Study of longitudinal fluctuations of the Sherrington-Kirkpatrick model

We study finite-size corrections to the free energy of the Sherrington-Kirkpatrick spin glass in the low temperature phase. We investigate the role of longitudinal fluctuations in these corrections, neglecting the transverse contribution. In particular, we are interested in the exponent $\alpha$ defined by the relation $f-f_{\mathrm{\infty }}\sim N^{-\alpha }$. We perform both an analytical and numerical estimate of the analytical result for $\alpha$. From both the approaches we get the result: $\alpha =0.8$.