Sequential growth of self-avoiding walks is not uniform ($\mathrm{Pr}(\text{path})\propto {\prod }_{j}1/a_j$)

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 $1/(4\cdot 3^3)$, whereas the path $(0,0),(1,0),(1,1),(0,1),(-1,1)$ has probability $1/(4\cdot 3^2\cdot 2)$. 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.