The free energy of an enriched continuous random energy model in the weak correlation regime

Recently, there has been an attempt to rewrite the free energy of spin glass models as a solution to a Hamilton-Jacobi equation. In the talk, I will overview the efforts and the results that several people have achieved. Then, I will talk about an on-going work with Alexander Alban and Justin Ko where we try to apply this framework to the branching random walk setting. In the weak correlation regime, where the CREM’s covariance function A is bounded above by the identity, we show that the free energy is given by the Hopf formula. In contrast, we provide an example outside of the weak regime, where A is a two-segment linear function exceeding the identity. Here, the Hamilton-Jacobi framework no longer applies, and a formal application of the associated variational principle fails to yield the correct free energy.