Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

Abstract

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.

From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

“…To use Theorem 2, we have to verify Assumptions 2 and 3. The former is rather easy and can be found in Lemma 11 in [8]. The latter is verified as follows: we first fix r 0 > 0 sufficiently large so that supp W ⊂ B(0, r 0 /4).…”

Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential

“…For the lower estimate, we take R, R 1 and β as in the proof of Proposition 2.2 in [4] and restrict the integral as…”

“…This paper is a continuation of [4]. We consider the initial value problem of the heat equation with a random potential…”

“…[1][2][3][4]9]. Also, the initial value problem (1.2) itself is called the "parabolic Anderson problem" in literature (see e.g.…”

The Parabolic Anderson Model

“…The PAM with this potential is analysed in [Fuk09a,FukUek10,FukUek11]. OE0; 1/ is a single-site potential, and .Á z / z2Z d is a sequence of centred R d -valued random variables.…”