In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II. We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$. We denote $\widehat{G}_h : x \longmapsto G_h(x + u_h) - \min G_h$. Show that $$\psi_n(h) = -G_h(u_h) - \frac{1}{2n}\ln\left(2\mathrm{e}^{\beta}\pi\beta\right) + \frac{1}{n}\ln\left(\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t\right)$$
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We denote $\widehat{G}_h : x \longmapsto G_h(x + u_h) - \min G_h$. Show that
$$\psi_n(h) = -G_h(u_h) - \frac{1}{2n}\ln\left(2\mathrm{e}^{\beta}\pi\beta\right) + \frac{1}{n}\ln\left(\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t\right)$$