6,821 research outputs found
Dirichlet random walks
This article provides tools for the study of the Dirichlet random walk in
. By this we mean the random variable
where is Dirichlet distributed and where
are iid, uniformly distributed on the unit sphere of
and independent of In particular we compute explicitely in
a number of cases the distribution of Some of our results appear already
in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010,
2011). In these cases, our proofs are much simpler from the original ones,
since we use a kind of Stieltjes transform of instead of the Laplace
transform: as a consequence the hypergeometric functions replace the Bessel
functions. A crucial ingredient is a particular case of the classical and non
trivial identity, true for : We extend these results to a study of
the limits of the Dirichlet random walks when the number of added terms goes to
infinity, interpreting the results in terms of an integral by a Dirichlet
process. We introduce the ideas of Dirichlet semigroups and of Dirichlet
infinite divisibility and characterize these infinite divisible distributions
in the sense of Dirichlet when they are concentrated on the unit ball of
{4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms,
random flight, distributions in a ball, hyperuniformity, infinite divisibility
in the sense of Dirichlet.
{4mm}\noindent \textsc{AMS classification}: 60D99, 60F99
Random walks in random Dirichlet environment are transient in dimension
We consider random walks in random Dirichlet environment (RWDE) which is a
special type of random walks in random environment where the exit probabilities
at each site are i.i.d. Dirichlet random variables. On , RWDE are
parameterized by a -uplet of positive reals. We prove that for all values
of the parameters, RWDE are transient in dimension . We also prove that
the Green function has some finite moments and we characterize the finite
moments. Our result is more general and applies for example to finitely
generated symmetric transient Cayley graphs. In terms of reinforced random
walks it implies that directed edge reinforced random walks are transient for
.Comment: New version published at PTRF with an analytic proof of lemma
Random Dirichlet environment viewed from the particle in dimension
We consider random walks in random Dirichlet environment (RWDE) which is a
special type of random walks in random environment where the exit probabilities
at each site are i.i.d. Dirichlet random variables. On , RWDE
are parameterized by a 2d-uplet of positive reals called weights. In this
paper, we characterize for the weights for which there exists an
absolutely continuous invariant probability for the process viewed from the
particle. We can deduce from this result and from [27] a complete description
of the ballistic regime for .Comment: 18 pages. arXiv admin note: text overlap with arXiv:1205.5709 by
other authors without attributio
A family of random walks with generalized Dirichlet steps
We analyze a class of continuous time random walks in
with uniformly distributed directions. The steps performed by these processes
are distributed according to a generalized Dirichlet law. Given the number of
changes of orientation, we provide the analytic form of the probability density
function of the position reached, at time
, by the random motion. In particular, we analyze the case of random walks
with two steps.
In general, it is an hard task to obtain the explicit probability
distributions for the process . Nevertheless,
for suitable values for the basic parameters of the generalized Dirichlet
probability distribution, we are able to derive the explicit conditional
density functions of . Furthermore, in some
cases, by exploiting the fractional Poisson process, the unconditional
probability distributions are obtained. This paper extends in a more general
setting, the random walks with Dirichlet displacements introduced in some
previous papers
Random walks in a Dirichlet environment
This paper states a law of large numbers for a random walk in a random iid
environment on , where the environment follows some Dirichlet
distribution. Moreover, we give explicit bounds for the asymptotic velocity of
the process and also an asymptotic expansion of this velocity at low disorder.Comment: Change in theorem
A zero-one law for random walks in random environments on with bounded jumps
This paper has two main results, which are connected through the fact that
the first is a key ingredient in the second. Both are extensions of results
concerning directional transience of nearest-neighbor random walks in random
environments to allow for bounded jumps. Zerner and Merkl proved a 0-1 law for
directional transience for planar random walks in random environments. We
extend the result to non-planar i.i.d. random walks in random environments on
with bounded jumps. Sabot and Tournier characterized directional
transience for a given direction for nearest-neighbor random walks in Dirichlet
environments on , . We extend this characterization to
random walks in Dirichlet environments with bounded jumps.Comment: 27 pages, 6 figure
Random walks in Dirichlet environment: an overview
Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in
Random Environment (RWRE) on where the transition probabilities are
i.i.d. at each site with a Dirichlet distribution. Hence, the model is
parametrized by a family of positive weights ,
one for each direction of . In this case, the annealed law is that
of a reinforced random walk, with linear reinforcement on directed edges. RWDE
have a remarkable property of statistical invariance by time reversal from
which can be inferred several properties that are still inaccessible for
general environments, such as the equivalence of static and dynamic points of
view and a description of the directionally transient and ballistic regimes. In
this paper we give a state of the art on this model and several sketches of
proofs presenting the core of the arguments. We also present new computation of
the large deviation rate function for one dimensional RWDE.Comment: 35 page
- …