The slide presentation suggested by Victor Miller is about a related problem, in which the probabilities pertain to the connections between the lattice points instead of the lattice points themselves. The expected component size function is qualitatively similar, but not the same function. On Tue, May 22, 2012 at 9:01 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Correction --- M.F.Sykes. But now I've read the subject line, I realise you probably already knew about all this! WFL
On 5/23/12, Fred lunnon <fred.lunnon@gmail.com> wrote:
I don't know the answer (though maybe I once did); but there used to be a minor industry devoted to questions like this, called "percolation theory". One of the more prolific authors was J.M.Sykes.
A web search should turn something up ...
WFL
On 5/22/12, Allan Wechsler <acwacw@gmail.com> wrote:
If the cells of a square grid are randomly colored black and white, with p being the probability of a cell being black, then for p small enough we can ask the expected size F(p) of the black polyomino including a given cell. (If that cell happens to be white, the size is zero.)
The reference cell is white with probability (1-p); this contributes 0*(1-p)=0 to the expected polyomino size.
It is black, and all its neighbors are white, with probability p(1-p)^4; this contributes p - 4p^2 + 6p^3 - 4p^4 + p^5 to the expected size.
All bigger polyominoes only contribute quadratic or higher terms, so we know the power series for F(p) begins 0 + p + ...
If you perform a census of the ways the starting cell can be part of a domino, you learn that the next term must be 4p^2.
Repeating this exercise with the trominoes, if I have done it right, gives us the cubic term, 12p^3.
I conjecture that all the coefficients of this power series are positive. If I did my counting right, the quartic coefficient is 24.
The sequence incipit 0, 1, 4, 12, 24, only yields one hit on OEIS, and it isn't this. I tried to calculate the quintic coefficient and made a blunder somewhere.
This must be known stuff. What are some more terms of this series?
Each oriented polyomino with one distinguished cell contributes n p^n (1-p)^m to the power series, where n is the number of cells in the polyomino, and m is the number of cells immediately adjacent to it. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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