The Free Energy of an Enriched Continuous Random Energy Model in the Weak Correlation Regime

We revisit the proof of the limiting free energy of the continuous random energy model (CREM) using the Hamilton--Jacobi approach for mean-field disordered systems. To achieve this, we introduce an enriched model that interpolates between the CREM and the Ruelle probability cascade. We focus on the weak correlation regime, where the CREM's covariance function $A$ is bounded above by the identity function. In the weak correlation regime, we show that the free energy is given by the Hopf formula. The resulting expression is independent of $A$, confirming that in this regime the free energy does not depend on the precise form of the covariance function. Outside of the weak correlation regime, the Hamilton--Jacobi framework no longer applies. Moreover, we provide an example where a formal application of the associated variational principle fails to yield the correct free energy.