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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 …

The step distribution $\mu$ of your random walk is finitely supported and symmetric, so that all standard results are applicable. In particular, your random walk is recurrent for any value of the para …

The right setup here is that of topological Markov chains. This is essentially the same as a directed graph, i.e., a (finite, for simplicity) set of states (vertices) $A$ endowed with a $\{0,1\}$-valu …

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 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 …

This is nothing but an ordinary Markov chain whose state space is the set of oriented edges of the graph with the transition probabilities determined by the configuration of two adjacent oriented edge …

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 …

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 …

There are explicit examples of radially symmetric random walks on free groups for which the Martin boundary contains potentials, see Theorems 2 and 4 in Cartwright and Sawyer. Concerning your second …

In general there is no reason for the coincidence of the measure classes of the harmonic measures of the original and of the reflected random walks. It they do coincide, then this indicates that the r …

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 …

As it has been proved in http://www.ams.org/mathscinet-getitem?mr=2191210, if the set of minimal harmonic functions is closed, then the group action on it is topologically amenable. Groups which admit …

The answers to all these questions are "yes" and are more or less obvious from the definitions of the corresponding properties. This is the reason why they don't appear in the literature in an explici …

The way your statement is formulated, it is wrong. The simplest counterexample is the function $f(x,y)=xy$ on $\mathbb Z^2$ which vanishes on the ball of radius 1. In Heilbronn's pioneeriing 1949 pape …

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 …