Survival probability of random walks leaping over traps

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. 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 ℓ and exhibits an interesting regime when ℓ → 0 as it tends to the ratio ℓ/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.

Critical current fluctuations in graphene Josephson junctions

We have studied 1/f noise in critical current $$I_c$$ I c in h-BN encapsulated monolayer graphene contacted by NbTiN electrodes. The sample is close to diffusive limit and the switching supercurrent with hysteresis at Dirac point amounts to $$\simeq 5$$ ≃ 5 nA. The low frequency noise in the superconducting state is measured by tracking the variation in magnitude and phase of a reflection carrier signal $$v_{rf}$$ v rf at 600–650 MHz. We find 1/f critical current fluctuations on the order of $$\delta I_c/I_c \simeq 10^{-3}$$ δ I c / I c ≃ 10 - 3 per unit band at 1 Hz. The noise power spectrum of critical current fluctuations $$S_{I_c}$$ S I c measured near the Dirac point at large, sub-critical rf-carrier amplitudes obeys the law $$S_{I_c}/{I{_c}}^2 = a/f^{\beta }$$ S I c / I c 2 = a / f β where $$a\simeq 4\times 10^{-6}$$ a ≃ 4 × 10 - 6 and $$\beta \simeq 1$$ β ≃ 1 at $$f > 0.1$$ f > 0.1 Hz. Our results point towards significant fluctuations in $$I_c$$ I c originating from variation of the proximity induced gap in the graphene junction.

Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.