111 research outputs found

### Spectral Analysis of a Self-Similar Sturm-Liouville Operator

In this text we describe the spectral nature (pure point or continuous) of a
self-similar Sturm-Liouville operator on the line or the half-line.
This is motivated by the more general problem of understanding the spectrum
of Laplace operators on unbounded finitely ramified self-similar sets. In this
context, this furnishes the first example of a description of the spectral
nature of the operator in the case where the so-called "Neumann-Dirichlet"
eigenfunctions are absent.Comment: 20 pages, 1 figur

### Random Dirichlet environment viewed from the particle in dimension $d\ge 3$

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 ${\mathbb Z}^d$, RWDE
are parameterized by a 2d-uplet of positive reals called weights. In this
paper, we characterize for $d\ge 3$ 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 $d\ge 3$.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1205.5709 by
other authors without attributio

### Markov chains in a Dirichlet Environment and hypergeometric integrals

The aim of this text is to establish some relations between Markov chains in
Dirichlet Environments on directed graphs and certain hypergeometric integrals
associated with a particular arrangement of hyperplanes. We deduce from these
relations and the computation of the connexion obtained by moving one
hyperplane of the arrangement some new relations on important functionals of
the Markov chain.Comment: 6 pages, preliminary not

### Ballistic random walks in random environment at low disorder

We consider random walks in a random environment of the type p_0+\gamma\xi_z,
where p_0 denotes the transition probabilities of a stationary random walk on
\BbbZ^d, to nearest neighbors, and \xi_z is an i.i.d. random perturbation. We
give an explicit expansion, for small \gamma, of the asymptotic speed of the
random walk under the annealed law, up to order 2. As an application, we
construct, in dimension d\ge2, a walk which goes faster than the stationary
walk under the mean environment.Comment: Published at http://dx.doi.org/10.1214/009117904000000739 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Random walks in Dirichlet environment: an overview

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in
Random Environment (RWRE) on $\Bbb{Z}^d$ 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 $(\alpha_i)_{i=1, \ldots, 2d}$,
one for each direction of $\Bbb{Z}^d$. 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

### Random walks in a Dirichlet environment

This paper states a law of large numbers for a random walk in a random iid
environment on ${\mathbb Z}^d$, 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

### Markov chains in a Dirichlet Environment and hypergeometric integrals

6 pages, preliminary note.International audienceThe aim of this text is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce from these relations and the computation of the connexion obtained by moving one hyperplane of the arrangement some new relations on important functionals of the Markov chain

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