Researching a question related to closed walks on graphs, I have come across the following problem. Let $G$ be a connected graph on $n$ vertices and $k=O(\log(n))$. Pick a random closed walk on $G$ as follows: Pick a uniformly random vertex $v$ of $G$ and then pick a uniformly random closed walk $w$ of length $2k$ starting and ending at $v$. I am interested in showing that with high probability the support of $w$, i.e., the number of vertices it passes through, is $\Omega(k^{\alpha})$ for some constant $\alpha>0$. More formally, is it true that $$\Pr[\text{support}(w)<k^{\alpha}]\to 0 \text{ as } n\to \infty?$$

It is not very important to me that $k=O(\log(n))$, so anything without this restriction would also be of interest!

Generally, I have had a hard time finding litterature dealing with properties of random closed walks. Is there any I have overlooked?