We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8 000 sites. We consider the effect of an exponential decay of the interactions with the distance, J(r) = J0 exp (-ar), with a > 0, and observe that this system seems to have critical exponents γ and ν which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio γ/ν remains essentially the same. We find numerical evidences (although not conclusive, due to the small range of system size) that the specific heat on this random system behaves as a power-law for a=0 and as a logarithmic divergence for a=0.5 and a=1.0
We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to $128000\ensuremath{\approx}{50}^{3}$ sites which are linked together according to the Voronoi-Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and ${\ensuremath{\varphi}}^{4}$ field theory on three-dimensional regular cubic lattices.
The single-cluster Monte Carlo algorithm and the reweighting technique are used to simulate the 3D-ferromagnetic Ising model on three dimensional Voronoi-Delaunay lattices. It is assumed that the coupling factor
J varies with the distance
r between the first neighbors as
J(r)∝e−ar, with
a≥0. The critical exponents
γ/ν,
β/ν, and
ν are calculated, and according to the present estimates for the critical exponents, we argue that this random system belongs to the same universality class of the pure three-dimensional ferromegnetic Ising model.
Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 10–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
Abstract Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 20–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.
We study the two-dimensional anisotropic three-state Potts ferromagnet on the square lattice using Wolff single-cluster Monte Carlo simulations and finite-size scaling. For coupling ratios
λ=Jy/Jx=0.5,
0.75, and
1, finite-size scaling of the correlation ratio yields critical behavior consistent with the two-dimensional three-state Potts universality class. The scaling of the leading Fisher zeros gives a correlation-length exponent consistent with
ν=5/6, while their cumulative density is consistent with the expected specific-heat exponent
α=1/3. We further characterize the anisotropy at criticality using directional correlation ratios
Rx and
Ry together with directional FK wrapping probabilities. For
λ=0.5 on a physically square lattice,
Rx and
Ry approach distinct critical values while yielding a common correlation-length exponent. From the wrapping probabilities, we independently determine an effective aspect ratio
ρe□=0.6413(5), in close agreement with the theoretical value
ρe□,th≃0.64150030 obtained from the isoradial representation. Using the theoretical value to set the physical aspect ratio restores directional equivalence,
Rx≃Ry, and brings the overall correlation ratio toward the isotropic-square reference. The results show that spatial anisotropy changes the effective critical geometry without altering the bulk three-state Potts universality class.