Site percolation

In site percolation, vertices are declared open or closed and one studies connected components of the induced open subgraph.

For independent site percolation on the triangular lattice, the critical probability is $p_c=1/2$. Self-duality, crossing estimates, and exclusion of an infinite critical cluster identify the threshold.

Conditioned on the sites revealed by an exploration interface, every unrevealed site retains its independent Bernoulli law. The explored boundary supplies boundary conditions for subsequent crossing events.

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.