An effective medium approach to the asymptotics of the statistical moments of the parabolic Anderson model and Lifshitz tails

TL;DR

This paper develops an effective medium approach to analyze the asymptotic behavior of moments in the parabolic Anderson model, connecting probabilistic and spectral theories, and introduces a correction to Lifshitz tails under certain conditions.

Contribution

It provides a unified method to study the transition of moments and density of states from classical to quantum regimes using an effective medium approach.

Findings

Established a connection between probabilistic and spectral analyses of PAM.

Derived a logarithmic correction to Lifshitz tails for fat-tailed potentials.

Proved asymptotic behavior of moments and density of states in different regimes.

Abstract

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 $< u(0,t) >$ 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.