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This problem was first considered and solved by Sunada, see his 1983 paper Mean-value theorems and ergodicity of certain geodesic random walks. Alas, the authors of the quoted arxiv paper were not awa …

According to the general theory any irreducible finite state space Markov chain has a unique stationary measure, and the empirical frequencies along a.e. sample path converge to this measure. The chai …

You haven't defined what "the lazy random walk" is. Since you refer to the vertex degrees, I presume that you mean that the transition probabilities are $$ p(x,y) = \begin{cases} \frac12 \;, & \text{i …

This is a particular case of random walks with internal degrees of freedom (aka semi-Markov random walks or covering Markov chains). In the case when the translation group is just $\mathbb Z$ (like in …

It is always transient in dimensions 3 and higher - see Theorem T1 in Section 8 of Spitzer's classical book "Principles of random walk" (2nd ed.).

For nearest neighbour random walks on certain free products the rate of escape (or, if you wish, drift) was explicitly calculated by Mairesse and Matheus. In particular, their formula (26) gives an ex …

No - you are completely off. There is no reason whatsoever for the harmonic measures of the original and of the reflected random walks to be equivalent (unless these random walks coincide, i.e., unles …

In my opinion, the closest to a "master theorem" is the criterion due to Terry Lyons, according to which a reversible Markov chain on a countable state space (in particular, the simple random walk on …

I am not aware of any other argument. Conceptually it is actually quite simple. The main idea consists in changing the viewpoint and considering, instead of the simple random walk on a single tree, th …

EDIT Sorry - first mistook "transitive" for "transient". Here is an example of a simple random walk on a group. The idea, indeed, is that there is a small subset of the group near the identity with a …

I don't think much is currently known. Concerning $(\ast\ast)$, there are also finitely presented groups with this property (although they are essentially obtained from wreath products) - see Erschler …

First about the background, which you present in a somewhat misleading form. The "brute force" argument of Avez you are referring to works just for finitely supported measures. On the other hand (and …

No. A counterexample can be obtained in the following way. Let us begin with an integer line $\mathbb Z$. Then take a sequence of finite graphs $\Gamma_n$, and for each $n\in\mathbb Z$ attach the grap …

Unfortunately, the probabilistic argot can be sometimes not so easy to understand for outsiders. Also, I think that your notation is unnecessarily complicated. So, what do we have here? The state spa …

You got all the references and some of the notions you mention wrong. First about the "fundamental inequality". It was first obtained by Guivarc'h in late 70s (although in a somewhat different form), …