  We introduce the quantum Levy walk to study transport and decoherence in a
quantum random model. We have derived from second order perturbation theory the
quantum master equation for a \textit{Levy-like particle}that moves along a
lattice through hopping scale-free while interacting with a thermal bath of
oscillators. The general evolution of the quantum Levy particle has been solved
for different preparations of the system. We examine the evolution of the
quantum purity, the localized correlation, and the probability to be in a
lattice site, all them leading to important conclusions concerning quantum
irreversibility and decoherence features. We prove that the quantum thermal
mean-square displacement is finite under a constraint that is different when
compared to the classical Weierstrass random walk. We prove that when the
mean-square displacement is infinite the density of state has a complex
null-set inside the Brillouin zone. We show the existence of a critical
behavior in the continuous eigenenergy which is related to its
non-differentiability and self-affine characteristics. In general our approach
allows to study analytically quantum fluctuations and decoherence in a
long-range hopping model.
