The way we did this in the MC simulations of magnets was to
"renormalize"
the lattice using "block spins." A block spin is the net result of
adding up
all of the elements in (for instance) a 3x3 block. It works for this
lattice too,
just using B and W, and the result just being B or W. Just call B +1
and W -1,
and the new color is just the sign of the sum.
Is there any chance you would take the whole lattice and renormalize it
repeatedly this way?
Thanks,
David
On 20, Feb 2007, at 7:32 PM, Chris Fant wrote:
this is very interesting. Can you compute some properties, like the
distribution of cluster sizes, or diagrams for cluster size /
boundary size
pairs? I don't know much about fractals, but does this picture
have some
fractal properties, too?
Here's some numbers (on a torus board) :
BoardSize, BoardArea, ChainCount, MinChainArea, MaxChainArea,
AvgChainArea, AvgChainPerimeter, AvgChainAreaToPerimeterRatio, Seconds
100, 10000, 21, 32, 3904, 476.2, 347.6, 1.0368,
200, 40000, 82, 27, 9411, 487.8, 364.0,
1.0341, 1
300, 90000, 186, 22, 10866, 483.9, 353.0,
1.0483, 2
400, 160000, 332, 17, 20828, 481.9, 355.7,
1.0143, 5
500, 250000, 568, 16, 19744, 440.1, 328.1,
1.0148, 8
600, 360000, 769, 14, 23761, 468.1, 346.3,
1.0278, 11
700, 490000, 1045, 16, 23250, 468.9, 343.8,
1.0423, 22
800, 640000, 1426, 15, 46446, 448.8, 334.6,
1.0211, 33
900, 810000, 1747, 14, 25748, 463.7, 340.4,
1.0284, 57
1000, 1000000, 2152, 15, 30207, 464.7, 339.4,
1.0233, 65
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