Some new results on 1-d self-repelling random walks

I will present two new results in the context of the title. Both are joint work with B. Veto.

1. In earlier work a limit theorem with $t^{2/3}$ scaling was established for a class of self repelling random walks on $\mathbb Z$ with long memory, where the self-interaction was defined in terms of the local time spent on unoriented edges. For combinatorial reasons this proof was not extendable to the natural case when the self-repellence is defined in trems of local time on sites. Now we prove a similar result for a *continuous time* random walk on $\mathbb Z$, with self-repellence defined in terms of local time on sites.

2. Defining the self-repelling mechanism in terms of the local time on *oriented edges* results in totally different asymptotic behaviour than the unoriented cases. We prove limit theorems for this random walk with long memory.