An object moves on a square grid, taking one step chosen uniformly randomly from one of the 4 adjacent squares at each time step.

The possible range after N steps is a diamond shaped region, reflecting the 4-fold symmetry of the orthogonal step directions.

However, the probability distribution of where the object is after N time steps, as N grows large, we conjecture to be circularly symmetric. (Based on the normal approximation to binomial distribution, and exp(x^2+y^2) being circularly symmetric.) This is surprising: additional symmetry, an infinite amount of it, seems to have appeared out of nowhere.

The actual probability distribution is a checkerboard because of parity. Modification: the object does not move with probability p, or moves to an adjacent square with probability (1-p)/4.

Also consider a grid of hexagons.

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