Function to create random matrix, then a simulation. *EDIT* Create two functions

Update: Animation of fire-spreading:

ClearAll[initState, vNNeighbors, step, iterationList]

initState[b_, i_, j_, pos_: Automatic] := ReplacePart[
   RandomChoice[{1/(1 + b), b/(1 + b)} -> {0, 1}, {i, j}], 
  (pos /. Automatic -> RandomChoice[Tuples[{Range@i, Range@j}]]) -> 2]

vNNeighbors[dim_: {30, 30}] := AdjacencyList[NearestNeighborGraph@Tuples@Range@dim, #]&

step = MapAt[Min[2, 2 #] &, #, vNNeighbors[][Position[#, 2]]] &;

iterationList[nmax_: Automatic][b_, i_, j_, pos_: Automatic] := 
   NestList[step, initState[b, i, j, pos], nmax /. Automatic -> Max[i, j]]

Example:

SeedRandom[1]
startpos = {15, 5};
{i, j} = {30, 30};

Manipulate[ListAnimate[MatrixPlot[#, Mesh -> All, Frame -> False] & /@ 
    iterationList[][b, i, j, startpos], Paneled -> False],
 {{b, .6}, .6, 1.6, .2}]

enter image description here

Alternatively, you can use FixedPointList to generate the list of arrrays:

ClearAll[FPList]
FPList[b_, i_, j_, pos_: Automatic] := FixedPointList[step, initState[b, i, j, pos]]

SeedRandom[1]
Manipulate[ListAnimate[MatrixPlot[#, Mesh -> All, Frame -> False] & /@ 
   FPList[b, i, j, startpos], Paneled -> False], 
 {{b, .6}, .6, 1.6, .2}]

enter image description here

Original answer:

ClearAll[initState]
initState[b_, i_, j_, pos_: Automatic] := ReplacePart[
   RandomChoice[{1/(1 + b), b/(1 + b)} -> {0, 1}, {i, j}], 
  (pos /. Automatic -> RandomChoice[Tuples[{Range@i, Range@j}]]) -> 2]

Default position of 2 is random:

SeedRandom[1]
MatrixPlot[initState[.6, 30, 30], Mesh -> All]

enter image description here

Position of 2 is {10, 15}

SeedRandom[1]
MatrixPlot[initState[.6, 30, 30, {10, 15}], Mesh -> All]

enter image description here

Animations:

Position of 2 changes randomly in every iteration:

SeedRandom[1]
frames1 = Table[MatrixPlot[initState[b, 30, 30], Mesh -> All, 
    Frame -> False], {b, Range[.6, 1.8, .2]}];

Export["animation1.gif", frames1]

enter image description here

Position of 2 is remains fixed at a random value over iterations:

SeedRandom[1]
frames2 = With[{pos2 = RandomChoice[Tuples[{Range@30, Range@30}]]}, 
   Table[MatrixPlot[initState[b, 30, 30, pos2], Mesh -> All, 
     Frame -> False], {b, Range[.6, 1.8, .2]}]];

Export["animation2.gif", frames2]

enter image description here

Tags:

Matrix

Random

Function Construction

Iterators

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