Annealed Lyapunov Exponents Research Articles

The relation between quenched and annealed Lyapunov exponents in random potential on trees

Our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which are defined by a sequence of independent and identically distributed non-negative random variables. To determine the risk of taking a walk in these potentials we consider the decay of the Green function. There are two possible tools to describe this decay: The quenched Lyapunov exponent and the annealed Lyapunov exponent. It turns out that on the integers and on regular trees we can state a precise relation between these two.

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

AbstractWe are interested in the analysis of Gibbs measures defined on two independent Brownian paths in ℝd interacting through a mutual self‐attraction. This is expressed by the Hamiltonian with two probability measures μ and ν representing the occupation measures of two independent Brownian motions. We will be interested in a class of potentials V that are singular, e.g., Dirac‐ or Coulomb‐type interactions in ℝ3, or the correlation function of the parabolic Anderson problem with white noise potential.The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self‐interacting case introduced in [27], owing to the lack of shift‐invariant structure in the mutual interaction. We prove a strong large‐deviation principle for the product measures of two Brownian occupation measures in such a compactification and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents, and a strict ordering of them implies the intermittency effect present in the smoothened model. © 2017 Wiley Periodicals, Inc.

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete Laplacian, and \xi is a dynamic random environment that drives the equation. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\kappa, split into two at rate \xi \vee 0, and die at rate (-\xi) \vee 0. Our focus is on the Lyapunov exponents \lambda_p(\kappa) = \lim_{t\to\infty} \frac{1}{t} \log \E([u(0,t)]^p)^{1/p}, p \in \N, and \lambda_0(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t). We investigate what happens when \kappa\Delta is replaced by \Delta^\cK, where \cK = \{\mathcal{K}(x,y)\colon\,x,y\in\Z^d,\,x \sim y\} is a collection of random conductances between neighbouring sites replacing the constant conductances \kappa in the homogeneous model. We show that the associated annealed Lyapunov exponents are given by the formula \lambda_p(\cK) = \sup\{\lambda_p(\kappa) \colon\,\kappa\in\Supp(\cK)\}, where \Supp(\cK) is the set of values taken by the \cK-field. We also show that for the associated quenched Lyapunov exponent this formula only provides a lower bound. Our proof is valid for three classes of reversible \xi, and for all \cK satisfying a certain clustering property, namely, there are arbitrarily large balls where \cK is almost constant and close to any value in \Supp(\cK). What our result says is that the Lyapunov exponents are controlled by those pockets of \cK where the conductances are close to the value that maximises the growth in the homogeneous setting.

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

Dans cet article on étudie l’équation parabolique d’Anderson $\partial u(x,t)/\partial t=\kappa\varDelta u(x,t)+\xi(x,t)u(x,t)$, $x\in\mathbb{Z}^{d}$, $t\geq0$, où les champs $u$ et $\xi$ sont à valeurs dans $\mathbb{R}$, $\kappa\in[0,\infty)$ est la constante de diffusion, et $\varDelta $ est le laplacien discret. Le champ $\xi$ joue le rôle d’environnement aléatoire dynamique et dirige l’équation. La condition initiale $u(x,0)=u_{0}(x)$, $x\in\mathbb{Z}^{d}$, est choisie positive et bornée. La solution de l’équation parabolique d’Anderson décrit l’évolution d’un champ de particules effectuant des marches aléatoires simples avec un branchement binaire : les particules sautent au taux $2d\kappa$, se divisent en deux au taux $\xi\vee0$, et meurent au taux $(-\xi)\vee0$. Notre but est de prouver un certain nombre de propriétés basiques de la solution $u$ sous des conditions sur $\xi$ qui sont aussi faibles que possible. Ces propriétés vont servir d’impulsion pour de futur améliorations. Tout au long de cet article nous supposons que $\xi$ est stationnaire et ergodique sous les translations en espace et en temps, n’est pas constant et satisfait $\mathbb{E}(|\xi(0,0)|)<\infty$, où $\mathbb{E}$ représente l’espérance par rapport à $\xi$. Sous une hypothèse très faible sur les queues de la distribution de $\xi$, nous montrons que la solution de l’équation parabolique d’Anderson existe et est unique pour tout $\kappa\in[0,\infty)$. Notre principal objet d’intérêt est l’exposant de Lyapunov quenched $\lambda_{0}(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t)$. Il a été prouvé dans Gärtner, den Hollander et Maillard (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) que cet exposant existe et est constant $\xi$-a.s., satisfait $\lambda_{0}(0)=\mathbb{E}(\xi(0,0))$ et $\lambda_{0}(\kappa)>\mathbb{E}(\xi(0,0))$ pour $\kappa\in(0,\infty)$, et est tel que $\kappa\mapsto\lambda_{0}(\kappa)$ est globalement lipschitzienne sur $(0,\infty)$ à l’extérieur de n’importe quel voisinage de $0$ où il est fini. Sous certaines conditions faibles de mélange en espace-temps sur $\xi$, nous montrons les propriétés suivantes : (1) $\lambda_{0}(\kappa)$ ne dépend pas de la condition initiale $u_{0}$; (2) $\lambda_{0}(\kappa)<\infty$ pour tout $\kappa\in[0,\infty)$; (3) $\kappa\mapsto\lambda_{0}(\kappa)$ est continue sur $[0,\infty)$ mais pas lipschitzienne en $0$. Nous conjecturons en outre : (4) $\lim_{\kappa\to\infty}[\lambda_{p}(\kappa)-\lambda_{0}(\kappa)]=0$ pour tout $p\in\mathbb{N}$, où $\lambda_{p}(\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\mathbb{E}([u(0,t)]^{p})$ est le $p$-ième exposant de Lyapunov annealed. (Dans (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) les propriétés (1), (2) et (4) n’ont pas été abordées, tandis que la propriété (3) a été prouvée sous des hypothèses beaucoup plus restrictives sur $\xi$.) Finalement, nous prouvons que nos conditions faibles de mélange en espace-temps sur $\xi$ sont satisfaites par plusieurs systèmes de particules en interaction.

Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form \beta V, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), and \beta\ > 0. When the potential is integrable, the asymptotic behaviours as \beta\ tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small \beta. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -\Delta\ + \beta V.

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.

Lyapunov exponents of Green’s functions for random potentials tending to zero

We consider quenched and annealed Lyapunov exponents for the Green's function of $-\Delta+\gamma V$, where the potentials $V(x), x\in\Z^d$, are i.i.d. nonnegative random variables and $\gamma>0$ is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like $c\sqrt{\gamma}$ as $\gamma$ tends to 0. Here the constant $c$ is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wei-Min Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.

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 $\mathbb{Z}^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\mathbb{Z}^d$ in Bernoulli equilibrium. This model describes the evolution of a reactant $u$ under the influence of a catalyst $\xi$. <br /><br /> In Gärtner, den Hollander and Maillard [3] 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 critical dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in $d\geq 4$, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for $\kappa$.

Coincidence of Lyapunov exponents for random walks in weak random potentials

We investigate the free energy of nearest-neighbor random walks on ℤd, endowed with a drift along the first axis and evolving in a nonnegative random potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensions d≥4, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.

Lyapunov exponents for the one-dimensional parabolic Anderson model with drift

We consider the solution to the one-dimensional parabolic Anderson model with homogeneous initial condition, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for all positive real $p$. These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p$ tends to 0. Furthermore, we show that the solution is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of the solution under the corresponding Gibbs measure. In our context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears as the drift parameter or diffusion constant increase, respectively.

Intermittency on catalysts: symmetric exclusion

We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst". <br> In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument. <br> In G&#228;rtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.

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 $\kappa \to\infty$.

On the Intermittency of the Light Propagation in Disordered Optical Materials

We consider propagation of light through an ensemble of N 1 statistically independent optical fibers of length L whose refraction coefficient is a random function of length. We introduce the generalized transmission coefficient |t(k, L)|p for energy k and study its quenched and annealed Lyapunov exponents. For small disorder we calculate the Lyapunov exponents in asymptotic form.

Brownian Directed Polymers in Random Environment

We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms of the Brownian motion. We mainly discuss: (i) The normalized partition function, its positivity in the limit which characterizes the phase diagram of the model. (ii) The existence of quenched Lyapunov exponent, its positivity, and its agreement with the annealed Lyapunov exponent; (iii) The longitudinal fluctuation of the free energy, some of its relations with the overlap between replicas and with the transversal fluctuation of the path. The model considered here, enables us to use stochastic calculus, with respect to both Brownian motion and Poisson process, leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram. We also relate our model to some formulation of the Kardar-Parisi-Zhang equation, more precisely, the stochastic heat equation. Our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super-diffusivity in dimension one. We also obtain an almost sure large deviation principle for the polymer measure.