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Critical point, continued

April 3, 2013

It turns out I should have read Renfrey B. Potts’ 1952 paper more carefully, since equation (9) gives the general form of \beta_{\textrm{critical}} for a regular 2D lattice, where r is the number of unique configurations:

\log(1 + \sqrt{r})

Marin & Robert (2007; sect. 8.3) applied the hidden Potts model to a satellite image of Lake Menteith, in Scotland (56^{\circ}10'36"N \;\;\; 4^{\circ}17'39"W):

lm3 <- as.matrix(read.table("Menteith"))
y <- as.vector(lm3)
image(1:100,1:100,t(lm3[100:1,]),col=gray(256:1/256),xlab="",ylab="",main="Lake of Menteith")

Lake Menteith

Where r=6, \beta_{crit} = \log(1 + \sqrt{6}) = 1.238226\dots

library(PottsUtils)
library(bayesImageS)

k <- 6
bcrit <- log(1 + sqrt(k))
beta <- seq(0,3.5,by=0.1)
counts <- matrix(0,length(beta),1000)
neigh <- getNeighbors(matrix(1,100,100), c(2,2,0,0))
block <- getBlocks(matrix(1,100,100), 2)
for (i in 1:length(beta)) {
  result <- mcmcPottsNoData(beta[i],k,neigh,block,11000)
  counts[i,] <- result$sum[10001:11000]
}
x <- rep(beta,each=1000)
y <- as.vector(t(counts))
plot(x,y,xlab=expression(beta),ylab="identical neighbours",main=paste0("Critical temperature of the 2D Potts model, k=",k))
abline(v=2/bcrit,col="red")
abline(v=bcrit,col="blue")

Critical temperature of the 2D Potts model, k=6
The phase transition at the critical point (the blue line) is more pronounced than in my last post, for k=2. As noted by Marin & Robert, for β>2 all of the pixels are given the same label. The slight variability beyond that point is due to very slow mixing of the MCMC samples at those temperatures. I might try using the Swendsen-Wang algorithm instead, to see if that produces a clearer picture.

References

Marin J M & Robert C P (2007) Bayesian Core: A Practical Approach to Computational Bayesian Statistics Springer-Verlag: New York

Potts R B (1952) Some generalized order-disorder transformations. Proc. Camb. Phil. Soc. 48 106

Swendsen R H & Wang J (1987) Nonuniversal critical dynamics in Monte Carlo simulations Phys. Rev. Lett. 58(2): 86–88

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