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Author: Metzger, Bernd × End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × 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 cluster size distributions in a continuous classical many-body system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jansen, Sabine; König, Wolfgang; Metzger, Bernd
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.