132 research outputs found

    The spread of a rumor or infection in a moving population

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    We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on Zd\mathbb{Z}^d, with jump rate DAD_A. These particles are called AA-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with NA(x,0−)N_A(x,0-) AA-particles at xx, and that the NA(x,0−),x∈ZdN_A(x,0-),x\in\mathbb{Z}^d, are i.i.d., mean-μA\mu_A Poisson random variables. In addition, there are BB-particles which perform continuous-time simple random walks with jump rate DBD_B. We start with a finite number of BB-particles in the system at time 0. BB-particles are interpreted as individuals who have heard a certain rumor or who are infected. The BB-particles move independently of each other. The only interaction is that when a BB-particle and an AA-particle coincide, the latter instantaneously turns into a BB-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if B~(t):={x∈Zd:\widetilde{B}(t):=\{x\in\mathbb{Z}^d: a BB-particle visits xx during [0,t]}[0,t]\} and B(t)=B~(t)+[−1/2,1/2]dB(t)=\widetilde{B}(t)+[-1/2,1/2]^d, then we investigate the asymptotic behavior of B(t)B(t). Our principal result states that if DA=DBD_A=D_B (so that the AA- and BB-particles perform the same random walk), then there exist constants 0<Ci<∞0<C_i<\infty such that almost surely C(C2t)⊂B(t)⊂C(C1t)\mathcal{C}(C_2t)\subset B(t)\subset \mathcal{C}(C_1t) for all large tt, where C(r)=[−r,r]d\mathcal{C}(r)=[-r,r]^d. In a further paper we shall use the results presented here to prove a full ``shape theorem,'' saying that t−1B(t)t^{-1}B(t) converges almost surely to a nonrandom set B0B_0, with the origin as an interior point, so that the true growth rate for B(t)B(t) is linear in tt. If DA≠DBD_A\ne D_B, then we can only prove the upper bound B(t)⊂C(C1t)B(t)\subset \mathcal{C}(C_1t) eventually.Comment: Published at http://dx.doi.org/10.1214/009117905000000413 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Stability of the Greedy Algorithm on the Circle

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    We consider a single-server system with service stations in each point of the circle. Customers arrive after exponential times at uniformly-distributed locations. The server moves at finite speed and adopts a greedy routing mechanism. It was conjectured by Coffman and Gilbert in~1987 that the service rate exceeding the arrival rate is a sufficient condition for the system to be positive recurrent, for any value of the speed. In this paper we show that the conjecture holds true

    A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains

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    We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time tt, we have an "aggregate" consisting of Z∩[0,R(t)]\Bbb{Z}\cap[0,R(t)] [with R(t)R(t) a positive integer]. We also have N(i,t)N(i,t) particles at ii, i>R(t)i>R(t). All these particles perform independent continuous-time symmetric simple random walks until the first time t′>tt'>t at which some particle tries to jump from R(t)+1R(t)+1 to R(t)R(t). The aggregate is then increased to the integers in [0,R(t′)]=[0,R(t)+1][0,R(t')]=[0,R(t)+1] [so that R(t′)=R(t)+1R(t')=R(t)+1] and all particles which were at R(t)+1R(t)+1 at time t′−t'{-} are removed from the system. The problem is to determine how fast R(t)R(t) grows as a function of tt if we start at time 0 with R(0)=0R(0)=0 and the N(i,0)N(i,0) i.i.d. Poisson variables with mean μ>0\mu>0. It is shown that if μ<1\mu<1, then R(t)R(t) is of order t\sqrt{t}, in a sense which is made precise. It is conjectured that R(t)R(t) will grow linearly in tt if μ\mu is large enough.Comment: Published in at http://dx.doi.org/10.1214/07-AOP379 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Multi-Particle Diffusion Limited Aggregation

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    We consider a stochastic aggregation model on Z^d. Start with particles located at the vertices of the lattice, initially distributed according to the product Bernoulli measure with parameter \mu. In addition, there is an aggregate, which initially consists of the origin. Non-aggregated particles move as continuous time simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is referred to as multi-particle diffusion limited aggregation. Our main result states that if on d>1 the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms, i.e. if F(t) denotes the point of the aggregate furthest away from the origin at time t>0, then there exists a constant c>0 so that |F(t)|>ct, for all t eventually. The key conceptual element of our analysis is the introduction and study of a new growth process. Consider a first passage percolation process, called type 1, starting from the origin. Whenever type 1 is about to occupy a new vertex, with positive probability, instead of doing it, it gives rise to another first passage percolation process, called type 2, which starts to spread from that vertex. Each vertex gets occupied only by the process that arrives to it first. This process may have three phases: an extinction phase, where type 1 gets eventually surrounded by type 2 clusters, a coexistence phase, where infinite clusters of both types emerge, and a strong survival phase, where type 1 produces an infinite cluster that successfully surrounds all type 2 clusters. Understanding the behavior of this process in its various phases is of mathematical interest on its own right. We establish the existence of a strong survival phase, and use this to show our main result.Comment: More thorough explanations in some steps of the proof
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