J. Appl. Prob. 47, 562–571 (2010) Printed in England  Applied Probability Trust 2010 NONHOMOGENEOUS RANDOM WALKS SYSTEMS ON Z ELCIO LEBENSZTAYN ∗ ∗∗ and FÁBIO PRATES MACHADO, ∗ ∗∗∗ University of São Paulo MAURICIO ZULUAGA MARTINEZ, ∗∗∗∗ Federal University of Pernambuco Abstract 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 1 but not fast enough. Keywords: Random walk; epidemic model; frog model 2010 Mathematics Subject Classification: Primary 60K35 Secondary 60G50 1. Introduction We study a random walks system on Z whose parameters are the integers L ≥ 1 and N ≥ 1, and a sequence {q n } n≥1 of real numbers in (0, 1). At time 0, there are N particles at each vertex of N ={1, 2,... }. All particles are inactive, except for those placed at the vertex 1. The active particles move as discrete-time independent nearest-neighbor random walks on Z, and activate the inactive particles encountered during their trajectories. Each active particle has a lifetime which depends on the past of the process, in the sense that its displacement lasts until it reaches a total of L jumps without activating any particle. When several active particles simultaneously jump on an inactive particle, we consider all of them responsible for the activation. We suppose that the jump probabilities of the active particles depend on their initial position: particles initially placed at position n, in the event of activation, jump at each step to the right with probability 1 - q n or to the left with probability q n . This model, named the particle process on Z, was first introduced in [5]. Its dynamics describe the evolution of a virus in a population with an infinite number of individuals (say, computers) connected in line. Under this interpretation, the active particles are viruses which create N replicas every time a new computer is infected. Once this happens, this computer Received 27 April 2009; revision received 11 March 2010. ∗ Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, Brazil. ∗∗ Email address: elcio@ime.usp.br ∗∗∗ Email address: fmachado@ime.usp.br ∗∗∗∗ Postal address: Department of Statistics, Federal University of Pernambuco, Cidade Universitária, CEP 50740- 540, Recife, PE, Brazil. Email address: zuluaga@ime.usp.br 562 available at https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1276784909 Downloaded from https://www.cambridge.org/core. IP address: 192.126.205.106, on 25 Apr 2020 at 03:04:12, subject to the Cambridge Core terms of use,