Asymptotic behavior of quantum walks on the line

TL;DR

This paper derives detailed asymptotic formulas for transition probabilities in discrete quantum walks on the line, revealing oscillations, large deviation behaviors, and boundary effects involving Airy functions.

Contribution

It provides new explicit asymptotic expressions for quantum walk transition probabilities depending on position, including boundary phenomena and large deviation estimates.

Findings

Oscillating behavior within the support of the limit distribution

Explicit large deviation rate functions outside the support

Airy function description near the boundary of the support

Abstract

This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the `normalized' position of the walk. When the position is in the support of the weak-limit distribution obtained by Konno, one observes, in addition to the limit distribution itself, an oscillating phenomenon in the leading term of the asymptotic formula. When the position lies outside of the support, one can establish an asymptotic formula of large deviation type. The rate function, which expresses the exponential decay rate, is explicitly given. Around the boundary of the support of the limit distribution (called the `wall'), the asymptotic formula is described in terms of the Airy function.