## 2 A Note on Criteria for the Seismic Triggering of Landslides  
As is well known, to induce a landslide on a particular slope, the ground acceleration must exceed the critical acceleration (\(A_{c}\)) for a finite length of time [3]. Therefore, any study that aims to evaluate the triggering of landslide events under seismic conditions should first verify whether seismic activity is likely to produce acceleration levels above a certain critical value at the site under study. As stated previously, the quantification of \(A_{c}\) typically requires knowledge of site-specific geotechnical parameters. Over time, researchers have explored the relationship between earthquake-induced landslides and ground motion parameters using data from various historical earthquakes [1; 3; 4; 8; 18; 19; 20; 35; 36; 37]. There is a general consensus that landslides can be triggered by very low acceleration values. In particular, [PERSON] and [PERSON] [3] proposed the value \(A_{c}=0.05\) g as an approximate lower bound for the seismic stability of susceptible slopes, whatever the mechanism of failure. This critical acceleration value of 0.05 g is in agreement with observations from the Wenchuan earthquake, for which a PGA value of 0.07 g has been identified as the minimum threshold value associated with a significant increase in local landslide density, although landslides have also been observed for acceleration levels as low as about 0.03 g [20]. Moreover, it is consistent with the findings of [PERSON] [37], which show an empirical range of minimum PGA values needed to trigger small failures on very susceptible slopes between 0.02 g and 0.11 g.  
In the present study, for all nodes considered in the probabilistic seismic hazard assessment of Italy [26; 27], the surface peak ground acceleration (\(a_{max}\), defined as \(a_{g}\times S\), where \(a_{g}\) is the PGA on rock and \(S\) is a site term that accounts for stratigraphic and topographic amplification according to the Italian building code [29]) was compared to the critical acceleration threshold defined by [PERSON] and [PERSON] [3]. We recall that the values of \(a_{g}\) considered here refer to the larger horizontal component of motion (i.e., the larger peak acceleration of the two horizontal components of motion was considered) [38]. The condition \(a_{max}\geq 0.05\) g was checked using the same \(a_{max}\) database provided by [PERSON] et al. [39; 40] for the three return periods considered. The results, which are not reported here for the sake of brevity, indicate that applying a criterion based on critical acceleration does not provide informative results. This is because \(a_{max}\) values consistently exceed the critical acceleration threshold of 0.05 g. For a 475-year return period, interested readers may refer to the map in Figure 2 of [PERSON] et al. [39], from which it can be deduced that \(a_{max}\) exceeds 0.05 g almost everywhere. Therefore, while the condition on critical acceleration should be the primary criterion for assessing the triggering of earthquake-induced landslides, it is not useful to the scope of the present work (however, it may be useful when considering shorter return periods) and, consequently, it has been disregarded in favor of the criterion described below. A study of worldwide historical earthquakes revealed that the extent of earthquake-induced landslide activity depends on both earthquake magnitude and source-to-site distance [1; 8], so that below a certain minimum magnitude (4.0 for disrupted slides and falls, 4.5 for coherent slides, and 5.0 for lateral spreads and flows) and beyond a specific source-to-site distance, earthquake-induced landslides would rarely occur unless they are related to specific tectonic environments (e.g., volcanic areas) [41; 42; 43]. Specifically, [PERSON] [1] proposed different upper-bound curves for different types of earthquake-induced landslides (Figure 2 in [1]).