Random interfaces generated by the addition of structures of variable size

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 the interface generically tends toward a self-affine random curve. 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 toward a fractional Brownian motion. Our results reveal the existence of a new class of random interfaces whose properties appear to be robust. The mechanism that generates correlations at large distances 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.