Behavior in Different Phases

• High-Temperature Phase ($T>T_{f}=b_{M}=1/{\sqrt {2\log 2}}$):

In this regime, the weight of the excited states diverges. This indicates that the ground state is not deep enough to render the system glassy.

• Low-Temperature Phase ($T<T_{f}=b_{M}=1/{\sqrt {2\log 2}}$):

In this regime, the integral is finite:

$\int _{0}^{\infty }dx\,e^{(1/b_{M}-\beta )x}/b_{M}={\frac {1}{\beta b_{M}-1}}={\frac {T}{T_{f}-T}}$

This result implies that below the freezing temperature $T_{f}$, the weight of all excited states is of the same order as the weight of the ground state. Consequently, the ground state is occupied with a finite probability, reminiscent of Bose-Einstein condensation.