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Ising and Potts Models

Let $ G = (V,E)$ be a graph representing a lattice (or a rectangular grid). and $ \Lambda = \{-1,+1\}$ be the states of spin ``up'' and ``down.'' The Hamiltonian is given by

$\displaystyle H(x) = -J \sum_{\{v,w\}\in E} x_v x_w - h\sum_{v\in V} x_v

where $ J$ and $ h$ 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 $ h=0$, the Ising model has the following site update:

The chance for the site $ v$ getting $ +1$ increases as the number of neighbors being $ +1$. In this sense the model is called an attractive spin system.

Critical temperature. When the temperature $ T = 1/\beta'$ 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 $ T_c$, and that $ T_c = 2/\ln(1+sqrt{2}) \approx 2.27$ when the graph $ G$ is a 2-dimensional grid.

Potts Model. Let $ \Lambda = \{1,\ldots,M\}$. Then the Ising model is generalized with the Hamiltonian

$\displaystyle H(x) = -J \sum_{\{v,w\}\in E} \delta(x_v, x_w) - h\sum_{v\in V}
\delta(x_v, f_v)


$\displaystyle \delta(\lambda, \lambda') = \begin{cases}
1 & \mbox{ if $\lambda = \lambda'$; } \\
0 & \mbox{ if $\lambda \neq \lambda'$. }

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

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