Fluctuations in the composite regime of a disordered growth model

We continue to study a model of disordered interface growth in two dimensions. The interface is given by a height function on the sites of the one--dimensional integer lattice and grows in discrete time: (1) the height above the site $x$ adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above $x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution $F$, and that the initial state is such that the origin is far above the other sites. Provided that the tails of the distribution $F$ at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of $p_x$. In the quenched case, the said fluctuations are asymptotically normal, while in the annealed case they satisfy the appropriate extremal limit law.