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## Ising and Potts Models

Let be a graph representing a lattice (or a rectangular grid). and be the states of spin up'' and down.'' The Hamiltonian is given by

where and represent the strength of interaction between neighbors and that of external magnetic field. The GRF with this Hamiltonian is called an Ising model.

Attractive spin system. Assuming , the Ising model has the following site update:

• Set with probability

;

• Set with probability

.

The chance for the site getting increases as the number of neighbors being . In this sense the model is called an attractive spin system.

Critical temperature. When the temperature is low, spins are aligned, creating a large cluster of aligned states. As the temperature gets higher, the randomness takes over. In the Ising model it is known that this phase transition'' occurs at the critical temperature , and that when the graph is a 2-dimensional grid.

Potts Model. Let . Then the Ising model is generalized with the Hamiltonian

where

Here is a configuration of external reference. This GRF is called a Potts model.