  We consider the random deposition of objects of variable width and height
over a line. The successive additions of these structures create a random
interface. We focus on the regime of heavy tailed distributions of the
structure width. When the structure center is chosen at random, the problem is
exactly solvable and we prove that the interfaces generically tend towards
self-affine random curves. The asymptotic behavior reached after a large number
of iterations is universal in the sense that it depends on only three
parameters: the shape of the added structure at its maximum, the power-law
exponent of the width distribution and the exponent that relates height and
width. The parameter space displays several transitions that separate different
asymptotic behaviors. In particular for a set of parameters, the interface
tends towards a fractional Brownian motion. Our results reveal the existence of
a new class of random interfaces which properties appear to be robust. The
mechanism that generates correlations at large distance is identified and it
explains the appearance of such correlations in several situations of interest
such as the physics of earthquakes or the propagation of energy through a
diffusive medium.
