MCMC

Ising and Potts Models

Let 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

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 $x_v \gets +1$ with probability
$\pi(+1 \vert (x_w)_{w\neq v})$ $= \displaystyle\left[ 1 + \exp\left(-2\theta \sum_{w:\{v,w\}\in E} x_w \right) \right]^{-1}$ ;
Set $x_v \gets -1$ with probability
$\pi(-1 \vert (x_w)_{w\neq v})$ $= \displaystyle\left[ 1 + \exp\left(2\theta \sum_{w:\{v,w\}\in E} x_w \right) \right]^{-1}$ .

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

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

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

where

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

Here

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