  We investigate, both analytically and with numerical simulations, a Monte
Carlo dynamics at zero temperature, where a random walker evolving in
continuous space and discrete time seeks to minimize its potential energy, by
decreasing this quantity at each jump. The resulting dynamics is universal in
the sense that it does not depend on the underlying potential energy landscape,
as long as it admits a unique minimum; furthermore, the long time regime does
not depend on the details of the jump distribution, but only on its behaviour
for small jumps. We work out the scaling properties of this dynamics, as
embodied by the walker probability density. Our analytical predictions are in
excellent agreement with direct Monte Carlo simulations.
