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.

It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In $N$ steps, the mean of the record distribution grows as the $\sqrt{4N/\pi }$ while the standard deviation grows as $\sqrt{(2-4/\pi )N}$, so the distribution is non-self-averaging. The mean shortest and longest duration records grow as $\sqrt{N/\pi }$ and $0.626508\dots N$, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based on both moments and wrapping probabilities, yielding pc=0.526\,565\ifmmode\pm\else\textpm\fi{}0.000\,005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent $\nu${} is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P$\infty${} (size of largest cluster), of the susceptibility, and of the second moment of finite clusters, where discontinuities appear at the threshold. The critical cluster-size distribution does not follow a consistent power law for the range of system sizes we study (L$\le${}8192) but may approach a power law with $\tau${}>2 for larger L.

We perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii. As a function of volumetric proportion $v$, the percolation threshold ${\varphi }_c(v,\lambda )$ is approximately symmetric about $v=0.5$ for a fixed ratio $\lambda$ of the radii. However, the difference from symmetry is strongly statistically significant. We also improve, by an order of magnitude, the measurement of the percolation threshold for disks of equal radius: ${\varphi }_c^{*}=0.6763475\left(6\right)$.