Dependent percolation

Dependent percolation allows the open states of different vertices or edges to be statistically dependent.

On a finite graph, the random-cluster model assigns an edge configuration $\omega$ probability proportional towhere $o,c,k$ count open edges, closed edges, and open connected components.

The Edwards-Sokal coupling assigns one common random spin to every random-cluster component. For $q=2$ it couples the random-cluster model to the Ising model.

A heat-bath Markov chain repeatedly chooses one coordinate and redraws it from its conditional distribution given all other coordinates. The target distribution is reversible and stationary for these updates.

For a random field $(Y_x)$, level-set percolation studies the random vertex set $\{x:Y_x\ge a\}$ as the threshold $a$ varies.

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.