Random Dirichlet environment viewed from the particle in dimension d≥3d\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

A note on edge oriented reinforced random walks and RWRE

This work introduces the notion of edge oriented reinforced random walk which proposes in a general framework an alternative understanding of the annealed law of random walks in random environment.Comment: 5 page

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

IDEAS project - Professional advice network study in Ethiopia

The IDEAS project sought to improve the health and survival of mothers and babies through generating evidence to inform policy and practice. This collection contains quantitative data, collection tools and documentation associated with a social network analysis study of the professional advice networks used by health care workers in Ethiopia. The cross-sectional, mixed-methods observational network study compared professional advice networks of 160 healthcare workers in 8 primary health care units (PHCUs) across four regions of Ethiopia; Amhara, Oromia, SNNP and Tigray. PHCUs include a health centre and typically 5 satellite health posts. Data captured included health care worker advice seeking and giving for the provision of four areas along the continuum of maternal and newborn care: antenatal care, childbirth care, postnatal care and newborn care. Additional information captured regarded professional advice exchange beyond the roster of health care workers in the PHCU. Network metrics were qualitatively compared to continuum of care coverage data as a secondary analysis. Twenty semi-structured qualitative interviews of purposively selected subjects followed the collection of quantitative network data to interpret and explain network roles and patterns observed

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

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

The spectra of the laplacians of fractal graphs not satisfying spectral decimation

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (C-2)(3) to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory

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