132 research outputs found
The spread of a rumor or infection in a moving population
We consider the following interacting particle system: There is a ``gas'' of
particles, each of which performs a continuous-time simple random walk on
, with jump rate . These particles are called -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
-particles at , and that the ,
are i.i.d., mean- Poisson random variables. In addition, there are
-particles which perform continuous-time simple random walks with jump rate
. We start with a finite number of -particles in the system at time 0.
-particles are interpreted as individuals who have heard a certain rumor or
who are infected. The -particles move independently of each other. The only
interaction is that when a -particle and an -particle coincide, the
latter instantaneously turns into a -particle. We investigate how fast the
rumor, or infection, spreads. Specifically, if
a -particle visits during
and , then we investigate the
asymptotic behavior of . Our principal result states that if
(so that the - and -particles perform the same random walk), then there
exist constants such that almost surely
for all large ,
where . In a further paper we shall use the results
presented here to prove a full ``shape theorem,'' saying that
converges almost surely to a nonrandom set , with the origin as an
interior point, so that the true growth rate for is linear in . If
, then we can only prove the upper bound 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
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
We consider the following problem in one-dimensional diffusion-limited
aggregation (DLA). At time , we have an "aggregate" consisting of
[with a positive integer]. We also have
particles at , . All these particles perform independent
continuous-time symmetric simple random walks until the first time at
which some particle tries to jump from to . The aggregate is
then increased to the integers in [so that
] and all particles which were at at time are
removed from the system. The problem is to determine how fast grows as a
function of if we start at time 0 with and the i.i.d.
Poisson variables with mean . It is shown that if , then
is of order , in a sense which is made precise. It is conjectured
that will grow linearly in if 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
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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