Further results on the free energy of the random field Ising chain in the case of centered disorder

We study the expansion of the limiting free energy density of the random field Ising chain with centered IID disorder and homogeneous coupling parameter, when the latter goes to infinity. We extend the first order result of [Collin, 2025] to the general case of finite second moment. Furthermore, we identify the value of the unknown constant appearing in the main theorem of [Collin, Giacomin, Greenblatt, Hu, 2025], which is the first non-universal constant in the expansion of the limiting free energy density.