774 research outputs found
Queueing with neighbours
In this paper we study asymptotic behaviour of a growth process generated by
a semi-deterministic variant of cooperative sequential adsorption model (CSA).
This model can also be viewed as a particular queueing system with local
interactions. We show that quite limited randomness of the model still
generates a rich collection of possible limiting behaviours
On the generalization of the GMS evolutionary model
We study a generalization of the evolution model proposed by Guiol, Machado
and Schinazi (arXiv:0909.2108). In our model, at each moment of time a random
number of species is either born or removed from the system; the species to be
removed are those with the lower fitnesses, fitnesses being some numbers in
. We show that under some conditions, a set of species approaches (in
some sense) a sample from a uniform distribution on for some , and that the total number of species forms a recurrent process in most
other cases
Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects
We study stability of a growth process generated by sequential adsorption of
particles on a one-dimensional lattice torus, that is, the process formed by
the numbers of adsorbed particles at lattice sites, called heights. Here the
stability of process, loosely speaking, means that its components grow at
approximately the same rate. To assess stability quantitatively, we investigate
the stochastic process formed by differences of heights.
The model can be regarded as a variant of a Polya urn scheme with local
geometric interaction
VRRW on complete-like graphs: Almost sure behavior
By a theorem of Volkov (2001) we know that on most graphs with positive
probability the linearly vertex-reinforced random walk (VRRW) stays within a
finite "trapping" subgraph at all large times. The question of whether this
tail behavior occurs with probability one is open in general. In his thesis,
Pemantle (1988) proved, via a dynamical system approach, that for a VRRW on any
complete graph the asymptotic frequency of visits is uniform over vertices.
These techniques do not easily extend even to the setting of complete-like
graphs, that is, complete graphs ornamented with finitely many leaves at each
vertex. In this work we combine martingale and large deviation techniques to
prove that almost surely the VRRW on any such graph spends positive (and equal)
proportions of time on each of its nonleaf vertices. This behavior was
previously shown to occur only up to event of positive probability (cf. Volkov
(2001)). We believe that our approach can be used as a building block in
studying related questions on more general graphs. The same set of techniques
is used to obtain explicit bounds on the speed of convergence of the empirical
occupation measure.Comment: Published in at http://dx.doi.org/10.1214/10-AAP687 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Markov chains in a field of traps
A general criterion is given for when a Markov chain trapped with probability
p(x) in state x will be almost surely trapped. The quenched (state x is a trap
forever with probability p(x)) and annealed (state x traps with probability
p(x) on each visit) problems are shown to be equivalent
Turning a coin over instead of tossing it
Given a sequence of numbers in , consider the following
experiment. First, we flip a fair coin and then, at step , we turn the coin
over to the other side with probability , . What can we say about
the distribution of the empirical frequency of heads as ?
We show that a number of phase transitions take place as the turning gets
slower (i.e. is getting smaller), leading first to the breakdown of the
Central Limit Theorem and then to that of the Law of Large Numbers. It turns
out that the critical regime is . Among the scaling limits,
we obtain Uniform, Gaussian, Semicircle and Arcsine laws
On a class of random walks in simplexes
We study the limit behaviour of a class of random walk models taking values
in the -dimensional unit standard simplex, , defined as follows.
From an interior point , the process chooses one of the vertices of
the simplex, with probabilities depending on , and then the particle
randomly jumps to a new location on the segment connecting to the
chosen vertex. In some specific cases, using properties of the Beta
distribution, we prove that the limiting distributions of the Markov chain are,
in fact, Dirichlet. We also consider a related history-dependent random walk
model in based on an urn-type scheme. We show that this random walk
converges in distribution to the arcsine law.Comment: final versio
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