Computational hardness of sampling beyond the shattering phase transition

Demonstrate the computational intractability of sampling from the Gibbs measure of mixed p-spin spherical spin glasses beyond the shattering (dynamical) phase transition, i.e., prove that sampling is fundamentally hard in the regime where the dynamical phase transition predicts slow dynamics.

Background

Beyond predictions for mixing behavior of Langevin dynamics, a stronger conjecture posits an inherent computational barrier to sampling from the Gibbs measure in the low-temperature (shattered) regime. Prior works have provided evidence of failure for certain classes of algorithms but a complete hardness result remains conjectural.

Establishing such hardness would align algorithmic limitations with the geometric complexity (shattering) of the Gibbs measure and solidify the link between thermodynamic transitions and computational barriers.