Random real trees

We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galton-Watson trees, and the calculation of their fractal dimension. We then consider spatial trees, which combine the genealogical structure of a real tree with spatial displacements, and we explain their connections with superprocesses. In the last section, we deal with a particular conditioning problem for spatial trees, which is closely related to asymptotics for random planar quadrangulations.Comment: 25 page

Bessel processes, the Brownian snake and super-Brownian motion

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension -5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.Comment: Submitted to the special volume of S\'eminaire de Probabilit\'es in memory of Marc Yo

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, \emph{i.e.} conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation $h$ of angle $\rho$, the rigid rotation of angle $\rho$ is in the closure of the conjugacy class of $h$. We also prove that pseudo-rotations are not persistent in $C^r$ topology for any $r\geq 0$.Comment: 25 page

Onset of the nonlinear dielectric response of glasses in the two-level system model

We have calculated the real part $\chi'$ of the nonlinear dielectric susceptibility of amorphous insulators in the kHz range, by using the two-level system model and a nonperturbative numerical quantum approach. At low temperature $T$, it is first shown that the standard two-level model should lead to a \textit{decrease} of $\chi'$ when the measuring field $E$ is raised, since raising $E$ increases the population of the upper level and induces Rabi oscillations canceling the ones induced from the ground level. This predicted $E$-induced decrease of $\chi'$ is at \textit{odds} with experiments. However, a \textit{good agreement} with low-frequency experimental nonlinear data is achieved if, in our fully quantum simulations, interactions between defects are taken into account by a new relaxation rate whose efficiency increases as $\sqrt{E}$, as was proposed recently by Burin \textit{et al.} (Phys. Rev. Lett. {\bf 86}, 5616 (2001)). In this approach, the behavior of $\chi'$ at low $T$ is mainly explained by the efficiency of this new relaxation channel. This new relaxation rate could be further tested since it is shown that it should lead: \textit{i)} to a completely new nonlinear behavior for samples whose thickness is $\simeq 10$ nm; \textit{ii)} to a decrease of nonequilibrium effects when $E$ is increased.Comment: latex Sept02.tex, 5 files, 4 figures, 17 pages, submitted to Eur. Phys. J. B. Text change

Bacteriostatic conformal coating for electronic components

Coating for electronic components used in space applications has bacteriostatic qualities capable of hindering bacterial reproduction, both vegetative and sporulative viable microorganisms. It exhibits high electrical resistivity, a low outgassing rate, and is capable of restraining electronic components when subjected to mechanical vibrations

Bacteriostatic conformal coating and methods of application Patent

Development of bacteriostatic conformal coating and methods of applicatio