RandomLattice[p_, n_] := Module[{g, img0, img1, topcolors},
   g = Flatten[Table[
      Table[
       If[RandomReal[] < p,
        {{i, j} -> pt},
        {}
        ],
       {pt, {{i + 1, j}, {i, j + 1}}}
       ],
      {i, n}, {j, n}
      ]];
   img0 = 
    GraphPlot[g, 
     VertexCoordinateRules -> (Flatten[ 
        Table[{i, j} -> {i, -j}, {i, n + 1}, {j, n + 1}]]),
     VertexRenderingFunction -> ({} &),
     PlotStyle -> Directive[Black]];
   img1 = ImageData[Binarize[ImageCrop[img0]]];
   Colorize[MorphologicalComponents[Binarize[ImageCrop[img0]]]]
   ];
Manipulate[
 RandomLattice[p, n],
 {{n, 50}, 0, 100, 1},
 {{p, 0.5}, 0, 1}
 ]