Random energy model

The solution of the Sherrington-Kirkpatrick (SK) model is challenging. To make progress, we first study the Random Energy Model (REM), introduced by B. Derrida. This model simplifies the problem by neglecting correlations between the $M=2^{N}$ configurations and assuming that the energies $E_{\alpha }$ are independent and identically distributed (i.i.d.) random variables. Here, "independent" means that the energy of one configuration does not influence the energy of another, e.g., a configuration identical to the previous one except for a spin flip. "Identically distributed" indicates that all configurations follow the same probability distribution.

Energy Distribution: Show that the energy distribution is given by:

$p(E_{\alpha })={\frac {1}{\sqrt {2\pi \sigma _{M}^{2}}}}\exp \left(-{\frac {E_{\alpha }^{2}}{2\sigma _{M}^{2}}}\right)$

and determine that:

$\sigma _{M}^{2}=N={\frac {\log M}{\log 2}}$

.

In the following, we present the original solution of the model. Here, we characterize the glassy phase by analyzing the statistical properties of the smallest energy values among the $M=2^{N}$ configurations. To address this, it is necessary to make a brief detour into the theory of extreme value statistics for i.i.d. random variables.