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I am supposed to write up a report on a percolation theory project which I worked on, over the summers. I'm trying to describe my mathematical model below:

We report some results regarding certain specific properties of a probabilistic model of two-dimensional square lattices. In our mathematical model, a site is considered to be "occupied" with a probability $p$, or "empty" with a probability $1-p$. We represent "occupied" cells with "black" (or $1$) and "empty" cells with "white" (or $0$). If any cell lies in the Von Neumann neighborhood of a certain cell and has the same color as of that cell, it is always said to belong to the same cluster as that of the central cell. Moreover, we do not allow for intertwining of black and white clusters, that is, if any one of the four second-nearest neighbors of a central "occupied" cell is "occupied", but the two nearest neighbors (of the "central" cell) sharing an edge with both of them are "empty", then those two "diagonally" connected cells (which were second-nearest neighbors) are said to belong to the same "black" cluster with a probability of $q$ (and in case they do belong to the same "black" cluster, the two "empty" cells along the other diagonal, will definitely not be "diagonally connected" i.e. won't be considered to belong to the same "white" cluster). However, it is important to note that in large lattices, we can instead assign a connection probability of $1-q$ to "empty" cells, which are second nearest neighbors of each other (and each having two "occupied" nearest neighbor cells which share an edge with both of them, respectively). For practical purposes, this model would be nearly equivalent (in terms of the "nature" of the clusters and their size distributions) to the previous one, when averaged over a large number of lattice configurations, for some given value(s) of $p$ and $q$. Both these models are, of course, similar to the popular site percolation models on $\mathbb {Z}^2$, but the popular models hardly take into account variable diagonal connection probabilities i.e. $q$. One of our primary investigations was related to spotting the variation in the nature of the "Euler number" graph, that is, the $\chi(p) \ [=N_B(p)-N_W(p)]$ vs. $p$ graph, for different values of $q$, where $N_B(p)$ is the number of black clusters and $N_W(p)$ is the number of white clusters, at a probability $p$. We also investigated the variation of site percolation threshold with $q$.

This part is supposed to be present in the "abstract" section of the report. I know that it looks rather convoluted but this is the best (and most descriptive version) I could come up with, to explain my mathematical model. It would be very helpful if someone could suggest possible modifications which I could make to the language so as to make it more succinct and precise.

P.S: I'm not sure whether these type of questions are allowed on Math SE, but well, it seems like the best place on the internet to try my luck anyway. So thanks in advance! By the way, if you want any clarification(s) to be made please feel free to ask in the comments.

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  • $\begingroup$ (I'm still chewing on the actual question, but just out of curiosity: was this at CEE, or another academic 'summer camp'?) $\endgroup$ – Steven Stadnicki Aug 3 '18 at 17:32
  • $\begingroup$ @StevenStadnicki Not really a "summer camp" :). The work was done at the CMPRC which is a research center at my home university. $\endgroup$ – Blue Aug 3 '18 at 17:34
  • $\begingroup$ Very very cool! (The ideas themselves also look really interesting, though I'm still trying to give a good reading to the presentation...) $\endgroup$ – Steven Stadnicki Aug 3 '18 at 17:43
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In my opinion, the explanation about handling diagonally adjacent cells could stand to be simplified.

We report some results regarding certain specific properties of a probabilistic model of two-dimensional square lattices. In our mathematical model, a site is considered to be "occupied" with a probability $p$, or "empty" with a probability $1−p$. We represent "occupied" cells with "black" (or 1) and "empty" cells with "white" (or 0). We partition the sites into clusters, so that two adjacent cells are in the same cluster whenever they are the same color. Furthermore, any two diagonally adjacent black cells, such that the two cells they orthogonally neighbor are white, are deemed to be in the same cluster with probability $q$.

The language is less precise, but if this is an abstract, then it should be more focused on giving a quick exposition rather than a technically correct one.

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