  We show that in generic one-dimensional Hamiltonian lattices the diffusion
coefficient of the maximum Lyapunov exponent diverges in the thermodynamic
limit. We trace this back to the long-range correlations associated with the
evolution of the hydrodynamic modes. In the case of normal heat transport, the
divergence is even stronger, leading to the breakdown of the usual
single-function Family-Vicsek scaling ansatz. A similar scenario is expected to
arise in the evolution of rough interfaces in the presence of a suitably
correlated background noise.
