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We show that the onset and rise of QCD mini-jets provide the dynamicalmechanism behind the appearance of a soft edge in pp collisions around ISRenergies and thus such a soft edge is built in our mini-jet model with softgluon resummation. Here the model is optimized for LHC at $\sqrt{s} = 7, 8 TeV$and predictions made for higher LHC and cosmic ray energies. Further, weprovide a phenomenological picture to discuss the breakup of the totalcross-section into its elastic, uncorrelated and correlated inelastic pieces inthe framework of a one-channel eikonal function.

Soft edge of hadron scattering and mini-jet models for the total and inelastic pp cross-sections at LHC and beyond