We study the fluctuations of the two-points correlation function in one-dimensional disordered spin models. Unlike the fluctuations of the free energy, these survive in the thermodynamic limit. The correlation functions fluctuate wildly among different samples, or for a given sample among different sites. We show how these fluctuations can be characterised starting from the moments. This leds us to study a set of generalised correlations lengths #xi#_q. The moments may also be calculated within the transfer-matrix formalism and provide insight on disorder-induced fluctuations. We show that the #xi#_q's can be computed in Monte Carlo simulations or real experiments. In general they do not exhibit a trivial dependence on q. We discuss the crossover of the correlation decay-rate at large distances to dominance by the most probable value given by #xi#_0, and the relation with the finite volume fluctuations of the free energy. Finally we sketch how to extend our arguments to dimensions two and three. This work was done in collaboration with G. Paladin, S. Nicolis and A. Vulpiani.
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