Hypergraph Counting and Mixed $p$-Spin Glass Models under Replica Symmetry

Understand

We study the fluctuation problems at high temperature in the general mixed $p$-spin glass models under the weak external field assumption: $h= \rho N^{-\alpha}, \rho>0, \alpha \in [1/4,\infty]$.

• By extending the cluster expansion approach to this generic setting, we convert the fluctuation problem as a hypergraph counting problem and thus obtain a new multiple-transition phenomenon.
• A by-product of our results is a new critical inverse temperature obtained from optimal second moment estimates.
• In particular, all our fluctuation results hold up to the threshold.