  We investigate modifications of the discrete-time lattice action, for a
quantum mechanical particle in one spatial dimension, that vanish in the
na\"ive continuum limit but which, nevertheless, induce non-trivial effects due
to quantum fluctuations. These effects are seen to modify the geometry of the
paths contributing to the path-integral describing the time evolution of the
particle, which we investigate through numerical simulations. In particular, we
demonstrate the existence of a modified lattice action resulting in paths with
any fractal dimension, d_f, between one and two. We argue that d_f=2 is a
critical value, and we exhibit a type of lattice modification where the
fluctuations in the position of the particle becomes independent of the time
step, in which case the paths are interpreted as superdiffusive L\'{e}vy
flights. We also consider the jaggedness of the paths, and show that this gives
an independent classification of lattice theories.
