Diffractive processes at the LHC

We present a model of high energy soft $pp$ interactions that has multi s- and t-channel components, which has been tuned to describe all the available data. The t-channel components allow matching of the soft to the hard (QCD) Pomeron. Absorptive effects are found to be large, and, for example, suppress the prediction of the total $pp$ cross section to about 90 mb at the LHC. We use the model to calculate the survival probability, S^2, of the rapidity gaps in the exclusive process pp \to p+H+p, a process with great advantages for searching for the H \to \bb signal. We consider both eikonal and enhanced rescattering.