We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average
(n=1), and others. In addition, for
q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.
We study bond percolation in two dimensions between random site variables having one out of
q colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature
T in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several
q-state Potts Hamiltonians. For high
T, these transitions fit, irrespective of
q, in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$,
q-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at
q=2. For
q<2 the Potts critical state appears to enhance percolation, for
q>2 it appears to suppress it. Remarkably, for
q=2 the percolation line coincides with the only flow line extending to
T>Tc from the critical fixed point associated with Potts clusters.
The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu shows little dependence on the disorder amplitude r, while the dynamic exponent z and static exponent 1/nu vary with the strength of disorder.
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.
We study the effect of varying strength $\ensuremath{\delta}$ of bond randomness on the phase transition of the three-dimensional Potts model for large
q. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder $\ensuremath{\delta}>{\ensuremath{\delta}}_{t}$ this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that ${\ensuremath{\delta}}_{t}$ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.10(2)$ and ${\ensuremath{\nu}}_{t}=0.67(4)$. We claim these exponents are
q independent for sufficiently large
q. In the second-order transition regime the critical exponents ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.60(2)$ and ${\ensuremath{\nu}}_{t}=0.73(1)$ are independent of the strength of disorder.