# Critical point, continued

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

Marin & Robert (2007; sect. 8.3) applied the hidden Potts model to a satellite image of Lake Menteith, in Scotland ():

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")

Where ,

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")

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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