Critical behavior and lack of self-averaging in the dynamics of the random Potts model in two dimensions

We study the dynamics of the q-state random bond Potts ferromagnet on the square lattice at its critical point by Monte Carlo simulations with single spin-flip dynamics. We concentrate on $q=3\mathrm{}$ and $q=24\mathrm{}$ and find, in both cases, conventional, rather than activated, dynamics. We also look at the distribution of relaxation times among different samples, finding different results for the two q values. For $q=3\mathrm{}$ the relative variance of the relaxation time $\tau$ at the critical point is finite. However, for $q=24\mathrm{}$ this appears to diverge in the thermodynamic limit and it is $\ln \tau$ which has a finite relative variance. We speculate that this difference occurs because the transition of the corresponding pure system is second order for $q=3\mathrm{}$ but first order for $q=24.\mathrm{}$