Cover times and ranges of elephant random walks

Source abstract

We study cover times on discrete tori and ranges on $\mathbb{Z}^d$ for elephant random walks with memory parameter $p\in[0,1)$. In dimension one there is a phase transition at $p=3/4$. Away from criticality we determine the exact first-order asymptotics of the mean cover time, while at $p=3/4$ the mean is of order $L^2/\sqrt{\log L}$, a factor $\sqrt{\log L}$ larger than the natural fluctuation scale $L^2/\log L$. We identify the limiting distribution under each of the three natural fluctuation normalizations. In dimensions $d\geq2$, for every fixed $p<1$, the cover time has the same order as for simple random walk, in expectation and with high probability. The leading constants also agree when $p<(2d+1)/(4d)$, and at equality when $d\geq3$. For the range on $\mathbb{Z}^d$, we prove $L^1$ convergence to the simple random walk asymptotics when $d=2$ and $p<5/8$, and for every $p<1$ when $d\geq3$. Finally, we give a cover-time upper bound for generalized step-reinforced random walks on finite groups in terms of a conditional $L^\infty$ mixing profile.