Concentration and fluctuation phenomena in the localized phase of the pinning model

We revisit and expand the analysis in [15] of the localized phase of disordered pinning models. The arguments are developed for i.i.d. site disorder on which we assume only that the moment generating function is bounded in a neighborhood of the origin. Quantitative $C^\infty$ estimates on the free energy density are established, showing in particular that the regularity class is at least Gevrey-3. After explaining how a quenched concentration bound and the quenched Central Limit Theorem (CLT) on the number of the pinned sites, i.e., the contact number, can be extracted from the regularity estimates on the free energy, we establish a quenched Local CLT for the same quantity. The centering in these CLTs is random in the sense that it is disorder dependent and a concentration bound and the CLT are established also for the centering sequence, as well as a Hardy-Littlewood random walk type-estimate for its fluctuations.