tokens: , d v1 ) and ( V v2 , d v2 ) (not shown here). Figure 8. Left-hand panel: 2D distribution ( V v3 , d v3 ). There is a linear trend that is correctly described by the linear relation given by equation ( 16 , dashed line). Right-hand panel: the correlation disappears after correcting the LOS displacements with this equation. This is a statistical demonstration of the v-RSD off-centring effect. From this analysis, we arrive at the third important conclusion of this work: void centres shift preferentially along the LOS when they are mapped from r - into z -space, and this displacement can be statistically quantified by means of equation ( 16 ). These results give support to the v-RSD off-centring effect postulated in Section 4.5 . 5.3 Cross-correlations Since the ratio (or equivalently δ R v ) characterizes the change of volume in voids, the statistical analysis of Section 5.1 gave support to the t-RSD expansion effect postulated in Section 4.2 . On the other hand, as the displacement and velocity characterize the movement of voids, the statistical analysis of Section 5.2 gave support to the v-RSD off-centring effect postulated in Section 4.5 . It only remains to test if both effects are statistically independent by looking for cross-correlations between these quantities. Fig. 9 shows the 2D LOS distributions (left-hand panel) and (right-hand panel). The horizontal lines are the theoretical predictions (dashed) and (solid). No correlations can be seen, giving support to the postulated independence. It is worth mentioning that the analogue POS distributions (not shown here): , , , and show a similar behaviour. These results allow us to interpret the large-scale dynamics of the whole region containing the void (v-RSD) as decoupled from the dynamics of the galaxies at scales of the void radius (t-RSD). This also suggests that potential distortion patterns in observations due to these two effects can be treated separately. This is the fourth important conclusion of this work. Figure 9. 2D distributions (left-hand panel) and (right-hand panel). The horizontal lines are the theoretical predictions (equation 7 , dashed) and (equation 10 , solid). No correlations can be seen in both panels, suggesting that the t-RSD expansion effect and the v-RSD off-centring effect are statistically independent. 6 IMPACT ON THE VOID SIZE FUNCTION This section has a double intention. On the one hand, we will finish the analysis of the last section incorporating the additional AP change of volume not treated yet. On the other hand, we will study the impact of all the z -space effects on the void size function. For this reason, we turn now to the FC void catalogues, fully affected by z -space systematics (see Table 1 ). 6.1 Generalities of the <xref ref-type="bibr" rid="bib69">VSF</xref> modelling We start with some generalities concerning the VSF modelling. The real-space VSF can be modelled using the excursion set formalism combined with the spherical expansion of matter
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