  In comparison with Derrida's REM, we investigate the influence of the
so-called decoration processes arising in the limiting extremal processes of
numerous log-correlated Gaussian fields. In particular, we focus on the
branching Brownian motion and two specific quantities from statistical physics
in the vicinity of the critical temperature. The first one is the
two-temperature overlap, whose behavior at criticality is smoothened by the
decoration process - unlike the one-temperature overlap which is identical -
and the second one is the temperature susceptibility, as introduced by Sales
and Bouchaud, which is strictly larger in the presence of decorations and
diverges, close to the critical temperature, at the same speed as for the REM
but with a different multiplicative constant. We also study some general
decorated cases in order to highlight the fact that the BBM has a critical
behavior in some sense to be made precise.
