Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels

We study the one-dimensional elephant random walk with two memory channels introduced by Saha [Phys.\ Rev.\ E $\mathbf{\text{106}}$, L062105 (2022)] with memory parameter $p\in (0,1)$. Maulik, Roy and Sadhukhan [arXiv:2509.10225] proved recurrence for $p\le 11\mathrm{/}16$ and transience for $p>7\mathrm{/}8$, leaving the range $11\mathrm{/}16<p\le 7\mathrm{/}8$ open. We close this gap by proving that $p=11\mathrm{/}16$ is the exact recurrence–transience threshold and by determining how fast the walk escapes from the origin throughout the transient regime. For $11\mathrm{/}16<p<7\mathrm{/}8$, we also show that the scaling limit obtained by Maulik, Roy and Sadhukhan is almost surely nonzero. At $p=7\mathrm{/}8$, we prove that the walk is transient with zero asymptotic velocity and escapes at the scale $n\mathrm{/}\sqrt{\log n}$.