Fábio P. Machado

Phase Transition for the Frog Model

We study a system of simple random walks on graphs, known as frog model. This model can be described as follows: There are active and sleeping particles living on some graph G. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active. Phase transition results and asymptotic values for critical parameters are presented for Z^d and regular trees.

LIMIT THEOREMS FOR A GENERAL STOCHASTIC RUMOUR MODEL

We study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour. We refer to this special kind of stifler as an uninterested individual. Our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does, so that particular cases are the classical Daley–Kendall and Maki–Thompson models. We prove a Law of Large Numbers and a Central Limit Theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it.

Branching random walk in random environment on trees

We study a supercritical branching random walk on a rooted tree with random environment. We are interested in the case where both the branching and the step transition parameters are random quantities. Criteria of (strong) recurrence and (strong) transience are presented for this model.

RUMOR PROCESSES ON N

We study four discrete time stochastic systems on

On a link between a species survival time in an evolution model and the Bessel distributions

\bbN

Recurrence and transience of multitype branching random walks

modeling processes of rumour spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumour. The appetite in spreading or hearing the rumour is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on those random variables distribution - whether the probability of having an infinite set of individuals knowing the rumour is positive or not.

Random walks systems with killing on ℤ

We consider a stochastic model for species evolution. A new species is born at rate lambda and a species dies at rate mu. A random number, sampled from a given distribution F, is associated with each new species at the time of birth. Every time there is a death event, the species that is killed is the one with the smallest fitness. We consider the (random) survival time of a species with a given fitness f. We show that the survival time distribution depends crucially on whether f f_c where f_c is a critical fitness that is computed explicitly.

Dispersion as a survival strategy

We study a discrete time Markov process with particles being able to perform discrete time random walks and create new particles, known as branching random walk (BRW). We suppose that there are particles of different types, and the transition probabilities, as well as offspring distribution, depend on the type and the position of the particle. Criteria of (strong) recurrence and transience are presented, and some applications (spatially homogeneous case, Lamperti BRW, many-dimensional BRW) are studied.

Estimates for the spreading velocity of an epidemic model

We study random walks systems on ℤ whose general description follows. At time zero, there is a number 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.

Random Walks Systems with Finite Lifetime on \mathbb {Z}

We consider stochastic growth models to represent population subject to catastrophes. We analyze the subject from different set ups considering or not spatial restrictions, whether dispersion is a good strategy to increase the population viability. We find out it strongly depends on the effect of a catastrophic event, the spatial constraints of the environment and the probability that each exposed individual survives when a disaster strikes.