Large deviations of correlation functions in random magnets

We present a large deviations theory of the spin-spin correlation functions in the random field Ising model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: The phase transition is driven by a few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero-temperature critical point the number of spin pairs correlated over a distance is shown to be no longer exponential, but only linear in the spins separation.