percolation theory by benjamin hansen: slides

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

Differences Among These Graphs?

With Largest Cluster Highlighted

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

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

Definitions

Square Lattice, Z2

Sites and BondsOpen and Closed

Dual Lattice, Z2∗

Bond Configuration, p

Definitions

Square Lattice, Z2

Dual Lattice, Z2∗

Definitions

Square Lattice, Z2

Dual Lattice, Z2∗

Definitions

Square Lattice, Z2

Dual Lattice, Z2∗

Definitions

Square Lattice, Z2

Dual Lattice, Z2∗

More Definitions

Walk (Open Walk)Path (Open Path)µd : The number ofpaths of length dCycle (Open Cycle)Open ClusterInfinite Cluster

More Definitions

Boundary of a Finite Open Cluster

Theorem:We can draw an open boundary in the dual lattice aroundevery finite open cluster.

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.

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.

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 .

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.

Count the expected number of dual cycles around our line,Ep(Ym).Counting dual cycles is hard! Counting paths is easier!

Some of the things we will end up including:

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) ≤∞∑

µ2dd(1− p)2d ≤ 43

∞∑d=m

d [3(1− p)]2d <∞

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.

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.

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.

Summary

PercolationRandomness Due to the MediumA Rapid Transitions in a Random SystemReally, Really Big Networks

ApplicationsChemistry and Material ScienceEpidemiologyImpressing Your Barista

Any Questions?

Summary

PercolationRandomness Due to the MediumA Rapid Transitions in a Random SystemReally, Really Big Networks

ApplicationsChemistry and Material ScienceEpidemiologyImpressing Your Barista

Any Questions?

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

percolation theory by benjamin hansen: slides

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