
Sites and BondsOpen and Closed
Dual Lattice, Z2∗
Bond Configuration, p
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Definitions
Square Lattice, Z2
Sites and BondsOpen and Closed
Dual Lattice, Z2∗
Bond Configuration, p
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Definitions
Square Lattice, Z2
Sites and BondsOpen and Closed
Dual Lattice, Z2∗
Bond Configuration, p
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Definitions
Square Lattice, Z2
Sites and BondsOpen and Closed
Dual Lattice, Z2∗
Bond Configuration, p
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
More Definitions
Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Boundary of a Finite Open Cluster
Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Bounding the Critical Probability: pH
Let pH be the value for which if p > pH , the probability theorigin is part of an infinite cluster, θ(p), is greater than zeroand if p < pH , then θ(p) = 0.
We will show: 1/3 ≤ pH ≤ 2/3.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Lower Bound: 1/3 ≤ pH
Two battling forces: As we increase d ,The number of paths, µd , increasesThe probability that a path is open decreases.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Who Wins?
Let N(d) be the number of open paths of length d . As dincreases, for p < 1/3:
θ(p) ≤ P(N(d) ≥ 1) Infinite cluster⇒ path of length d
≤ E(N(d))∑
p(x) ≤∑
p(x)x for x ≥ 1
= µdpd Expectation is linear
≤ 4(3d−1)pd µd ≤ 4(3d−1)
=43(3p)d → 0
Probability a path isn’t open beats the number of paths.Hence 1/3 ≤ pH .
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
If p > 2/3, it’s easy to have open paths in Z2 and hard topaths in the dual.Assume we have a open line of length m.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Count the expected number of dual cycles around our line,Ep(Ym).Counting dual cycles is hard! Counting paths is easier!
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Some of the things we will end up including:
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Some of the things we will end up including:
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Some of the things we will end up including:
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Some of the things we will end up including:
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Expected Number of Open Dual Cycles
Every length, 2d , of a cycle has at most d choices for astarting location. For one length d :µd x(# of starting locations)x(probability path is open)
Ep(Ym) ≤∞∑
d=m
µ2dd(1− p)2d ≤ 43
∞∑d=m
d [3(1− p)]2d <∞
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
Crank m up!
Ep(Ym) ≤43
∞∑d=m
d [3(1− p)]2d < 1
For long enough line, the expected number of dual cycles isless than one! Ep(Y ) < 1⇒ P(No Dual Cycles!)> 0.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Positive Probability of No Dual Cycles
No dual cycles⇒ no boundary.No boundary⇒ our open cluster is infinite!
Hold the phone, assuming the bonds in that line are open.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Proof of Lower BoundProof of Upper Bound
Proof of Upper Bound: pH ≤ 2/3
If our path is of length m, then our line will be open withprobability pm > 0.Two Events:
No dual cyclesThe line is open
These events are independent!So we have some chance of having an infinite cluster. SopH ≤ 2/3.Hence 1/3 ≤ pH ≤ 2/3.
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Summary
PercolationRandomness Due to the MediumA Rapid Transitions in a Random SystemReally, Really Big Networks
ApplicationsChemistry and Material ScienceEpidemiologyImpressing Your Barista
Any Questions?
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Summary
PercolationRandomness Due to the MediumA Rapid Transitions in a Random SystemReally, Really Big Networks
ApplicationsChemistry and Material ScienceEpidemiologyImpressing Your Barista
Any Questions?
Benjamin Hansen Percolation Theory
History and OverviewUnique Boundary of a Finite Open Cluster
Bounding the Critical Probability: pHSummary
Bibliography
Bollobás, Bela, and Oliver Riordan. Percolation.Cambridge: Cambridge University Press, 2006.
Hammersley, J.M. Origins of Percolation Theory. Annalsof the Israel Physical Society, Vol. 5. 1983
Benjamin Hansen Percolation Theory