= Sequential growth of self-avoiding walks is not uniform
{title2=$\Pr(\text{path})\propto\prod_j1/a_j$}
Choosing uniformly among currently unused neighbours gives a <self-avoiding walk> with a path-dependent probability: $a_j$ is the number of choices available at step $j$. In the square lattice, a four-step straight path has probability $1/(4\cdot3^3)$, whereas the path $(0,0),(1,0),(1,1),(0,1),(-1,1)$ has probability $1/(4\cdot3^2\cdot2)$. There is no trapping at that length, and these unequal probabilities disprove uniformity.
When a trapped attempt is restarted, the final law is the one-attempt law conditional on successful completion. This divides successful-path probabilities by a common normalizing constant and does not remove their unequal products. The local uniformity of each move therefore does not imply uniformity over complete paths.
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