Asymptotic shape for the chemical distance and first-passage percolation in random environment

The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on \Zd to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given.Comment: redaction du 10 avril 200

Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

Starting with a monodisperse configuration with $n$ size-1 particles, an additive Marcus-Lushnikov process evolves until it reaches its final state (a unique particle with mass $n$). At each of the $n-1$ steps of its evolution, a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.Comment: 28 pages, 1 figur

Continuous first-passage percolation and continuous greedy paths model: linear growth

We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.Comment: 13 pages, two appendice

Does Eulerian percolation on Z2Z^2 percolate ?

Eulerian percolation on Z 2 with parameter p is the classical Bernoulli bond percolation with parameter p conditioned on the fact that every site has an even degree. We first explain why Eulerian percolation with parameter p coincides with the contours of the Ising model for a well-chosen parameter $\beta$(p). Then we study the percolation properties of Eulerian percolation.Comment: This improves the previous version, only the status of one value for p is unknow

Limit law of the standard right factor of a random Lyndon word

Consider the set of finite words on a totally ordered alphabet with $q$ letters. We prove that the distribution of the length of the standard right factor of a random Lyndon word with length $n$, divided by $n$, converges to: $$\mu(dx)=\frac1q \delta_{1}(dx) + \frac{q-1}q \mathbf{1}_{[0,1)}(x)dx,$$ when $n$ goes to infinity. The convergence of all moments follows. This paper completes thus the results of~\cite{Bassino}, giving the asymptotics of the mean length of the standard right factor of a random Lyndon word with length $n$ in the case of a two letters alphabet

Continuum percolation in high dimensions

26 pages, 3 figuresConsider a Boolean model $\Sigma$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d.\ with common distribution $\nu$. The critical covered volume is the proportion of space covered by $\Sigma$ when the intensity $\lambda$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $\nu$ is a Dirac measure. In this paper, we prove that it is not the case at least in high dimension. To establish this result we study the asymptotic behaviour, as $d$ tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension

Progesterone receptors: a key for neuroprotection in experimental stroke

Progesterone receptors (PR) are expressed throughout the brain. However, their functional significance remains understudied. Here we report a novel role of PR as crucial mediators of neuroprotection using a model of transient middle cerebral artery occlusion and PR knockout mice. Six hours after ischemia, we observed a rapid increase in progesterone and 5-dihydroprogesterone, the endogenous PR ligands, a process that may be a part of the natural neuroprotective mechanisms. PR deficiency, and even haploinsufficiency, increases the susceptibility of the brain to stroke damage. Within a time window of 24 h, PR-dependent signaling of endogenous brain progesterone limits the extent of tissue damage and the impairment of motor functions. Longer-term improvement requires additional treatment with exogenous progesterone and is also PR dependent. The potent and selective PR agonist Nestorone is also effective. In contrast to progesterone, levels of the neurosteroid allopregnanolone, which modulates -aminobutyric acid type A receptors, did not increase after stroke, but its administration protected both wild-type and PR-deficient mice against ischemic damage. These results show that 1) PR are linked to signaling pathways that influence susceptibility to stroke, and 2) PR are direct key targets for both endogenous neuroprotection and for therapeutic strategies after stroke, and they suggest a novel indication for synthetic progestins already validated for contraception. Although allopregnanolone may not be an endogenous neuroprotective agent, its administration protects the brain against ischemicdamageby signaling mechanisms not involving PR. Collectively, our data clarify the relative roles of PR and allopregnanolone in neuroprotection after stroke.Fil: Liu, Ailing. Institut National de la Santé et de la Recherche Médicale; Francia. Université Paris Sud; FranciaFil: Margaill, Isabelle. Institut National de la Santé et de la Recherche Médicale; Francia. Universite de Paris V; Francia. University Paris Descartes; FranciaFil: Zhang, Shaodong. Institut National de la Santé et de la Recherche Médicale; Francia. Université Paris Sud; FranciaFil: Labombarda, Maria Florencia. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; Argentina. Universidad de Buenos Aires. Facultad de Medicina. Departamento de Bioquímica Humana; ArgentinaFil: Coqueran, Bérard. Institut National de la Santé et de la Recherche Médicale; Francia. Universite de Paris V; FranciaFil: Delespierre, Brigitte. Université Paris Sud; Francia. Institut National de la Santé et de la Recherche Médicale; FranciaFil: Liere, Philippe. Université Paris Sud; Francia. Institut National de la Santé et de la Recherche Médicale; FranciaFil: Marchand Leroux, Catherine. Institut National de la Santé et de la Recherche Médicale; Francia. Universite de Paris V; FranciaFil: O’Malley, Bert W.. Baylor College of Medicine;Fil: Lydon, John P.. Baylor College of Medicine;Fil: de Nicola, Alejandro Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; ArgentinaFil: Sitruk Ware, Regine. The Rockefeller University; Estados UnidosFil: Mattern, Claudia. MetP Pharma AG; SuizaFil: Plotkine, Michel. Universite de Paris V; FranciaFil: Schumacher, Michael. Institut National de la Santé et de la Recherche Médicale; Francia. Université Paris Sud; FranciaFil: Guennoun, Rachida. Université Paris Sud; Francia. Institut National de la Santé et de la Recherche Médicale; Franci