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One quick remark: the result in Doyle and Snell cannot be sharp. Indeed, take any graph and on each vertex add a finite but huge (and larger and larger as you go away from a merked origin vertex) tree …

At the intuitive level, it would seem that the process should behave as if it had finite memory, and be diffusive in the long term: as soon as $p>0$ I would expect a central limit theorem. Of course t …

That will depend on your exact setup, i.e. on what you mean exactly by a SAW ... If the question is "take a simple random walk, how long will it remain self-avoiding", this will behave like a geomet …

Following up on Nathanael's answer: let me give a more precise statement about the dynamics as I understand it. At each step, choose whether you want to pop or push/unpush, then pick a location (or tw …

Say that a graph $G$ has the Liouville property if all bounded (discrete-)harmonic functions on it are constant. If it is possible to couple random walks on $G$ started from any two starting points …

There is a problem in your definition of transience: the standard definition of the self-avoiding walk is the uniform (counting) measure on the set of walks of a given length, say $n$, and one is inte …

As noted by Yoav above, and by myself in the comments to the linked questions, you need to make a distinction between a simple random walks restricted to make steps into previously unvisited sites (th …

Giving appropriate meaning to "generating $G$", here is one thing that could give some information: start from $v_0$ and walk until the walk has come back $k$ times to $v_0$; or possibly, instead, wal …