# Support of random closed walk in arbitrary graph

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)

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?

Such a bound for regular graphs valid for $$\alpha<1/4$$ was recently obtained in "Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs" https://arxiv.org/pdf/2007.12819.pdf Theorem 1.3