  We study the energy minimization for a particle in a quadratic well in
presence of short-ranged heavy-tailed disorder, as a toy model for an elastic
manifold. The discrete model is shown to be described in the scaling limit by a
continuum Poisson process model which captures the three universality classes.
This model is solved in general, and we give, in the present case (Frechet
class), detailed results for the distribution of the minimum energy and
position, and the distribution of the sizes of the shocks (i.e. switches in the
ground state) which arise as the position of the well is varied. All these
distributions are found to exhibit heavy tails with modified exponents. These
results lead to an "exotic regime" in Burgers turbulence decaying from a
heavy-tailed initial condition.
