Feller property and infinitesimal generator of the exploration process

We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale

Orbital frustration at the origin of the magnetic behavior in LiNiO2

We report on the ESR, magnetization and magnetic susceptibility measurements performed over a large temperature range, from 1.5 to 750 K, on high-quality stoichiometric LiNiO2. We find that this compound displays two distinct temperature regions where its magnetic behavior is anomalous. With the help of a statistical model based on the Kugel'-Khomskii Hamiltonian, we show that below T_of ~ 400 K, an orbitally-frustrated state characteristic of the triangular lattice is established. This then gives a solution to the long-standing controversial problem of the magnetic behavior in LiNiO2.Comment: 5 pages, 5 figures, RevTex, accepted in PR

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

Salto de Truel

En este salto sobre el rio Tarn (Francia) se aprovechan las aguas del rio con un desnivel comprendido entre el del remanso del salto de La Jourdanie (situado aguas abajo) y el nivel de restitución de aguas del salto de Pinet (emplazado aguas arriba)

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small correction

Searching for Stable Na-ordered Phases in Single Crystal Samples of gamma-NaxCoO2

We report on the preparation and characterization of single crystal gamma phase NaxCoO2 with 0.25 < x < 0.84 using a non-aqueous electrochemical chronoamperemetry technique. By carefully mapping the overpotential versus x (for x < 0.84), we find six distinct stable phases with Na levels corresponding to x ~ 0.75, 0.71, 0.50, 0.43, 0.33 and 0.25. The composition with x ~0.55 appears to have a critical Na concentration which separates samples with different magnetic behavior as well as different Na ion diffusion mechanisms. Chemical analysis of an aged crystal reveals different Na ion diffusion mechanisms above and below x_c ~ 0.53, where the diffusion process above x_c has a diffusion coefficient about five times larger than that below x_c. The series of crystals were studied with X-ray diffraction, susceptibility, and transport measurements. The crystal with x = 0.5 shows a weak ferromagnetic transition below T=27 K in addition to the usual transitions at T = 51 K and 88 K. The resistivity of the Curie-Weiss metallic Na0.71CoO2 composition has a very low residual resistivity, which attests to the high homogeneity of the crystals prepared by this improved electrochemical method. Our results on the various stable crystal compositions point to the importance of Na ion ordering across the phase diagram.Comment: 9 pages, 9 figure