Exponential tail estimates for quantum lattice dynamics

TL;DR

This paper establishes exponential tail estimates for the probability distribution of a quantum particle's scaled position on a lattice over large times, showing rapid decay outside the velocity support and providing uniform rate estimates.

Contribution

It introduces a simple method to estimate exponential decay rates of tail probabilities in quantum lattice dynamics, applicable to both discrete and continuous time.

Findings

Probability outside velocity support decays exponentially with time

Rate function vanishes near boundary as power 3/2 of the distance

Method is demonstrated through multiple examples

Abstract

We consider the quantum dynamics of a particle on a lattice for large times. Assuming translation invariance, and either discrete or continuous time parameter, the distribution of the ballistically scaled position $Q(t)/t$ converges weakly to a distribution that is compactly supported in velocity space, essentially the distribution of group velocity in the initial state. We show that the total probability of velocities strictly outside the support of the asymptotic measure goes to zero exponentially with $t$, and we provide a simple method to estimate the exponential rate uniformly in the initial state. Near the boundary of the allowed region the rate function goes to zero like the power 3/2 of the distance to the boundary. The method is illustrated in several examples.