Survival probability of random walks leaping over traps

Abstract

We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length $\ell$ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance $L$. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate of the survival probability depends in a non-trivial way on the trap length $\ell$ and exhibits an interesting regime when $\ell \to 0$ as it tends to the ratio $\ell /L$, which is reminiscent of strongly chaotic deterministic systems. We generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump. In this case, we find that the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers.

arXiv comment29 pages, 5 figures

References in corpus (26)

• Fractional calculus and continuous-time finance
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Fractional calculus and continuous-time finance II: the waiting-time distribution
• A continuous time random walk model for financial distributions
• First-passage times for random walks in bounded domains
• Random Convex Hulls and Extreme Value Statistics
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Exact Asymptotics for One-Dimensional Diffusion with Mobile Traps
• Trapping Reactions with Randomly Moving Traps: Exact Asymptotic Results for Compact Exploration
• Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps
• Survival probability of a particle in a sea of mobile traps: A tale of tails
• Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap
• Survival of classical and quantum particles in the presence of traps
• Exit time distribution in spherically symmetric two-dimensional domains
• Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates
• Formal Solution of a Class of Reaction-Diffusion Models: Reduction to a Single-Particle Problem
• Extreme times in financial markets
• Mean first-passage time to a small absorbing target in an elongated planar domain
• Sinai model in presence of dilute absorbers
• One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas
• Mean Exit Time and Survival Probability within the CTRW Formalism
• Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps
• Residual mean first-passage time for jump processes: theory and applications to Lévy flights and fractional Brownian motion
• Perturbation theory for the one-dimensional trapping reaction