## 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

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

- Set
with probability
;

- Set
with probability
.

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

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