9 - Propagators, mostly replicon

Summary

In the previous chapter we have started to look at the difficult task of inverting the Gaussian fluctuation matrix around a Parisi-like RSB equilibrium Q_ab. As we have seen, the core of the difficulty is manifest in the unitarity equations (8.27). In Section 8.4 we have shown that in the near infrared regime these equations become linearly coupled equations that can easily be solved to get the propagators. In general, however, the unitarity equations contain nontrivial integrals over replica overlaps (see Eq. (8.37)). To overcome this problem and solve the equations in full generality one resorts once again to the technique of the Replica Fourier Transform. This is what we shall do in this chapter. Even if the complete resolution of the unitarity equations is beyond the scope of this book, we wish at least to compute explicitly the propagators in the most dangerous replicon sector, giving a few hints on what can be done for the other sectors. With this aim in mind, we proceed in the following way. In Section 9.1 we return to the RFT technique which was used in Chapter 6 to diagonalize the Hessian (in the absence of RSB) and we generalize it for R steps of RSB. We exhibit how it works on a simple toy model and in Section 9.2 we use it to obtain the propagators in the replicon sector.