Narrow Results

We investigate a novel (d + 1)-dimensional discrete erosion model for d = 1, 2 and 3. The dynamics of the model is controlled by the physically motivated erosion mechanism. The coarse grained nature of this erosion process has been well compared with the Kardar–Parisi–Zhang (KPZ) equation. The kinetic roughening of the discrete model shows the same scaling behavior as that of the KPZ equation in the dimensions d = 1, 2. Moreover, in this present discrete model in (3 + 1)-dimension almost smooth interface has been obtained with vanishingly small roughness exponent, indicating the model belongs to the weak coupling regime of KPZ universality class.

In this present numerical work, we report a discrete erosion kind of model in (1 + 1)-dimension. Erosion and re-deposition phenomena with probabilities p and q(= 1 - p) are considered as two tunable parameters, which control the overall kinetic roughening behavior of the interface. Redeposition or diffusion dominated erosion like kinetic roughening model gives rise to nonuniversal growth exponent, which varies continuously with respect to erosion probability. However, universal character is restored for the roughness exponent with the value of 0.5 in (1 + 1)-dimension with respect to p. Due to nonuniversal nature of growth exponent, we observe a significant modification to the scaling behavior of surface width with respect to erosion probability. For low erosion probability (≲ 0.1) a power law like divergence has been observed of the correlation growth time. This can be argued as limiting behavior of a generalized functional behavior of crossover time with erosion probability.

We investigate the inactive–active phase transition in an array of additive (exclusive-or) cellular automata (CA) under noise. The model is closely related with the Domany-Kinzel (DK) probabilistic cellular automaton (PCA), for which there are rigorous as well as numerical estimates on the transition probabilities. Here, we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram’s enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.

We introduce a minesweeper percolation model, in which the system configuration is obtained via an automatic minesweeper process. For a variety of candidate networks with different lattice configurations, our process gives rise to a second-order phase transition. Using Monte Carlo simulation, we identify the critical points implied by giant components. A set of critical exponents are extracted to characterize the nature of the minesweeper percolation transition. The determined universality class shows a clear difference from the traditional percolation transition. A proper mine density of the minesweeper game should be set around the critical density.

We study the scaling properties of a hybrid deposition model that considers random deposition (RD) and ballistic deposition (BD) with next-nearest-neighbor aggregation. The BD occurs with probability p and competes with RD with probability $(1-p)$. It is found that the dynamics of the growing interface follows the same universality class as that of the correlated BD and obeys the generalized scaling ansatz of Family–Vicsek, as proposed by Horowitz and Albano. The deposition process leads to porous structure in the bulk. We have shown that the saturated porosity scales as a power law with the probability of mixing p, which can be described well by extending the scaling arguments as that for the surface.

We investigated two simple SOS growth models where the roughness of the profile is softened by surface relaxation or by refuse. In the first case, a particle can relax until it reaches a local minimum and this process is described by a linear equation. In the second case, a particle can be evaporated and this process is described by the KPZ equation. In this work, we have mixed both the models in the same growth process. Numerical simulations in (1+1)-dimensions show a process that for short times behaves as the surface relaxation model and for intermediate times, behaves as the refuse model. Numerical simulations indicate for this crossover a dynamical exponent z≈1.4, indicating that it is in the KPZ universality class.

In this work we study the spectrum of the lowest screening masses for Yang–Mills theories on the lattice. We used the SU(2) gauge group in (3 + 1) dmensions. We adopted the multiple exponential method and the so-called "variational" method, in order to detect possible excited states. The calculations were done near the critical temperature of the confinement-deconfinement phase transition. We obtained values for the ratios of the screening masses consistent with predictions from universality arguments. A Monte Carlo evolution of the screening masses in the gauge theory confirms the validity of the predictions.

We study the critical properties of the Bak–Sneppen coevolution model on scale-free networks by Monte Carlo method. We report the distribution of the avalanche size and fractal activity through the branching process. We observe that the critical fitness f_c(N) depends on the number of the node such as f_c(N) ~ 1/log(N) for both the scale-free network and the directed scale-free network. Near the critical fitness many physical quantities show power-law behaviors. The probability distribution P(s) of the avalanche size at the critical fitness shows a power-law like P(s) ~ s^-τ with τ = 1.53(5) regardless of the scale-free network and the directed scale free network. The probability distribution P_f(t) of the first return time also shows a power-law such as P_f(t) ~ t^-τ_f. The critical exponents τ_f are equivalent for both the scale-free network and the directed scale-free network. We obtain the critical exponents as τ_f = 1.776(5) at the scalinge regime. The directionality of the network does not change the universality on the network.

We review multifractal surfaces focusing on a comparison between Lucena and Stanley approaches. The multifractal model as presented by Stanley is basically a multinomial measure over a standard partition of a square. The Lucena approach is geometric oriented, the multifractality is not imposed over a regular lattice, but the lattice itself follows a partition where area tiles obey a multifractal distribution. The non-trivial tilling has a distribution of neighbors of lattice elements that shows a fat tail. Despite the strong differences in the two bidimensional models, both Liacir and Stanley multifractals can be reduced to the same object in one dimension. The message of this article is that there is no unique multifractal object in two dimensions and, as a consequence, we should caution about algorithms that estimate multifractal spectrum in two dimensions because it is not clear what kind of multifractality is being measured. Finally we propose a mixed Lucena-Stanley bimultifractal, an object that combines a multifractal measure over a geometric multifractal tilling.

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

The functional measure of the three-dimensional complex |ψ|⁴ theory allows for line-like topological excitations which can be related to vortex lines in super-fluid helium by universality arguments. Upon approaching the λ point, these lines proliferate and destroy the superfluidity. To study the phase transition from this geometrical point of view, we investigated the statistical properties of the emerging vortex-loop network in the vicinity of the critical point by means of high-precision Monte Carlo simulations. For comparison the standard magnetic properties of the system were considered as well. Using sophisticated embedded cluster update techniques we examined if both of them exhibit the same critical behaviour leading to the same critical exponents and therefore to a consistent description of the phase transition. We find that different definitions for constructing the vortex-loop network lead to slightly (but statistically significantly) different results in the thermodynamic limit, and that the percolation thresholds are close to but do not really coincide with the thermodynamic phase transition point.