About the AT line in Replica Symmetry Breaking assumption for spin glasses

Replica Symmetry Breaking is a fascinating phenomenon of spin glasses model which could have consequences also in other field of studies. Although there are several studies regarding the stability between the Replica Symmetric and first step of Replica Symmetry Breaking approximations, we have very few results for the following steps (apart from that one by Gardner for P-spin glasses in 1985 and Chen in 2017 and 2021). This is link to the fact that the classic method, based from the work by De Almeida and Thoules (from which the critical stability line takes its name), is cumbersome to generalise for the next assumptions. In this paper we devise a new straightforward method inspired to the work by Toninelli in 2002 to recover the critical line in order to inspect the stability first between the second and the first steps of Replica Symmetry Breaking and then, we generalise to Kth step, with K finite.