Hessian descent for spherical spin glasses with uniform log-Sobolev disorder

The present work concerns spherical spin glass models with disorder satisfying a uniform logarithmic Sobolev inequality. We show that the Hessian descent algorithm introduced by Subag can be extended to this setting, thanks to the abundance of small eigenvalues near the edge of the Hessian spectrum. Combined with the ground state universality recently proven by Sawhney and Sellke, this implies that when the model is in the full-RSB phase, the Hessian descent algorithm can find a near-minimum with high probability. Our proof consists of two main ingredients. First, we show that the empirical spectral distribution of the Hessian converges to a semicircular law via the moment method. Second, we use the logarithmic Sobolev inequality to establish concentration and obtain uniform control of the spectral edge.