Timeline for Sizes of connected components from a random choice in a grid
Current License: CC BY-SA 4.0
11 events
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| May 20, 2020 at 12:15 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
I realized that the counting of "components" that I used is slightly different from what the OP took. Corrected.
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| May 20, 2020 at 12:09 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
I realized that the counting of "components" that I used is slightly different from what the OP took. Corrected.
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| May 20, 2020 at 10:26 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
added 1 character in body
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| May 20, 2020 at 10:10 | comment | added | ofer zeitouni | Let's give a dictionary. Tile your nxn box by 1x1 boxes, whose bottom left corner indexes the boxes. The origin refers to this index. | |
| May 20, 2020 at 10:08 | comment | added | Wolfgang | So you'd shift the "cells of the original lattice" by 1/2 for that? (original lattice = the red dots in the picture) | |
| May 20, 2020 at 10:07 | comment | added | ofer zeitouni | @Wolfgang concerning your second comment, I understood what you meant, and my answer refers to it. You can think of the diagonals as forming a path, and I am counting how many cells of the original lattice are contained in each component. BTW, this model is not far from percolation models (except that there is some dependence here compared to the standard percolation model). | |
| May 20, 2020 at 10:06 | comment | added | Wolfgang | OK thanks for the clarification! A question like that had not at all come to my mind! | |
| May 20, 2020 at 10:05 | comment | added | ofer zeitouni | @Wolfgang There is only one connected component that contains the origin (I think of your nxn grid as centered around the origin, think of a box of side n/2). This is what I called "THE" connected component. It is a random variable, and I was discussing its tail. | |
| May 20, 2020 at 9:00 | comment | added | Wolfgang | Sure my wording may be a bit unfortunate. It is meant not to mix up those small "unit squares" (of the initial $n\times n$) with what I call "cells" (i.e. the small rotated white squares). | |
| May 20, 2020 at 8:54 | comment | added | Wolfgang | I have a hard time understanding several details. E.g. "the tail behavior of the size of the connected component ": What do you mean by "THE connected component"? The biggest one, which would be the pink one above? (very fishy) Or the average size? Do we even have the same idea of "component"? For me, each colored region is one component. And then "is the square A having (0,0) and (1,1) as vertices a component" ?? The (what I call) components are rotated by 45° and composed of 4, 8, 12, ... triangles each "half of A", so I don't understand what you mean. | |
| May 20, 2020 at 6:44 | history | answered | ofer zeitouni | CC BY-SA 4.0 |