Arakelov Self-intersection numbers of minimal regular models of modular curves $X_0(p^2)$

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^2)$ over $\mathbb{Q}$. The computation of the self-intersection numbers are used to prove effective Bogolomov conjecture for the semi-stable models of modular curves $X_0(p^2)$ and obtain a bound on the stable Faltings height for those curves in a companion article arXiv:1802.06968.