# Percolation/Site percolation/J

`groups=:[: +/\ 2 </\ 0 , *ooze=: [ >. [ +&* [ * [: ; [email protected][ <@(* * 2 < >./)/. +percolate=: ooze/\[email protected]|.^:2^:_@(* (1 + # {. 1:)) trial=: [email protected]([ >: ][email protected]\$0:)simulate=: %@[ * [: +/ (2 e. {:)@trial&15 15"0@#`

Explanation:

`groups` uses a sum fold on successive prefixes to identify regions of contiguous bits.

`   2 0 2 0 1 0 1 0 0 1 1 1 1 1 02 0 2 0 1 0 1 0 0 1 1 1 1 1 0   * 2 0 2 0 1 0 1 0 0 1 1 1 1 1 01 0 1 0 1 0 1 0 0 1 1 1 1 1 0   (0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 00 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0   (2 </\ 0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 01 0 1 0 1 0 1 0 0 1 0 0 0 0 0   ([: +/\ 2 </\ 0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 01 1 2 2 3 3 4 4 4 5 5 5 5 5 5`

`ooze` propagates 2s from its right argument to adjacent groups of non-zero locations in its left argument.

`   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 + 0 0 1 1 1 1 0 2 0 2 0 1 1 1 12 0 3 1 2 1 1 2 0 3 1 2 2 2 1   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1┌───┬───┬───┬─────┬───────────┐│0 0│1 1│0 0│0 0 0│1 1 1 1 1 1│└───┴───┴───┴─────┴───────────┘   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([: ; [email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 10 0 1 1 0 0 0 0 0 1 1 1 1 1 1   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ * [: ; [email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 10 0 2 0 0 0 0 0 0 1 1 1 1 1 0   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ +&* [ * [: ; [email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 11 0 2 0 1 0 1 0 0 2 2 2 2 2 0   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ >. [ +&* [ * [: ; [email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 12 0 2 0 1 0 1 0 0 2 2 2 2 2 0`

Note that u/. uses the key in the left argument to gather corresponding values from the right argument (which are then processed by `u`).

`   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([email protected][ </. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1┌───┬───┬───┬─────┬───────────┐│2 0│3 1│2 1│1 2 0│3 1 2 2 2 1│└───┴───┴───┴─────┴───────────┘   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([email protected][ <@(>./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1┌─┬─┬─┬─┬─┐│2│3│2│2│3│└─┴─┴─┴─┴─┘   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([email protected][ <@(2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1┌─┬─┬─┬─┬─┐│0│1│0│0│1│└─┴─┴─┴─┴─┘   2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([email protected][ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1┌───┬───┬───┬─────┬───────────┐│0 0│1 1│0 0│0 0 0│1 1 1 1 1 1│└───┴───┴───┴─────┴───────────┘`

We use +&* because 1 +&* 2 is 2, but also 2 +&* 2 is 2:

`   +&* table 0 1 2┌───┬─────┐│+&*│0 1 2│├───┼─────┤│0  │0 1 1││1  │1 2 2││2  │1 2 2│└───┴─────┘`

percolate changes 1s in the first row to 2s, then flips over that result and performs an ooze right fold on suffixes, and then does that again (which restores the original orientation and propagates 2s in all directions). And it repeats this until all changes have been propagated.

`   A1 1 1 0 0 0 0 0 1 1 1 1 0 1 01 1 1 1 0 1 0 0 1 1 0 0 0 1 00 1 1 0 0 0 1 1 0 0 1 1 0 1 00 1 1 1 0 1 1 1 0 1 1 0 0 1 01 0 1 1 0 1 1 1 1 0 1 1 0 1 10 0 1 0 0 1 1 0 1 0 1 1 1 0 10 1 1 0 1 0 1 0 0 1 0 1 1 0 01 0 1 0 1 1 0 0 0 0 1 0 1 0 01 1 1 0 1 0 0 0 0 0 0 0 0 1 11 1 0 1 0 1 1 0 1 1 1 1 1 1 11 1 0 1 1 1 0 0 1 0 1 0 0 0 11 0 1 1 1 1 1 0 1 0 0 1 1 0 01 1 1 1 0 1 1 1 1 1 0 0 1 0 11 1 1 1 1 0 0 1 1 1 1 0 0 1 01 1 1 0 1 1 0 0 1 0 1 0 0 1 1   (1 + # {. 1:) A2 1 1 1 1 1 1 1 1 1 1 1 1 1 1   (* (1 + # {. 1:)) A2 2 2 0 0 0 0 0 2 2 2 2 0 2 01 1 1 1 0 1 0 0 1 1 0 0 0 1 00 1 1 0 0 0 1 1 0 0 1 1 0 1 00 1 1 1 0 1 1 1 0 1 1 0 0 1 01 0 1 1 0 1 1 1 1 0 1 1 0 1 10 0 1 0 0 1 1 0 1 0 1 1 1 0 10 1 1 0 1 0 1 0 0 1 0 1 1 0 01 0 1 0 1 1 0 0 0 0 1 0 1 0 01 1 1 0 1 0 0 0 0 0 0 0 0 1 11 1 0 1 0 1 1 0 1 1 1 1 1 1 11 1 0 1 1 1 0 0 1 0 1 0 0 0 11 0 1 1 1 1 1 0 1 0 0 1 1 0 01 1 1 1 0 1 1 1 1 1 0 0 1 0 11 1 1 1 1 0 0 1 1 1 1 0 0 1 01 1 1 0 1 1 0 0 1 0 1 0 0 1 1   |. (* (1 + # {. 1:)) A1 1 1 0 1 1 0 0 1 0 1 0 0 1 11 1 1 1 1 0 0 1 1 1 1 0 0 1 01 1 1 1 0 1 1 1 1 1 0 0 1 0 11 0 1 1 1 1 1 0 1 0 0 1 1 0 01 1 0 1 1 1 0 0 1 0 1 0 0 0 11 1 0 1 0 1 1 0 1 1 1 1 1 1 11 1 1 0 1 0 0 0 0 0 0 0 0 1 11 0 1 0 1 1 0 0 0 0 1 0 1 0 00 1 1 0 1 0 1 0 0 1 0 1 1 0 00 0 1 0 0 1 1 0 1 0 1 1 1 0 11 0 1 1 0 1 1 1 1 0 1 1 0 1 10 1 1 1 0 1 1 1 0 1 1 0 0 1 00 1 1 0 0 0 1 1 0 0 1 1 0 1 01 1 1 1 0 1 0 0 1 1 0 0 0 1 02 2 2 0 0 0 0 0 2 2 2 2 0 2 0   ooze/\. |. (* (1 + # {. 1:)) A2 2 2 0 2 2 0 0 1 0 1 0 0 1 12 2 2 2 2 0 0 1 1 1 1 0 0 1 02 2 2 2 0 1 1 1 1 1 0 0 1 0 12 0 1 1 1 1 1 0 1 0 0 1 1 0 02 2 0 1 1 1 0 0 1 0 1 0 0 0 12 2 0 1 0 1 1 0 1 1 1 1 1 1 12 2 2 0 1 0 0 0 0 0 0 0 0 1 11 0 2 0 1 1 0 0 0 0 1 0 1 0 00 2 2 0 1 0 1 0 0 1 0 1 1 0 00 0 2 0 0 1 1 0 1 0 1 1 1 0 21 0 2 2 0 1 1 1 1 0 1 1 0 2 20 2 2 2 0 1 1 1 0 1 1 0 0 2 00 2 2 0 0 0 1 1 0 0 1 1 0 2 02 2 2 2 0 1 0 0 2 2 0 0 0 2 02 2 2 0 0 0 0 0 2 2 2 2 0 2 0   ooze/\[email protected]|.^:2 (* (1 + # {. 1:)) A2 2 2 0 0 0 0 0 2 2 2 2 0 2 02 2 2 2 0 1 0 0 2 2 0 0 0 2 00 2 2 0 0 0 1 1 0 0 1 1 0 2 00 2 2 2 0 1 1 1 0 1 1 0 0 2 01 0 2 2 0 1 1 1 1 0 1 1 0 2 20 0 2 0 0 1 1 0 1 0 1 1 1 0 20 2 2 0 1 0 1 0 0 1 0 1 1 0 02 0 2 0 1 1 0 0 0 0 1 0 1 0 02 2 2 0 1 0 0 0 0 0 0 0 0 1 12 2 0 2 0 2 2 0 1 1 1 1 1 1 12 2 0 2 2 2 0 0 1 0 1 0 0 0 12 0 2 2 2 2 2 0 1 0 0 1 1 0 02 2 2 2 0 1 1 1 1 1 0 0 1 0 12 2 2 2 2 0 0 1 1 1 1 0 0 1 02 2 2 0 2 2 0 0 1 0 1 0 0 1 1   ooze/\[email protected]|.^:2^:2 (* (1 + # {. 1:)) A2 2 2 0 0 0 0 0 2 2 2 2 0 2 02 2 2 2 0 1 0 0 2 2 0 0 0 2 00 2 2 0 0 0 1 1 0 0 1 1 0 2 00 2 2 2 0 1 1 1 0 1 1 0 0 2 01 0 2 2 0 1 1 1 1 0 1 1 0 2 20 0 2 0 0 1 1 0 1 0 1 1 1 0 20 2 2 0 1 0 1 0 0 1 0 1 1 0 02 0 2 0 1 1 0 0 0 0 1 0 1 0 02 2 2 0 1 0 0 0 0 0 0 0 0 2 22 2 0 2 0 2 2 0 2 2 2 2 2 2 22 2 0 2 2 2 0 0 2 0 1 0 0 0 12 0 2 2 2 2 2 0 2 0 0 1 1 0 02 2 2 2 0 2 2 2 2 2 0 0 1 0 12 2 2 2 2 0 0 2 2 2 2 0 0 1 02 2 2 0 2 2 0 0 2 0 2 0 0 1 1   ooze/\[email protected]|.^:2^:3 (* (1 + # {. 1:)) A2 2 2 0 0 0 0 0 2 2 2 2 0 2 02 2 2 2 0 1 0 0 2 2 0 0 0 2 00 2 2 0 0 0 1 1 0 0 1 1 0 2 00 2 2 2 0 1 1 1 0 1 1 0 0 2 01 0 2 2 0 1 1 1 1 0 1 1 0 2 20 0 2 0 0 1 1 0 1 0 1 1 1 0 20 2 2 0 1 0 1 0 0 1 0 1 1 0 02 0 2 0 1 1 0 0 0 0 1 0 1 0 02 2 2 0 1 0 0 0 0 0 0 0 0 2 22 2 0 2 0 2 2 0 2 2 2 2 2 2 22 2 0 2 2 2 0 0 2 0 2 0 0 0 22 0 2 2 2 2 2 0 2 0 0 1 1 0 02 2 2 2 0 2 2 2 2 2 0 0 1 0 12 2 2 2 2 0 0 2 2 2 2 0 0 1 02 2 2 0 2 2 0 0 2 0 2 0 0 1 1   ooze/\[email protected]|.^:2^:_ (* (1 + # {. 1:)) A2 2 2 0 0 0 0 0 2 2 2 2 0 2 02 2 2 2 0 1 0 0 2 2 0 0 0 2 00 2 2 0 0 0 1 1 0 0 1 1 0 2 00 2 2 2 0 1 1 1 0 1 1 0 0 2 01 0 2 2 0 1 1 1 1 0 1 1 0 2 20 0 2 0 0 1 1 0 1 0 1 1 1 0 20 2 2 0 1 0 1 0 0 1 0 1 1 0 02 0 2 0 1 1 0 0 0 0 1 0 1 0 02 2 2 0 1 0 0 0 0 0 0 0 0 2 22 2 0 2 0 2 2 0 2 2 2 2 2 2 22 2 0 2 2 2 0 0 2 0 2 0 0 0 22 0 2 2 2 2 2 0 2 0 0 1 1 0 02 2 2 2 0 2 2 2 2 2 0 0 1 0 12 2 2 2 2 0 0 2 2 2 2 0 0 1 02 2 2 0 2 2 0 0 2 0 2 0 0 1 1`

(^:_ repeats an operation until it reaches a fixed point. In other words, it's basically a while loop.)

`trial` generates an arbitrary connection matrix and runs percolate on it. You'll typically get a different result from each trial.

`   0.6 0: 15 150   0.6 (] [email protected]\$ 0:) 15 15  0.251168 0.233985   0.14541 0.219665  0.854985 0.0104645  0.449358 0.397573 0.0221767  0.701012      0.5692 0.320853    0.9029 0.0698325  0.439166  0.919973  0.31569  0.733443 0.397144  0.314414 0.0626569  0.125752 0.222472  0.859704  0.517682    0.249954 0.420026  0.974397  0.353249  0.199624 0.0632196 0.729989 0.0384723 0.962052  0.583483  0.794639  0.101726 0.987826 0.0711966 0.0199436    0.692024 0.281396   0.23614  0.445332  0.669898  0.817156 0.896721  0.128014 0.499834   0.86062  0.544871  0.296865 0.762644  0.583945  0.325163    0.701976 0.863764  0.631463  0.590224 0.0233733  0.577801 0.607615  0.893612 0.372065  0.692356  0.385318  0.948463 0.179848  0.482163   0.40125    0.690535  0.48328  0.936104  0.813984  0.319394  0.311636  0.27107  0.565229 0.152493  0.990904  0.856595 0.0755649 0.511731  0.636254 0.0199192   0.0299945 0.465337  0.472827  0.978434  0.207082  0.343524 0.388673  0.169093 0.133254  0.520914  0.736237  0.663096 0.757243  0.901397  0.367862     0.94831  0.45684 0.0569871 0.0211516  0.833437  0.745867  0.62734  0.790395 0.909088  0.359723  0.265804  0.176521  0.34355 0.0484308  0.775815    0.593603 0.023714  0.359704  0.621523  0.567126  0.866712 0.572275  0.679043  0.27958  0.649579   0.91871  0.229361 0.495236  0.646061  0.980648    0.563149 0.461033  0.630641  0.115773  0.4608410.00359967 0.198523  0.509623 0.995623  0.395574  0.130878  0.458647 0.304251  0.552992  0.491661   0.0503197 0.523992  0.554198  0.818936  0.392742  0.696368   0.3988  0.931908 0.776482   0.23695  0.686757  0.799919 0.471909  0.441598  0.145376    0.470236 0.728451  0.567937   0.26908  0.564895  0.826714 0.202945  0.673436 0.655122   0.61951 0.0459954  0.777457 0.968675  0.418669  0.862087    0.836782 0.274477  0.483467  0.345968  0.530893  0.294519 0.286444  0.522912 0.455377 0.0480425  0.789532   0.22362 0.116726  0.563241  0.107526 0.000663275  0.36509  0.140703  0.694587  0.533685  0.155445 0.320069  0.102258 0.844235  0.981544  0.306235  0.627662 0.638738  0.806254  0.753405    0.391762 0.201705   0.07424  0.707749  0.373488   0.58236 0.384308  0.636697 0.577945  0.953101  0.527706  0.992441 0.242658  0.514722  0.269795    0.621798 0.528643 0.0910917  0.072488  0.746536   0.6 ([ >: ] [email protected]\$ 0:) 15 151 1 0 1 1 0 1 1 1 1 1 1 1 0 10 1 1 1 1 1 1 0 0 1 0 0 1 1 10 1 1 1 1 1 0 1 1 1 0 0 0 0 01 0 1 0 1 0 1 1 0 0 0 1 0 1 10 1 0 0 1 1 1 0 1 0 0 0 1 1 11 0 0 1 0 1 0 1 1 1 0 1 0 1 10 1 0 1 1 1 0 1 0 1 1 1 1 1 11 1 1 0 1 0 1 1 0 1 0 1 1 1 00 0 1 1 1 1 1 1 1 1 1 1 0 1 11 1 1 1 0 1 1 1 1 1 1 1 1 1 01 1 0 0 1 0 1 1 0 1 0 1 1 1 11 1 1 0 1 0 1 1 1 1 1 1 1 0 10 1 1 0 1 1 1 1 0 1 1 1 0 0 01 1 0 1 1 1 1 1 1 1 0 1 1 0 10 1 1 1 1 0 1 0 0 1 1 0 0 1 0   0.6 ([ >: ] [email protected]\$ 0:) 15 150 1 0 1 0 0 1 1 0 1 1 1 1 1 10 1 1 1 0 1 1 1 1 1 1 0 0 1 01 1 1 0 0 1 0 1 1 0 1 0 1 0 01 0 1 0 1 1 1 0 0 1 0 1 1 1 01 1 1 1 0 0 0 1 1 0 0 1 0 1 10 1 1 1 0 1 1 1 0 1 0 1 1 0 11 0 1 0 0 1 1 0 1 1 1 1 1 0 11 1 1 0 1 1 1 0 1 0 0 1 0 1 01 1 1 0 1 0 1 1 1 1 1 1 0 1 11 0 1 1 1 0 1 0 1 1 1 1 1 0 11 0 1 0 1 1 1 1 1 1 1 1 1 0 11 0 1 0 1 0 1 1 1 1 0 1 0 0 11 1 0 1 0 1 1 1 1 1 1 0 1 1 00 1 1 1 0 0 1 1 0 1 1 0 0 0 01 1 0 1 0 1 1 1 1 0 0 1 1 1 0   0.6 [email protected]([ >: ] [email protected]\$ 0:) 15 150 0 0 0 2 2 2 2 2 0 2 0 0 0 20 1 0 2 0 2 2 0 2 0 2 2 0 0 01 0 2 2 2 2 2 2 2 0 0 2 0 0 10 2 2 0 2 0 2 0 2 0 2 2 2 0 12 2 2 2 0 2 2 2 2 0 2 2 0 0 12 2 2 0 1 0 2 2 0 0 2 2 2 2 00 0 2 2 0 0 2 2 2 0 2 2 0 2 20 2 2 0 0 2 2 2 2 2 0 2 2 0 20 2 0 0 0 2 0 0 0 0 0 2 2 0 00 2 0 2 2 2 2 0 2 2 2 2 0 0 00 2 2 2 2 2 0 0 2 2 2 0 1 0 22 0 2 0 0 0 2 2 0 0 0 0 0 0 22 2 2 2 2 2 2 2 2 2 2 2 0 2 20 0 2 2 2 2 0 2 2 0 0 2 2 2 00 1 0 0 2 0 2 2 0 0 1 0 2 0 1`

`simulate` runs a sequence of 15 by 15 trials and averages them. Since each trial is different, each run of simulate is different.

`   40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.50.1   40 ((2 e. {:)@trial&15 15"0@#) 0.50 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0   40 ((2 e. {:)@trial&15 15"0@#) 0.50 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0    40 ([: +/ (2 e. {:)@trial&15 15"0@#) 0.56   40 ([: +/ (2 e. {:)@trial&15 15"0@#) 0.54   40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.50.1   40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.50.125`

Note that the alternative implementation is different. (For example, there, `percolate` generates an arbitrary 15 by 15 connection matrix, and finds connections on that - rather like `trial` in this implementation.)