Trapping problems have a long-standing interest in the physics community and beyond: they have been studied with various dynamics for the particle and in a wide variety of static, dynamic and random environments. Nevertheless, traps are generally represented by point absorbers, either on a lattice or in continuous space, and their spatial extent is usually neglected. We try to answer the following natural question: "How is the survival probability of a particle affected in the presence of traps with a finite size?".
We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length l (over which the RWs can jump), 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. Its decay rate depends in a non-trivial way on the trap length l and exhibits an interesting regime when l tends to 0 as it tends to the ratio l/L, which is reminiscent of strongly chaotic deterministic systems. Then we generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump: in this case, 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.
It would be interesting to pursue further the connection with the classical billiard problems and investigate to what extent there are universal features with arbitrary jump distributions.
(Joint work with: Benjamin De Bruyne)