Robert M. Ziff

PRECISE DETERMINATION OF THE BOND PERCOLATION THRESHOLDS AND FINITE-SIZE SCALING CORRECTIONS FOR THE SC, FCC, AND BCC LATTICES

Extensive Monte-Carlo simulations were performed to study bond percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic (b.c.c.) lattices, using an epidemic kind of approach. These simulations provide very precise values of the critical thresholds for each of the lattices: pc(s.c.) = 0.2488126 ± 0.0000005, pc(f.c.c.) = 0.1201635 ± 0.0000010, and pc(b.c.c.) = 0.1802875 ± 0.0000010. For p close to pc, the results follow the expected finite-size and scaling behavior, with values for the Fisher exponent � (2.189 ±0.002), the finite-size correction exponent (0.64 ±0.02), and the scaling function exponent � (0.445 ± 0.01) confirmed to be universal.

Fast Monte Carlo algorithm for site or bond percolation

We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.

Efficient Monte Carlo algorithm and high-precision results for percolation.

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.

Efficient measurement of the percolation threshold for fully penetrable discs

We study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs. We use two different algorithms to efficiently simulate the frontier, including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice. We find that φc = 0.676 339 ± 0.000 004, thus providing an extra significant digit of accuracy to this constant. (Some figures in this article appear in colour in the electronic version; see www.iop.org)

Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm

Precise values for the critical threshold for the three-dimensional “Swiss cheese” continuum percolation model have been calculated using extensive Monte Carlo simulations. These simulations used a growth algorithm and memory blocking scheme similar to what we used previously in three-dimensional lattice percolation. The simulations yield a value for the critical number density nc=0.652 960±0.000 005, which confirms recent work but extends the precision by two significant figures.

Random sequential adsorption of unoriented rectangles onto a plane

Random sequential adsorption of nonoverlapping rectangles of arbitrary orientation onto a continuous plane was investigated by computer simulation. The approach to the jamming limit was found to obey Feder’s law for a wide range of rectangle aspect ratios. The coverage fraction at the jamming limit was found to depend upon the aspect ratio of the adsorbed rectangles, with a maximum in the jamming coverage occurring at aspect ratios ≊2.

The efficient determination of the percolation threshold by a frontier-generating walk in a gradient

The frontier in gradient percolation is generated directly by a type of self-avoiding random walk. The existence of the gradient permits one to generate an infinite walk on a computer of finite memory. From this walk, the percolation threshold pc for a two-dimensional lattice can be determined with apparently maximum efficiency for a naive Monte Carlo calculation (+or-N-12/). For a square lattice, the value pc=0.592745+or-0.000002 is found for a simulation of N=2.6*1011 total steps (occupied and blocked perimeter sites). The power of the method is verified on the Kagome site percolation case.

Coagulation processes with a phase transition

Abstract Smoluchowskis coagulation equation with a collection kernel K(x, y) ∼ (xy)ω with 1 2 describes a gelation transition (formation of an infinite cluster after a finite time tc (gel point)). For general ω and t > tc the size distribution is c(x, t) ∼ x−τ for x → ∞ with τ = ω + 3 2 . For ω = 1, we determine c(x, t) and the time dependent sol mass M(t) for arbitrary initial distribution in pre- and post-gel stage, where c(x, t) ∼ x − 5 2 exp (−x/x c ) for large x and t c(x, t) ∼ (− M ) 1 2 x − 5 2 for large xt and t > tc. Here xc is a critical cluster size diverging as (t - tc)−2 as t ↑ tc. For initial distributions such that c(x, 0) ∼ xp-2 as x → 0, we find M(t) ∼ t−p/(p+1) as t → ∞. New explicit post-gel solutions are obtained for initial gamma distributions, c(x, 0) ∼ xp-2e−px (p > 0) in the form of a power series (convergent for all t), and reducing for p = ∞ to the solution for monodisperse initial conditions. For p = 1, the solution is found in closed form.

On the stability of coagulation—fragmentation population balances

Abstract When coagulation and fragmentation both occur in a system, the competition between these processes may lead to a steady-state size distribution. We consider some specific moment solutions to a generalized coagulation-fragmentation population balance equation (in which multiple breakup is allowed) in order to determine when it is may be possible for such steady states to exist. Steady states occur for systems with homogeneous rate kernels of order β (fragmentation) and λ (coagulation) that satisfy β − λ + 1 > 0. Finally, we discuss the applicability of scaling to this generalized coagulation-fragmentation population balance.

Kinetics of gelation and universality

The exact solution (size distribution ck(t) and moments M, (t)) of Smoluchowskis coagulation equation (S-model) and of a modified equation (F-model) with a coagulation rate K,, = ij for i- and j-clusters is obtained for arbitrary Ck(0) in the sol (t (,) phases, where tc is the gel point. The behaviour of ck(t) and M,(t) is given for k -+CO, t -+ CO and t + I,. The critical exponents, critical amplitudes and scaling function that characterise the singularities near the non-equilibrium phase transition are calculated. For short-range ck(0) (i.e. all M, <a) the F-model belongs to the universality class of classical gelation theories and of bond percolation on Cayley trees; the S-model does not.