Generalizations of Parisi's replica symmetry breaking and overlaps in random energy models

The random energy model (REM) is the simplest spin glass model which exhibits replica symmetry breaking. It is well known since the 80's that its overlaps are non-selfaveraging and that their statistics satisfy the predictions of the replica theory. All these statistical properties can be understood by considering that the low energy levels are the points generated by a Poisson process with an exponential density. Here we first show how, by replacing the exponential density by a sum of two exponentials, the overlaps statistics are modified. One way to reconcile these results with the replica theory is to allow the blocks in the Parisi matrix to fluctuate. Other examples where the sizes of these blocks should fluctuate include the finite size corrections of the REM, the case of discrete energies and the overlaps between two temperatures. In all these cases, the blocks sizes not only fluctuate but need to take complex values if one wishes to reproduce the results of our replica-free calculations.