### Reinforced digging random walks on trees

Probability Seminar

22nd March 2024, 3:30 pm – 4:30 pm

Fry Building, 2.04

The branching-ruin number, which can be interpreted as a polynomial version of the branching number, was shown to be the critical parameter for recurrence and transience of once-reinforced random walks and M-digging random walks on tree graphs. In the talk, we will introduce a new type of random walks with reinforcement, which we call reinforced digging random walks (RDRW), which are self-interacting non-Markovian random processes in a random environment. In particular, we will prove that the phase transition of RDRW on trees depends on the relation between the branching-ruin number of a tree and a certain quantity that will be defined in the talk.

The model of RDRW is an attempt at generalising the class of reinforced random walks, and as it will be shown, once-reinforced random walks and M-digging random walks, in fact, are examples of RDRW and we will recover the phase transition results for them.

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