Survival probabilities in the Sisyphus random walk model with absorbing traps

Abstract

We analyze the dynamics of the Sisyphus random walk model, a discrete Markov chain in which the walkers may randomly return to their initial position x0. In particular, we present a remarkably compact derivation of the time-dependent survival probability function S(t;x0) which characterizes the random walkers in the presence of an absorbing trap at the origin. The survival probabilities are expressed in a compact mathematical form in terms of the x0-generalized Fibonacci-like numbers Gt(x0). Interestingly, it is proved that, as opposed to the standard random walk model in which the survival probabilities depend linearly on the initial distance x0 of the walkers from the trap and decay asymptotically as an inverse power of the time, in the Sisyphus random walk model the asymptotic survival probabilities decay exponentially in time and are characterized by a non-trivial (non-linear) dependence on the initial gap x0 from the absorbing trap. We use the analytically derived results in order to analyze the underlying dynamics of the ‘survival-game’, a highly risky investment strategy in which non-absorbed agents receive a reward P(t) which gradually increases in time.