These experiments focused on random walks on percolation clusters. These walks are of interest because the graphs of these clusters are not regular; some vertices have more neighbors than others, hence influencing where a walk can travel. We sought to look at the nature of these walks in terms of recurrence and transience. On an infinite graph, a recurrent walk is defined as one that, with probability 1, returns to its original position infinitely many times. Since we worked on finite graphs to model infinite graphs, we defined a recurrent walk as one that returned to its starting position at some point over the course of the walk. To simulate walks on percolation clusters in an infinite setting, we ran these walks on the maximum cluster of the percolated fractal graphs. We also wanted to look at the relative size of these walks compared to the cluster they were run on. To evaluate size, we looked at the number of distinct vertices that the walk hits compared to the number of vertices in the cluster. We also looked at how far the walks got by computing the distance at each step from the starting position and looking at the farthest point the walk traversed.

The aim of these experiments was to understand the nature of random walk on the Sierpinski Gasket percolated at different probabilities. We already know that the lattice graph yields recurrent walks at every percolation probability (though it is just barely recurrent at p = 1.0), so the hope of these experiments was to understand walks on SG by comparing data with data on the lattice. To minimize the effects of outliers in the data, for each probability we ran multiple walks (either 25 or 100) on 10 distinct clusters and then took averages over all of the data. Each walk had a length of 1,000 steps.