Percolation theory studies random subgraphs formed by independently or dependently declaring vertices or edges open.
In bond percolation with parameter , every edge of a graph is independently open with probability and closed with probability .
The percolation susceptibility is the expected size of the open cluster containing a specified root:
At critical planar percolation, the probability of an open crossing of a rectangle of any fixed aspect ratio stays bounded away from zero and one uniformly over its scale.
The Russo-Seymour-Welsh theorem and the Harris-FKG inequality imply for a scale-independent constant .
RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.
In site percolation, vertices are declared open or closed and one studies connected components of the induced open subgraph.
Dependent percolation allows the open states of different vertices or edges to be statistically dependent.
The critical threshold is the boundary between levels at which an unbounded superlevel component can occur and levels at which every superlevel component is almost surely finite.
A random field is finite-range dependent if collections indexed by sets farther apart than a fixed distance are independent. Sparse subsets of long paths then restore enough independence for path-counting arguments.
The one-arm event is the event that a specified root is joined through open vertices or edges to graph distance .
For a Boolean function of independent coordinates and a randomized decision tree that determines it, the OSSS inequality bounds its variance by a sum of coordinate influences weighted by their revealment probabilities.
A decision tree adaptively reveals input coordinates until their observed values determine the output. Its revealment for a coordinate is the probability that the coordinate is inspected.
Choosing an intermediate radius uniformly and exploring the open cluster meeting that sphere gives a decision tree for a one-arm event. Translation and a union bound control each revealment by a constant times the average of the one-arm probabilities over the possible radii.
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This field is likely both ugly and useless.
OK, in 2D they've achieved some cute rational number results. But still.
Percolation theory is a mathematical concept originally developed in the context of physics and materials science to study the behavior of connected clusters in a random medium. It explores how the properties of such clusters change as the density of the medium is varied. The theory has applications in various fields, including physics, chemistry, computer science, biology, and even social sciences.