How long does it take an Elephant Random Walk to forget its training

We study how conditioning affects the long-term behavior of the Elephant Random Walk, focusing on its first return to the origin, namely, the time it takes to forget the training. When the walker is conditioned to be at position $n$ at time $n$, this return time scales as $n^{(4-4p)\mathrm{/}(3-4p)}$ in the diffusive regime, and grows exponentially in the critical regime.