= Self-avoiding polygon
{title2=$x_0=x_M,\quad x_0,\ldots,x_{M-1}\text{ distinct}$}
A <self-avoiding polygon> on a lattice is a closed nearest-neighbour walk whose vertices are distinct except for the final repetition of the starting point. On the square lattice a nondegenerate polygon has even perimeter at least four, since each step changes the parity of the sum of the coordinates. One can generate such a polygon by generating a self-avoiding walk with $M-1$ edges, retaining it only when its endpoint is a neighbour of its origin, and appending the closing edge. The resulting distribution inherits the bias of the walk sampler unless an additional correction is applied.
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