Parabolic Anderson model with a finite number of moving catalysts

We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t = \kappa\Delta u + \xi u$ with $u\colon\, \Z^d\times [0,\infty)\to \R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$. In the present paper we focus on the case where $\xi$ is a system of $n$ independent simple random walks each with step rate $2d\rho$ and starting from the origin. We study the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$ and show that these exponents, as a function of the diffusion constant $\kappa$ and the rate constant $\rho$, behave differently depending on the dimension $d$. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents, depicting how the total mass of $u$ concentrates as $t\to\infty$. Our results are both a generalization and an extension of the work of G\"artner and Heydenreich 2006, where only the case $n=1$ was investigated.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 25 pages. Updated version following the referee's comment

Intermittency in a catalytic random medium

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\xi u$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$ is a space-time random medium. We focus on the case where $\xi$ is $\gamma$ times the random medium that is obtained by running independent simple random walks with diffusion constant $\rho$ starting from a Poisson random field with intensity $\nu$. Throughout the paper, we assume that $\kappa,\gamma,\rho,\nu\in (0,\infty)$. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$, and show that they display an interesting dependence on the dimension $d$ and on the parameters $\kappa,\gamma,\rho,\nu$, with qualitatively different intermittency behavior in $d=1,2$, in $d=3$ and in $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $\kappa\downarrow0$ and $\kappa \to\infty$.Comment: Published at http://dx.doi.org/10.1214/009117906000000467 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Micropatterned Electrostatic Traps for Indirect Excitons in Coupled GaAs Quantum Wells

We demonstrate an electrostatic trap for indirect excitons in a field-effect structure based on coupled GaAs quantum wells. Within the plane of a double quantum well indirect excitons are trapped at the perimeter of a SiO2 area sandwiched between the surface of the GaAs heterostructure and a semitransparent metallic top gate. The trapping mechanism is well explained by a combination of the quantum confined Stark effect and local field enhancement. We find the one-dimensional trapping potentials in the quantum well plane to be nearly harmonic with high spring constants exceeding 10 keV/cm^2.Comment: 21 pages, 6 figures, submitted to Phys. Rev.

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

Violator Spaces: Structure and Algorithms

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006; author spelling fixe

Drift mobility of long-living excitons in coupled GaAs quantum wells

We observe high-mobility transport of indirect excitons in coupled GaAs quantum wells. A voltage-tunable in-plane potential gradient is defined for excitons by exploiting the quantum confined Stark effect in combination with a lithographically designed resistive top gate. Excitonic photoluminescence resolved in space, energy, and time provides insight into the in-plane drift dynamics. Across several hundreds of microns an excitonic mobility of >10^5 cm2/eVs is observed for temperatures below 10 K. With increasing temperature the excitonic mobility decreases due to exciton-phonon scattering.Comment: 3 pages, 3 figure

Intermittency in a catalytic random medium

Article / Letter to editorMathematisch Instituu

Regulation of cargo transfer between ESCRT-0 and ESCRT-I complexes by flotillin-1 during endosomal sorting of ubiquitinated cargo

Ubiquitin-dependent sorting of membrane proteins in endosomes directs them to lysosomal degradation. In the case of receptors such as the epidermal growth factor receptor (EGFR), lysosomal degradation is important for the regulation of downstream signalling. Ubiquitinated proteins are recognised in endosomes by the endosomal sorting complexes required for transport (ESCRT) complexes, which sequentially interact with the ubiquitinated cargo. Although the role of each ESCRT complex in sorting is well established, it is not clear how the cargo is passed on from one ESCRT to the next. We here show that flotillin-1 is required for EGFR degradation, and that it interacts with the subunits of ESCRT-0 and -I complexes (hepatocyte growth factor-regulated tyrosine kinase substrate (Hrs) and Tsg101). Flotillin-1 is required for cargo recognition and sorting by ESCRT-0/Hrs and for its interaction with Tsg101. In addition, flotillin-1 is also required for the sorting of human immunodeficiency virus 1 Gag polyprotein, which mimics ESCRT-0 complex during viral assembly. We propose that flotillin-1 functions in cargo transfer between ESCRT-0 and -I complexes