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A strip is a locally finite graph which admits a quasi-transitive (i.e. finitley many orbits on vertices) action of $\mathbb Z$. A self avoiding walk is a walk which visits no vertex more than once.

Is it known that the random self-avoiding walk on any strip has positive speed? More precisely, let $W_n$ be a self avoiding walk of length $n$ chosen uniformly at random, and denote by $d_n$ the distance (in the graph metric) between the initial and terminal vertex of $W_n$. Are there $a,b>0$ such that $\operatorname P[d_n<an] < e^{-b n}$ for any large enough $n$?

Some special cases have been proven in "Janson, Svante, Random self-avoiding walks in some one-dimensional lattices, Ann. Discrete Math. 33, 91-109 (1987) ZBL0674.05043.", and more special cases can be proved along the same lines, but the proof does not immediately help with the general case.

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