Simulation with two random walkers on a lattice with periodic boundaries

This is not answer, but an extremely long comment.

I find this problem very interesting, but haven't been able to solve it. In my attempts, I developed a tool to visualize the the two-walker random walk. I am posting this tool because I think it might be useful to the OP or anyone else looking this problem for exploring what's going on.

steps = {{0, 1}, {1, 0}, {0, -1}, {-1, 0}};

simulation[n_, max_] :=
  Module[{index, w1, w2, pos1, pos2},
    index = 0;
    pos1 = RandomInteger[{0, n - 1}, 2];
    pos2 = RandomInteger[{0, n - 1}, 2];
    w1 = {pos1}; w2 = {pos2};
    While[pos1 != pos2 && index <= max,
      AppendTo[w1, pos1 = Mod[pos1 + RandomChoice[steps], n]];
      AppendTo[w2, pos2 = Mod[pos2 + RandomChoice[steps], n]];
      index++];
    {w1, w2}]

path[pts : {{_, _} ...}, color_?ColorQ, r_Real] := 
  {color, 
   {Opacity[.5], Line[Subsequences[pts, {2}]]},
   {Thick, Circle[pts[[1]], r]}, (* start marker *)
   Circle[pts[[-1]], 1.6 r]}     (* end marker *)

visualize[n_, max_: 500] :=
  Module[{w1, w2, r},
    r = .016 (n - 1);
    {w1, w2} = simulation[n, max];
    Graphics[
      {{PointSize[Scaled[.018]], 
         Point @ Flatten[CoordinateBoundsArray[{{0, n - 1}, {0, n - 1}}], 
       1]},                  (* lattice points *)
      {path[w1, Red, 1.4 r], (* red walker path *)
       path[w2, Blue, r]}},  (* blue walker path *)
      PlotRangePadding -> Scaled[.05]]]

Here are two interesting paths (from a code developer's point-of-view).

visualize[9]

The above walk is interesting because the walk ended at the starting point of the blue walker and demonstrates why the marker circles are scaled the way the are.

visualize[8]

This one is interesting because it is one of the relatively rare even lattice simulations to come to a successful conclusion.

I really hope someone will come up with an answer to this question soon because I spending too much time on and not getting much out it.

The discrepancy had to do with the way the walkers were subsequently moved. You have to either move synchronously (on separate threads perhaps) or you have to make it check for a specific case where the walkers are exactly one step away and their next step is towards each other. In other words, if they swap positions then there was a collision.

Thanks everyone.