Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.



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> Question 0: Is there a standard name for $G'$?
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Intuitively $G'$ is a thin subgraph, so for instance, even when $G$ is transient, $G'$ can be recurrent.



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> Question 1: Is there a counterexample? So, Is there a transient graph $G$ so that $G'$ is transient with positive probability?
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I'm also curious to know what happens when one iterates this procedure, $G,G',G'',\dots$. Does it eventually look like a path graph?



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> Question 2: What can one say about $G^{(n)}$ as $n\to \infty$?
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Random walk inside a random walk inside...
Question 0: $G'$ is known as the *trace* of the random walk. 


Question 1: $G'$ is always recurrent with probability one. This is [a result of Benjamini, Gurel-Gurevich, and Lyons](https://projecteuclid.org/euclid.aop/1175287760) from 2007.


Question 2: Since $G'$ is recurrent, with probability one we have $G^{(n)}=G'$ for all $n \geq 1$. 


