  We consider bosonic atoms in an optical lattice at integer filling, tuned to
the superfluid-Mott insulator critical point, and coupled to a single, mobile
impurity atom of a different species. This setup is inspired by current
experiments with quantum gas microscopes, which enable tracking of the impurity
motion. We describe the evolution of the impurity motion from quantum wave
packet spread at short times, to Brownian diffusion at long times. This
dynamics is controlled by the interplay between dangerously irrelevant
perturbations at the strongly-interacting field theory describing the
superfluid-insulator transition in two spatial dimensions.
