Spectral characterization of aging: the rem-like trap model

We review the aging phenomenon in the context of the simplest trap model, Bouchaud's REM-like trap model, from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalized exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a ``grand canonical'' representation of the model in terms of the Markov process on a Poisson point process. In this context we analyze the dynamics on various time scales.Comment: Published at http://dx.doi.org/10.1214/105051605000000359 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Isoperimetric inequalities and mixing time for a random walk on a random point process

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for $\alpha\in(0,d)$, that the random walk confined to a cubic box of side $L$ has a.s. Cheeger constant of order at least $L^{-1}$ and mixing time of order $L^2$. For the Poisson point process, we prove that at $\alpha=d$, there is a transition from diffusive to subdiffusive behavior of the mixing time.Comment: Published in at http://dx.doi.org/10.1214/07-AAP442 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fluctuation theorems for discrete kinetic models of molecular motors

Motivated by discrete kinetic models for non-cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi 1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti-Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in \cite{LLM1}. We also derive fluctuation theorems for the time-integrated cycle currents and discuss how the matrix approach of \cite{LLM1} can be extended to derive the above Gallavotti-Cohen symmetry for any Markov random walk on $\mathbb{Z}$ with periodic jump rates. Finally, we review in the present context some large deviation results of \cite{FS1} and give some specific examples with explicit computations.Comment: Modified Appendix B, added figure 13, minor modifications. 27 pages, 16 figure

Discrete kinetic models for molecular motors: asymptotic velocity and gaussian fluctuations

We consider random walks on quasi one dimensional lattices, as introduced in \cite{FS}. This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.Comment: 32 pages, 5 figures. modified comments concerning ref. [12] after private communications of the author (A.B. Kolomeisky

Connection probabilities in Poisson random graphs with uniformly bounded edges

We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with lower-bounded conductances. The latter is relevant for the analysis of conductivity by Mott variable range hopping in strongly disordered systems. By using the method of randomized algorithms developed by Duminil-Copin et al. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to $(\lambda-\lambda_c)$, $\lambda$ being the density of the Poisson point process and $\lambda_c$ being the critical density.Comment: 25 pages. corrected version. the proof of Lemma 3.4 is much shorter. the concept of good function is less restrictiv

Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the homogenized matrix $D$ of the single random walk, which can be degenerate. The above exclusion process is non-gradient but we avoid the standard non-gradient machinery. Our derivation is based on the duality between the simple exclusion process and the random walk and on our recent homogenization results for random walks in random environment presented in \cite{Fhom3}. Our hydrodynamic limit covers a large family of models, including e.g. simple exclusion processes built from random conductance models on $\mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.Comment: 29 pages, 1 figur

Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

We consider a lattice gas on the discrete d-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field E/N. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., E constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field E is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on E.Comment: Published in at http://dx.doi.org/10.1214/11-AAP805 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time scale separation in the low temperature East model: Rigorous results

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. Specifically we analyse time scale separation and dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scale associated to two length scales $d/q^\gamma$ and $d'/q^\gamma$ are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $d'/d$ is large enough independently of $q$. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. Finally we show that no form of time scale separation can occur for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was previously assumed in the physical literature based on numerical simulations.Comment: 6 pages, 0 figures; clarified q dependence of bounds, results unchange