Ionization feedback should impact the probability distribution function (PDF) of the column density around the ionized gas. We aim to quantify this effect and discuss its potential link to the Core and Initial Mass Function (CMF/IMF). We used in a systematic way Herschel column density maps of several regions observed within the HOBYS key program: M16, the Rosette and Vela C molecular cloud, and the RCW 120 H ii region. We fitted the column density PDFs of all clouds with two lognormal distributions, since they present a double-peak or enlarged shape in the PDF. Our interpretation is that the lowest part of the column density distribution describes the turbulent molecular gas while the second peak corresponds to a compression zone induced by the expansion of the ionized gas into the turbulent molecular cloud. The condensations at the edge of the ionized gas have a steep compressed radial profile, sometimes recognizable in the flattening of the power-law tail. This could lead to an unambiguous criterion able to disentangle triggered from pre-existing star formation. In the context of the gravo-turbulent scenario for the origin of the CMF/IMF, the double peaked/enlarged shape of the PDF may impact the formation of objects at both the low-mass and the high-mass end of the CMF/IMF. In particular a broader PDF is required by the gravo-turbulent scenario to fit properly the IMF with a reasonable initial Mach number for the molecular cloud. Since other physical processes (e.g. the equation of state and the variations among the core properties) have already been suggested to broaden the PDF, the relative importance of the different effects remains an open question.
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