Have there been any work analyzing some properties of random DFS walks? By that I mean a DFS search, which chooses the next node to visit with uniform probability. i.e, it still refrains from visiting any node already traversed.
For example, is there anything concerning the distribution of the nodes visited, given a uniformly chosen vertex as a starting point? I'm interested in degree bound graphs if that helps.
I've tried looking myself but haven't found anything.