Intermittency on catalysts

The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a \emph{time-dependent random medium}, describing the evolution of a ``reactant'' in the presence of a ``catalyst''. Three examples of catalysts are considered: (1) independent simple random walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is on the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant. It turns out that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be published by Cambridge University Pres

Intermittency on catalysts: three-dimensional simple symmetric exclusion

We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice Laplacian on $\Z^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\Z^d$ in Bernoulli equilibrium. This model describes the evolution of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$. In G\"artner, den Hollander and Maillard (2007) we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to\infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $\kappa$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $\kappa\to\infty$ in the \emph{critical} dimension $d=3$, which was left open in G\"artner, den Hollander and Maillard (2007) and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a \emph{Green} term, as in $d\geq 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $\kappa$.Comment: 38 page

Intermittency on catalysts: Voter model

In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\gamma\xi u$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $\kappa\in[0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in(0,\infty)$ is the coupling constant, and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. We focus on the case where $\xi$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $\nu_{\rho}$ or the equilibrium measure $\mu_{\rho}$, where $\rho\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $\kappa$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $\xi$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.Comment: Published in at http://dx.doi.org/10.1214/10-AOP535 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representationComment: 22 pages 0 figure The paper has been reorganized, splitting the results into two sections : results pertaining to Physics and results pertaining to Mathematic

New Results on the Helium Stars in the Galactic Center Using BEAR Spectro-Imagery

Integral field spectroscopy of the central parsec of the Galactic Center was obtained at 2.06 microns using BEAR, an imaging Fourier Transform Spectrometer, at a spectral resolution of 74 km/s. Sixteen stars were confirmed as helium stars by detecting the He I 2.058 microns line in emission, providing a homogeneous set of fully resolved line profiles. These observations allow us to discard some of the earlier detections of such stars in the central cluster and to add three new stars. The sources detected in the BEAR data were compared with adaptive optics images in the K band to determine whether the emission was due to single stars. Two sub-classes of almost equal number are clearly identified from the width of their line profiles, and from the brightness of their continuum. Most of the emission lines show a P Cygni profile. From these results, we propose that the latter group is formed of stars in or near the LBV phase, and the other one of stars at the WR stage. The division into two groups is also shown by their spatial distribution, with the narrow-line stars in a compact central cluster (IRS 16) and the other group distributed at the periphery of the central cluster of hot stars. In the same data cube, streamers of interstellar helium gas are also detected. The helium emission traces the densest parts of the SgrA West Mini-Spiral. Several helium stars have a radial velocity comparable to the velocity of the interstellar gas in which they are embedded. In the final discussion, all these findings are examined to present a possible scenario for the formation of very massive stars in the exceptional conditions of the vicinity of the central Black Hole

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices

A conjecture is given for the exact location of the multicritical point in the phase diagram of the +/- J Ising model on the triangular lattice. The result p_c=0.8358058 agrees well with a recent numerical estimate. From this value, it is possible to derive a comparable conjecture for the exact location of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in excellent agreement with a numerical study. The method is a variant of duality transformation to relate the triangular lattice directly with its dual triangular lattice without recourse to the hexagonal lattice, in conjunction with the replica method.Comment: 9 pages, 1 figure; Minor corrections in notatio

Feeling unsure: a lived experience of humanbecoming

The aim of this study was to explore the phenomenon of feeling unsure as viewed from the humanbecoming school of thought. From the humanbecoming perspective feeling unsure is a universal lived experience of health and quality of life. The purposes of this study were to understand the lived experience of feeling unsure from the humanbecoming perspective, to enhance understanding of the lived experience of feeling unsure as an essence of health and quality of life, to discover the structure of the lived experience of feeling unsure, to add to the body of knowledge on the phenomenon of feeling unsure, and to contribute to expand the theory of humanbecoming. The Parse research methodology was used to guide this study and answer the question: Whqt is the structure of the lived experience of feeling unsure? Ten persons living in community accepted to participate in this study. The processes of dialogical engagement, extraction-synthesis, and heuristic interpretation were used for data gathering and analysis. The central finding of this study is Feeling unsure is wavering irresolutely with discerning ponderings arising in venturing with trepidations, while revering alliances. The findings of this study emerged as new knowledge that extend the theory of humanbecoming and enhance the understanding of the lived experience of feeling unsure

Enseigner le français en contexte universitaire à des étudiants non francophones en mobilité : quelle place pour les littératures francophones ?

The role of literature in language teaching has been re-assessed ; it has been documented in numerous work, many of them paying a special attention to francophone literature, which seems to be able to function as a bridge between languages and cultures. This paper will analyze individual interviews conducted with teachers  working in French-language university centers. How do they define this corpus (including, or not, french «hexagonal» literature ?)  Which place and functions do they assign to theses litteratures in their teaching practices