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The phase transition of $M$-digging random on a general tree was studied byCollevecchio, Huynh and Kious (2018). In this paper, we study particularly thecritical $M$-digging random walk on a superperiodic tree that is proved to berecurrent. We keep using the techniques introduced by Collevecchio, Kious andSidoravicius (2017) with the aim of investigating the phase transition ofOnce-excited random walk on general trees. In addition, we prove if $\mathcalT$ is a tree whose branching number is larger than $1$, any multi-excitedrandom walk on $\mathcal{T}$ moving, after excitation, like a simple randomwalk is transient.

Phase transition for the once-excited random walk on general trees