Spin glasses in a field show a phase transition varying the distance among real replicas (and how to exploit it to find the critical line in a field). We discuss a phase transition in spin glass models which have been rarely considered in the past, namely the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e. at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the SherringtonKirkpatrick model in a field close to its critical point, we show that even in a paramagnetic phase the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de AlmeidaThouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finitedimensional spin glasses. This is a notoriously difficult computational problem, because of huge finitesize corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, hu003e0 and T_c(h)u003cTu003cT_c(h0), of the d1 spin glass model with longrange interactions decaying fast enough to be outside the regime of validity of the meanfield theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.

Numerical Assessment of the Percolation Threshold Using Complement Networks. Models of percolation processes on networks currently assume locally treelike structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold (p_c) and its modelpredicted value (pi _c). Here we show the existence of an empirical linear relation between (p_c) and (pi _c) across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of (pi _c). We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, (p_c), and that of its complement, (barp_c).