Experimental Mathematics - preliminary results for Project - Bobby Demarco & Kellen Myers
0. Goal and approach: Our goal in this project was to use experimental mathematics to begin to explore percolation in thin or otherwise predictably shaped regions of the plane. We worked under the assumption that percolation occurs top-to-bottom (and side-to-side) only, using the simplified model like in class that avoides "working backwards" to get from top to bottom. This assumption does not change scenarios in which we are in a very thin (width <5) strip, since we can only get this "working backwards" behavior if there is something like this in the lattice:
| 1 | 0 | 0 | 0 | 0 | |
| 1 | 0 | 1 | 1 | 1 | |
| 1 | 0 | 1 | 0 | 1 | |
| 1 | 0 | 1 | 0 | 1 | |
| 1 | 1 | 1 | 0 | 1 | |
| 0 | 0 | 0 | 0 | 1 |
A bit of thought and one can convince oneself that without five rows, this sort of configuration reduces to one that does not require "backwards" (ie against gravity) motion. In order to help accumulate varied and accurate results, we used several different algorithms in different computer algebra systems (Maple & Mathematica). The Maple source code can be found here and the Mathematica notebook can be found here.
1. Thin strips: For thin strips, a symbolic Monte Carlo algorithm in Mathematica was used to generate a graph of the probability of percolation (given p, the probability that any paritcular site was open, meaning 1). A random 0-1 matrix was generated, and its weight (how likely it is, with respect to p) was computed. It was then determined whether there is a percolation path of 1s through the matrix. The weight of the matrix was added to the "running total" denominator, and if it percolates, this weight was also added to the "running total" numerator. In the end, a rational function of p was produced. Below are some graphs of this probability for various sized strips.
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| 3×15; 10,000 iterations | 3×30; 10,000 iterations |
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| 4×20; 10,000 iterations | 20×4; 10,000 iterations |
2. Other rectanglar regions: A similar algorithm, in Maple, is used to symbolically compute the probability for square regions 1×1 through 5×5, and then for regions twice as tall (2×1, 4×2, 6×3). Although the graphs for the square regions appear to intersect at the same point, numerical comparisons indicate that the critical probability is actually decreasing (slightly) as the size of the region increases. It increases more noticably in the case where the region is taller. One interesting point is that the algorithm in Maple does not seem to induce instability past the critical probability in these cases, as (quite notacibly) the Mathematica algorithm does in the cases where the strip is much longer -- we have not yet determined if this is specific to each computer algebra system, to each algorithm, or if these instabilities crop up more noticably as the strip lengthens (which might be expected, with extremely high-degree rational functions).
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| square matrices | double-height (2n×n) matrices |