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Fluctuations and the rapidity dependence of charged particles spectra in fixed centrality bins in p$A$ collisions

We argue that large fluctuations in the saturation momentum are necessary to explain the ATLAS and ALICE data on pA collisions measured at the LHC. Using a form for the distribution of fluctuations motivated by theoretical studies of the non-linear evolution equations for the Color Glass Condensate, we find a remarkably good agreement between theory and the measured distributions. If the saturation momentum fluctuates, we argue that the cross section for a proton probe should also fluctuate, consistent with previous observations. Finally we discuss these results and their possible consistency with what is known about multiplicity fluctuations in such collisions, and attempts to describe the ridge effect from properties of the initial state.