Scaling of loop-erased walks in 2 to 4 dimensions

We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2,3, and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in $d=4$, simulations in lower dimensions were done for completeness and in order to test the algorithm. In $d=2$, we verify with high precision the prediction $D=5/4$, where the number of steps $n$ after erasure scales with the number $N$ of steps before erasure as $n\sim N^{D/2}$. In $d=3$ we again find a power law, but with an exponent different from the one found in the most precise previous simulations: $D = 1.6236\pm 0.0004$. Finally, we see clear deviations from the naive scaling $n\sim N$ in $d=4$. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure

Percolation Threshold, Fisher Exponent, and Shortest Path Exponent for 4 and 5 Dimensions

We develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created. The memory required to grow a cluster of mass s is of the order of $s^\theta$ bytes where $\theta$ ranges from 0.4 for 2-dimensional lattices to 0.5 for 6- (or higher)-dimensional lattices. We use this method to estimate $d_{\scriptsize min}$, the exponent relating the minimum path $\ell$ to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find $d_{\scriptsize min}=1.607\pm 0.005$ (4D) and $d_{\scriptsize min}=1.812\pm 0.006$ (5D). In order to determine $d_{\scriptsize min}$ to high precision, and without bias, it was necessary to first find precise values for the percolation threshold, $p_c$: $p_c=0.196889\pm 0.000003$ (4D) and $p_c=0.14081\pm 0.00001$ (5D) for site and $p_c=0.160130\pm 0.000003$ (4D) and $p_c=0.118174\pm 0.000004$ (5D) for bond percolation. We also calculate the Fisher exponent, $\tau$, determined in the course of calculating the values of $p_c$: $\tau=2.313\pm 0.003$ (4D) and $\tau=2.412\pm 0.004$ (5D)

Continuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability $p$. At $p=0.5$, the GAP becomes the random growth model and leads to the minority product rule at $p=1$. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with $0.5 \le p \le 1$ are always continuous and their critical exponents depend on $p$. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter $p$ in addition.Comment: 7 pages, 14 figures, accepted for publication in Eur. Phys. J.

Competitive random sequential adsorption of point and fixed-sized particles: analytical results

We study the kinetics of competitive random sequential adsorption (RSA) of particles of binary mixture of points and fixed-sized particles within the mean-field approach. The present work is a generalization of the random car parking problem in the sense that it considers the case when either a car of fixed size is parked with probability q or the parking space is partitioned into two smaller spaces with probability (1-q) at each time event. This allows an interesting interplay between the classical RSA problem at one extreme (q=1), and the kinetics of fragmentation processes at the other extreme (q=0). We present exact analytical results for coverage for a whole range of q values, and physical explanations are given for different aspects of the problem. In addition, a comprehensive account of the scaling theory, emphasizing on dimensional analysis, is presented, and the exact expression for the scaling function and exponents are obtained.Comment: 7 pages, latex, 3 figure

Universal crossing probability in anisotropic systems

Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a length-rescaling procedure. For strongly anisotropic systems in 1+1 dimensions, exact results are obtained for the random walk with absorbing boundary conditions, which can be considered as a linearized mean-field approximation for directed percolation. The bond and site directed percolation problem is itself studied numerically via Monte Carlo simulations on the diagonal square lattice with either free or periodic boundary conditions. A scale-invariant critical crossing probability is still obtained, which is a universal function of the effective aspect ratio r_eff=c r where r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.Comment: 7 pages, 4 figure

Proposal for a CFT interpretation of Watts' differential equation for percolation

G. M. T. Watts derived that in two dimensional critical percolation the crossing probability Pi_hv satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Pi_h, Pi_hv. We will show that this differential equation can be derived from a level three null vector condition of a rational c=-24 CFT and motivate how this solution may be fitted into known properties of percolation.Comment: LaTeX, 20p, added references, corrected typos and additional content

The critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model

We present a new way of probing the universality class of the site-diluted two-dimensional Ising model. We analyse Monte Carlo data for the magnetic susceptibility, introducing a new fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude and the sample dependent pseudo-critical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration $q$ of the empty sites for all investigated values of $q\le 0.25$. At the same time the average effective exponent $\gamma_{eff}$ is found to vary with the concentration $q$, which may be argued to be due to logarithmic corrections to the power law of the pure system. This corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.Comment: 6 pages, 4 figure

Gaussian fluctuations in an ideal bose-gas -- a simple model

Based on the canonical ensemble, we suggested the simple scheme for taking into account Gaussian fluctuations in a finite system of ideal boson gas. Within framework of scheme we investigated the influence of fluctuations on the particle distribution in Bose -gas for two cases - with taking into account the number of particles in the ground state and without this assumption. The temperature and fluctuation parameter dependences of the modified Bose- Einstein distribution have been determined. Also the dependence of the condensation temperature on the fluctuation distribution parameter has been obtained.Comment: arXiv admin note: text overlap with arXiv:cond-mat/040859

Dynamic Scaling in One-Dimensional Cluster-Cluster Aggregation

We study the dynamic scaling properties of an aggregation model in which particles obey both diffusive and driven ballistic dynamics. The diffusion constant and the velocity of a cluster of size $s$ follow $D(s) \sim s^\gamma$ and $v(s) \sim s^\delta$, respectively. We determine the dynamic exponent and the phase diagram for the asymptotic aggregation behavior in one dimension in the presence of mixed dynamics. The asymptotic dynamics is dominated by the process that has the largest dynamic exponent with a crossover that is located at $\delta = \gamma - 1$. The cluster size distributions scale similarly in all cases but the scaling function depends continuously on $\gamma$ and $\delta$. For the purely diffusive case the scaling function has a transition from exponential to algebraic behavior at small argument values as $\gamma$ changes sign whereas in the drift dominated case the scaling function decays always exponentially.Comment: 6 pages, 6 figures, RevTeX, submitted to Phys. Rev.

Kinetics of fragmentation-annihilation processes

We investigate the kinetics of systems in which particles of one species undergo binary fragmentation and pair annihilation. In the latter, nonlinear process, fragments react at collision to produce an inert species, causing loss of mass. We analyse these systems in the reaction-limited regime by solving a continuous model within the mean-field approximation. The rate of fragmentation, for a particle of mass $x$ to break into fragments of masses $y$ and $x-y$, has the form $x^{\lambda-1}$ ($\lambda>0$), and the annihilation rate is constant and independent of the masses of the reactants. We find that the asymptotic regime is characterized by the annihilation of small-mass clusters. The results are compared with those for a model with linear mass-loss (i.e.\ with a sink). We also study more complex models, in which the processes of fragmentation and annihilation are controlled by mutually-reacting catalysts. Both pair- and linear-annihilation are considered. Depending on the specific model and initial densities of the catalysts, the time-decay of the cluster-density can now be very unconventional and even non-universal. The interplay between the intervening processes and the existence of a scaling regime are determined by the asymptotic behaviour of the average-mass and of the mass-density, which may either decay indefinitely or tend to a constant value. We discuss further developments of this class of models and their potential applications.Comment: 16 pages(LaTeX), submitted to Phys. Rev.