This is a sequel to Random knight… don’t expect more than you usually get from sequels. I do not plan to do other chess pieces, although “random pawn” is an intriguing idea.

Queen begins its random walk at d1. After two moves, it can get anywhere on the board.

It is not surprising that the most likely outcome is the return to initial state. It is less obvious why the second most likely position is d3.

After 3 moves, the most likely positions are near the center of the board.

After 4 moves, a concentric-square pattern emerges.

After 10 moves, the distribution is visually identical to the steady-state distribution.

As was noted in the *Random knight* post, the steady state distribution is the normalized number of moves from a given position. Hence, all probabilities are rational. For the knight they happen to be unit fractions: 1/168, 1/112, 1/84, 1/56, 1/42. The queen probabilities are uglier: 3/208, 23/1456, 25/1456, 27/1456. Random queen is distributed more uniformly than random knight: it spends between 1.44% and 1.85% of its time in any given square. For the knight, the range is from 0.59% to 2.38%.

But the champion in uniform coverage of the chessboard is the rook: when moving randomly, it spends 1/64 of time on every square.

A random pawn will of course march down and be promoted, with every possible promotion equally likely. So its steady state distribution will be the average of the steady state distribution of the queen, rook, knight and (appropriately colored) bishop.

I agree. With the promotion taken into account, a random pawn is pretty uniform across the board. The most non-uniform piece is (of course) the bishop – but if only reachable squares are considered, then it’s the knight.