Phase Transition in the Random Energy Model

The Random Energy Model (REM) exhibits two distinct phases:

• High-Temperature Phase:

At high temperatures, the system is in a paramagnetic phase where the entropy is extensive, and the occupation probability of a configuration is approximately $\sim 1/M$.

• Low-Temperature Phase:

Below a critical freezing temperature $T_{f}$, the system transitions into a glassy phase. In this phase, the entropy becomes subextensive (i.e., the extensive contribution vanishes), and only a few configurations are visited with finite, $M$-independent probabilities.

Calculating the Freezing Temperature $T_{f}$

Thanks to the computation of ${\overline {n(x)}}$, we can identify the fingerprints of the glassy phase and calculate $T_{f}$. Let's compare the weight of the ground state against the weight of all other states:

${\frac {\sum _{\alpha }z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}{\frac {z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}e^{-\beta (E_{\alpha }-E_{\min })}\sim 1+\int _{0}^{\infty }dx\,{\frac {d{\overline {n(x)}}}{dx}}\,e^{-\beta x}=1+\int _{0}^{\infty }dx\,{\frac {e^{x/b_{M}}}{b_{M}}}\,e^{-\beta x}$