A Spectral Approach to Survival Bounds for the Elephant Random Walk

Abstract

We study a rate-one Poissonization of the one-dimensional elephant random walk killed upon reaching $\pm N$. For every fixed memory parameter $0<p<1$, we give a self-contained proof of two-sided exponential survival bounds on the rate scale $N^{-2}$, uniformly for sufficiently large $N$ and times $t\ge T{N}^3$. Our approach combines the classical Dirichlet Poincaré inequality with an $O(N/t)$ estimate on the memory-dependent perturbation and a quantitative initialization of the principal-eigenvector projection. The lower bound covers the critical and superdiffusive regimes $3/4\le p<1$ in this late-time range. The confined-path comparison used for initialization also gives a direct proof of the lower bound. The spectral argument controls the exponentially small forcing terms and jumps introduced by Poisson truncation.

arXiv comment15 pages