Sisyphus random walks in the presence of moving traps

TL;DR

The paper investigates how a slowly moving absorbing trap affects the late-time survival of Sisyphus random walkers with a restart mechanism. It derives a recurrence for the survival probability $S(t)$ that depends on the moving trap position $x_T(t)$ and analyzes asymptotics by positing a power-law tail $S(t)\sim \alpha t^{-{\beta}}$. The key finding is that when the trap advances as $x_T(t)=a\ln t+b$ (so the velocity $v_{ ext{trap}}=a/t$ vanishes), the survival probability decays as $S(t)\sim \alpha t^{-{\beta}}$ with $\beta=(1-q)\,q^{b}$ and $a=1/\ln(1/q)$, revealing a universal exponent with respect to $a$. This contrasts with exponential decay for static traps and with power-law tails emerging only under specific moving-trap dynamics in ordinary random walks, highlighting how restart can qualitatively alter absorption processes.

Abstract

It has recently been proved that, in the presence of a static absorbing trap, Sisyphus random walkers with a restart mechanism are characterized by {\it exponentially} decreasing asymptotic survival probability functions. Interestingly, in the present compact paper we prove analytically that, in the presence of a moving trap whose velocity approaches zero asymptotically in time as $v_{\text{trap}}\sim 1/t$, the survival probabilities of the Sisyphus walkers are dramatically changed into inverse {\it power-law} decaying tails.