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We describe in detail a new and highly 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 which 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.

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 = 0.59274621(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.Published as Phys. Rev. Lett. 85, 4104-4107 (2000).

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We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability 0 ≤ p ≤ 1, while with the complementary probability 1 − p, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution f 0 (η). We show that, for arbitrary p, the statistics of records up to step N is completely universal, i.e., independent of f 0 (η) for any N .We also compute the connected two-time correlation function C p (m 1 , m 2 ) of the record-breaking events at times m 1 and m 2 and show it is also universal for all p. Moreover, we demonstrate that C p (m 1 , m 2 ) < C 0 (m 1 , m 2 ) for all p, indicating that a nonzero p induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing p. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when p → 1, N → ∞ with the product t = (1 − p) N fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit.

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The equation that describes fragmentation kinetics, such as that which occurs in cluster breakup and polymer chain degradation (depolymerisation), is solved for models where the rate of breakup depends upon the size of the object breaking up. The resulting dynamic scaling behaviour is investigated. Both discrete and continuous models are considered.

The growth of two-dimensional lattice bond percolation clusters through a cooperative Achlioptas type of process, where the choice of which bond to occupy next depends upon the masses of the clusters it connects, is shown to go through an explosive, first-order kinetic phase transition with a sharp jump in the mass of the largest cluster as the number of bonds is increased. The critical behavior of this growth model is shown to be of a different universality class than standard percolation.