Quantum walks, introduced by Richard Feynman, are archetypical for studying quantum algorithms. A one-dimensional quantum walk exhibits a phase transition: the wave function oscillates inside some region but exponentially decreases outside. The large-time asymptotic behavior at the transition point is described by the Airy function, as proved by Sunada-Tate in 2012. For the first time, we prove a uniform asymptotic formula in the whole domain, where the difference between the average velocity and the speed of light is bounded from zero. We also prove a general theorem producing uniform asymptotic formulae of this kind from conformal mappings.

This is joint work with M. Drmota, F. Kuyanov, and A. Ustinov