# Self-avoiding walks on strips

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 $$\operatorname P[d_n for any large enough $$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.