Random walks systems with killing on Z

We study random walks systems on Z whose general description follows. At time zero, there is a number N >= 1 of particles at each vertex of N, all being inactive, except for those placed at the vertex one. Each active particle performs a simple random walk on Z and, up to the time it dies, it activates all inactive particles that it meets along its way. An active particle dies at the instant it reaches a certain fixed total of jumps (L >= 1) without activating any particle, so that its lifetime depends strongly on the past of the process. We investigate how the probability of survival of the process depends on L and on the jumping probabilities of the active particles.

NONHOMOGENEOUS RANDOM WALKS SYSTEMS ON Z

We consider a random walks system on Z in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on Z, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to I but not fast enough.

SCALING LIMIT FOR A DRAINAGE NETWORK MODEL

We consider the two-dimensional version of a drainage network model introduced ill Gangopadhyay, Roy and Sarkar (2004), and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed in Fontes, Isopi, Newman and Ravishankar (2002).

Cellular automaton model for molecular traffic jams

We consider the time evolution of an exactly solvable cellular automaton with random initial conditions both in the large-scale hydrodynamic limit and on the microscopic level. This model is a version of the totally asymmetric simple exclusion process with sublattice parallel update and thus may serve as a model for studying traffic jams in systems of self-driven particles. We study the emergence of shocks from the microscopic dynamics of the model. In particular, we introduce shock measures whose time evolution we can compute explicitly, both in the thermodynamic limit and for open boundaries where a boundary-induced phase transition driven by the motion of a shock occurs. The motion of the shock, which results from the collective dynamics of the exclusion particles, is a random walk with an internal degree of freedom that determines the jump direction. This type of hopping dynamics is reminiscent of some transport phenomena in biological systems.