Going Critical

Vidhi Lalchand
vr308@cam.ac.uk

(A 10 minute introduction to percolation theory)

17th May, 2021

In 1957 Hammersley and Broadbent considered the propagation of a “fluid” through a medium like a fractured rock, crystal or a maze.

At which p do we have a giant connected cluster?

Abstraction: 2d Lattice

In this short talk we take an informal and intuitive look at percolation theory, and its fascinating connection to vast array of natural and social phenomena.

"If the grounds are packed too tightly , water may not find a path through.When they are loose enough: drip!"

John Hammersley (1920-2004)

p=0.75

Percolation of fluid through a medium

p=0.3

Credit: Jen Christiansen, Scientific American. "The Math of Large connections"

probability of percolation

The percolation process is subject to the 0-1 law of probability theory

p < p_{c} \longrightarrow f(p) = 0 \\ p > p_{c} \longrightarrow f(p) = 1

\( L = 8\)

\( L = 100\)

Bush Fires

Percolation theory underpins a staggering number of physical and social phenomena.

p=0.53

p=0.593

http://cormas.cirad.fr/en/applica/fireautomata.htm

green - tree, red - tree on fire, grey - ash

Wireless Mesh Networks

Huynh H.N. (2019) Continuum Percolation and Spatial Point Pattern in Application to Urban Morphology. In: D'Acci L. (eds) The Mathematics of Urban Morphology. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12381-9_18

Flash crash

DJIA plunged 600 points for no specific reason, at 2:35 pm - May 6th, 2010

Vuorenmaa, Tommi; Wang, Liang (October 2013), "An Agent-Based Model of the Flash Crash of May 6, 2010, with Policy Implications", VALO Research and University of Helsinki, SSRN 2336772

Percolation & Pandemics

Insights from percolation are used to model \( R_{0} \) - albeit one has to juggle are large number of unknowns.

A Google scholar search of percolation and pandemics yields 18000 hits.

Great paper->

Does this have something to do with the power-law?

For \( p > p_{c} \), the mean cluster size diverges but just at the critcial threshold the size of the mean clusters exhibit a power-law like distribution

Power Laws, Pareto Distributions and Zipf's Law, MEJ Newman, Contemporary Physics, 2005