Gibbs Measure - Gibbs Measure On Lattices

Gibbs Measure On Lattices

What follows is a formal definition for the special case of a random field on a group lattice. The idea of a Gibbs measure is, however, much more general than this.

The definition of a Gibbs random field on a lattice requires some terminology:

• The lattice: A countable set .

• The single-spin space: A probability space .

• The configuration space:, where and .

• Given a configuration and a subset, the restriction of to is . If and, then the configuration is the configuration whose restrictions to and are and, respectively. These will be used to define cylinder sets, below.

• The set of all finite subsets of .

• For each subset, is the -algebra generated by the family of functions, where . This sigma-algebra is just the algebra of cylinder sets on the lattice.

• The potential: A family of functions such that
  1. For each, is -measurable.
  2. For all and, the series exists.

• The Hamiltonian in with boundary conditions, for the potential, is defined by

where .

• The partition function in with boundary conditions and inverse temperature (for the potential and ) is defined by

A potential is -admissible if is finite for all, and .

A probability measure on is a Gibbs measure for a -admissible potential if it satisfies the Dobrushin-Lanford-Ruelle (DLR) equations

for all and .