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

TL;DR

The paper investigates the time-dependent survival probability $S(t;x_0,q)$ for biased Sisyphus random walkers that restart at $x_0$ with probability $1-q$ and move toward an absorbing trap at $0$ with probability $q>1/2$. Using a recurrence and an asymptotic ansatz, the authors derive an exponential decay rate $\beta$ governed by the root of $\beta^{x_0+1}-\beta^{x_0}+q^{x_0}(1-q)=0$, yielding $S(t+1)/S(t)\to\beta\approx1-q^{x_0}(1-q)$ for large $x_0$. They show that, compared to standard biased walkers (no restart), there exists a $q$-dependent critical initial gap $x_0^{\mathrm{crit}}(q)$ such that, for $x_0>x_0^{\mathrm{crit}}(q)$, the Sisyphus walker has a larger late-time survival; this threshold diverges as $q\to1/2$, while for $q\gtrsim0.78$ the Sisyphus advantage holds even at $x_0=1$. The results combine exact recurrences, asymptotic analysis, and numerical validation, providing insight into how restart dynamics can modify absorbing-state survival in biased random walks.

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.