Finite-energy property (percolation)

A percolation measure has the finite-energy property if, conditional on all exterior states, every prescribed state of any finite collection of sites or edges has positive probability almost surely. This permits finite local modifications within events of positive probability. It is a hypothesis on the measure and does not follow merely from invariance under graph symmetries.

For $0<p<1$, prescribing any finite collection of Bernoulli percolation sites or edges has positive conditional probability given all states outside that collection.

For $0<p<1$, every prescribed state pattern on a finite edge set has positive conditional probability given all exterior states: independence makes it $p^a(1-p{)}^b$ for $a$ prescribed open and $b$ prescribed closed edges. Thus a positive-probability exterior event remains possible after any finite local surgery. This is the finite-energy property of Bernoulli percolation, and applies equally to dual bond percolation.