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We refine previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any dimension d ≥ 1 and requires only a moment condition slightly stronger than finite first moment. For heavy-tailed interarrival times, we prove a Complete Convergence Theorem and examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of distribution attracted to a stable law of index 1 for which the critical value vanishes. [ABSTRACT FROM AUTHOR]

We derive the existence of infinite level GREM-like K-processes by taking the limit of a sequence of finite level versions of such processes as the number of levels diverges. The main step in the derivation is obtaining the convergence of the sequence of underlying finite level clock processes. This is accomplished by perturbing these processes so as to turn them into martingales, and resorting to martingale convergence to obtain convergence for the perturbed clock processes; nontriviality of the limit requires a specific choice of parameters of the original process; we conclude the step by showing that the perturbation washes away in the limit. The perturbation is done by inserting suitable factors into the expression of the clocks, as well as rescaling the resulting expression suitably; the existence of such factors is itself established through martingale convergence. [ABSTRACT FROM AUTHOR]

We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution μ is heavier than t − α for some α < 1 (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if μ has decreasing hazard rate and tail bounded by t − α with α > 1 , then the critical value is positive in the one-dimensional case. A more robust and much simpler argument shows that the critical value is positive in any dimension whenever the interarrival distribution has a finite second moment. [ABSTRACT FROM AUTHOR]

We investigate a non-Markovian analogue of the Harris contact process in Z d : an individual is attached to each site x ∈ Z d , and it can be infected or healthy; the infection propagates to healthy neighbours just as in the usual contact process, according to independent exponential times with a fixed rate λ ; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that the resulting processes have a critical value equal to zero. [ABSTRACT FROM AUTHOR]

We study a pure death process. At each discrete time every individual dies or not independently of each other with a constant probability. We give examples showing that in a certain limit extinction happens along a path where one and only one individual is lost at a time. We also exhibit an example for which such a path goes from infinity to 0 in a finite time. This is what we call a process implosion. [ABSTRACT FROM AUTHOR]

We study the threshold θ ≥ 2 contact process on a homogeneous tree $$\mathbb T_b$$ of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point $$\lambda_{\rm c}^{\rm MF}(\kappa,\theta)$$ and for $$\lambda \geq\lambda_{\rm c}^{\rm MF}(\kappa,\theta)$$ it survives iff $$p \geq p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda)$$ , where this critical density satisfies $$0 < p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda) < 1$$ , $$ \lim_{\lambda \to \infty} p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda) = 0$$ . For large b, we show that the process on $$\mathbb T_b$$ has a qualitatively similar behavior when λ is small, including the behavior at and close to the critical point $$\lambda_{\rm c}(\mathbb T_b,\theta)$$ . In contrast, for large λ the behavior of the process on $$\mathbb T_b$$ is qualitatively distinct from that of the mean-field model in that the critical density has $$p_{\rm c}({\mathbb T}_b,\theta,\infty)\,:= \lim_{\lambda \to \infty} p_{\rm c}(\mathbb T_b,\theta,\lambda) > 0$$ . We also show that $$\lim_{b \to \infty} b \lambda_{\rm c}(\mathbb T_b,\theta) = \Phi_{\theta}$$ , where 1 < Φ2 < Φ3 < ..., $$\lim_{\theta \to \infty} \Phi_{\theta} = \infty$$ , and $$0 < \lim inf_{b \to \infty} b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty)\leq \lim sup_{b \to \infty}b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty) < \infty$$ . [ABSTRACT FROM AUTHOR]

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = p c+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = p c, based on SLE 6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees. [ABSTRACT FROM AUTHOR]

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all spacetime trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg. [ABSTRACT FROM AUTHOR]

Presents the Foreword to a volume of papers related to the 6th Brazilian School of Probability in Ubatuba-SP in August 2002. Range of subjects from mainstream of theoretical probability to current problems of applied probability; Number of participants over 110 from 15 countries; Expression of gratitude to committees, staff members agencies etc.