| This thesis contains work which lies at the interface between mathematics and physics. It is related to the analysis of mathematical statistical mechanics models, notably two-dimensional critical percolation, near-critical percolation, and invasion percolation. Also included is a study of a recently introduced model for rill erosion.;First we study invasion percolation in two dimensions. We compare connectivity properties of the origin's invaded region to those of the critical percolation cluster of the origin. To exhibit similarities, we show that for any k ≥ 1, the k-point function of the first so-called pond has the same asymptotic behaviour as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint pc-open clusters. Further, for k > 1, we compute the exact decay rate of the distributions of the radius and the volume of the kth pond and see that it is different than that of the radius of the critical cluster of the origin.;Next we compare the invasion to generalizations of Kesten's incipient infinite cluster (IIC). We show that the invasion percolation measure and the incipient infinite cluster measure are mutually singular. We then prove existence of multiplearmed IIC measures for any number of arms and for any color sequence. We use these measures to study the invaded region near so-called outlets and near edges in the invasion backbone far from the origin. To conclude our investigation of invasion percolation, we focus on asymptotic properties of the weights of these outlets.;In the second half of the thesis, we introduce a model for rill erosion. We start with a lattice of vertices and consider a dynamics on edges between them which makes the process highly non-Markovian. The behavior of nodes in the streams is similar to the behavior of Polya urns with time-dependent input. We use a combination of rigorous arguments and simulation results to show that the model exhibits many properties of rill erosion; in particular, nodes which are deeper in the network tend to switch less quickly. |
