Survival probabilities in biased random walks: To restart or not to restart? that is the question

TL;DR

This paper derives the analytical form of the survival probability for biased Sisyphus random walkers, revealing exponential decay at late times and a critical initial gap influencing survival chances compared to standard random walks.

Contribution

The paper provides an explicit analytical derivation of the survival probability function for biased Sisyphus random walkers, including the identification of a critical initial gap affecting survival.

Findings

Survival probability decays exponentially at late times.

Existence of a critical initial gap $x^{\text{crit}}_0(q)$ influencing survival.

Survival probability can be larger than that of standard random walkers for certain initial conditions.

Abstract

The time-dependent survival probability function $S(t;x_0,q)$ of biased Sisyphus random walkers, who at each time step have a finite probability $q$ to step towards an absorbing trap at the origin and a complementary probability $1-q$ to return to their initial position $x_0$, is derived {\it analytically}. In particular, we explicitly prove that the survival probability function of the walkers decays exponentially at asymptotically late times. Interestingly, our analysis reveals the fact that, for a given value $q$ of the biased jumping probability, the survival probability function $S(t;x_0,q)$ is characterized by a {\it critical} (marginal) value $x^{\text{crit}}_0(q)$ of the initial gap between the walkers and the trap, above which the late-time survival probability of the biased Sisyphus random walkers is {\it larger} than the corresponding survival probability of standard random walkers.