Unusual scaling in a discrete quantum walk with random long range steps

A discrete time quantum walker is considered in one dimension, where at each step, the translation can be more than one unit length chosen randomly. In the simplest case, the probability that the distance travelled is $\ell$ is taken as $P(\ell) = \alpha \delta(\ell-1) + (1-\alpha) \delta (\ell-2^n)$ with $n \geq 1$. Even the $n=1$ case shows a drastic change in the scaling behaviour for any $\alpha \neq 0,1$. Specifically, $\langle x^2\rangle \propto t^{3/2}$ for $0 < \alpha < 1$, implying the walk is slower compared to the usual quantum walk. This scaling behaviour, which is neither conventional quantum nor classical, can be justified using a simple form for the probability density. The decoherence effect is characterized by two parameters which vanish in a power law manner close to $\alpha =0$ and $1$ with an exponent $\approx 0.5$. It is also shown that randomness is the essential ingredient for the decoherence effect.