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End date: 2019 × Start date: 2010 × DDC: 510 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Lifshitz tails × Subject: large deviations ×

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    An effective medium approach to the asymptotics of the statistical moments of the parabolic Anderson model and Lifshitz tails : dedicated to Peter Stollmann on the occasion of his 50th birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Metzger, Bernd; Stollmann, Peter
    Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson model $H_omega= -Delta + V_omega $ on $l^2(bZ^d)$ has become in mathematical physics as well as in probability theory a paradigmatic example for the relevance of disorder effects. Here $Delta$ is the discrete Laplacian and $V_omega = V_omega(x): x in bZ^d$ is an i.i.d. random field taking values in $bR$. A popular model in probability theory is the parabolic Anderson model (PAM), i.e. the discrete diffusion equation $partial_t u(x,t) =-H_omega u(x,t)$ on $ bZ^d times bR_+$, $u(x,0)=1$, where random sources and sinks are modelled by the Anderson Hamiltonian. A characteristic property of the solutions of (PAM) is the occurrence of intermittency peaks in the large time limit. These intermittency peaks determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. The rigorous study of the relation between the probabilistic approach to the parabolic Anderson model and the spectral theory of Anderson localization is at least mathematically less developed. We see our publication as a step in this direction. In particular we will prove an unified approach to the transition of the statistical moments $langle u(0,t) rangle$ and the integrated density of states from classical to quantum regime using an effective medium approach. As a by-product we will obtain a logarithmic correction in the traditional Lifshitz tail setting when $V_omega$ satisfies a fat tail condition.
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    Large deviations for the local times of a random walk among random conductances in a growing box
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) König, Wolfgang; Wolff, Tilman
    We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Zd. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box. An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some p (2d d+2; 2). This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p = 2 to these values. As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.