## Potts Model

The Ising model is a special case of the Potts model since their Hamiltonian functions and satisfy when or accordingly as or 0. The Potts model in potts.r assumes . Thus, the Potts model with and the inverse temperature simulates the phase transition for Ising model.

> source("potts.r") > potts(m=2,tmax=1000,beta=0.88) > potts(m=2,tmax=1000,beta=0.89)

Reference to external force. An external force at each site is indicated by where the state 0 implies no external force at a particular site. Entire data of external force is represented in a matrix. To see the external force, use it as initial state

> potts(m=2,tmax=0,init.state=f1) > potts(m=5,tmax=0,init.state=f2)

Explore it. Download potts.r. Choose a different number of common states and inverse temperature , and see how the Gibbs sampler generates a GRF with or without existence of external force.

> source("potts.r") > potts(m=2,tmax=1000,beta=0.9) > potts(m=2,tmax=1000,beta=0.9,ref.state=f1) > potts(m=5,tmax=1000,beta=1.1) > potts(m=5,tmax=1000,beta=1.1,ref.state=f2)

Programming note.
An matrix of data `a` is created with

data.matrix = matrix(data=a, n, m)where the data is a sequence of values, and will be assigned from the first column to the last column of the matrix. For example, a matrix form of Potts state in grid is created and visualized as follows.

> s1 = matrix(data=c(1,2,3,3,1,2,2,3,1,1,2,2,1,1,1,2), 4, 4) > potts(m=3,tmax=0,init.state=s1)

Explore it. Create a matrix of external force, and run the Gibbs sampler for Potts model.

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