Spectral Signatures of Replica Symmetry Breaking in Optimization-Induced Random Matrices

Abstract

We study optimization-induced matrix ensembles generated by Gibbs measures. The same quenched disorder that weights configurations also supplies the matrix entries observed on them. For glassy Gibbs measures this raises a natural question: does the induced spectrum inherit the underlying glassy Gibbs geometry? In a dense tensor optimization model we find a selective answer. A single induced matrix has a universal leading bulk that washes out the glassy organization. The difference of two matrices built from independent thermal samples in the same disorder does not: its spectrum gives an explicit image of the glassy Gibbs geometry, encoded by the distribution of mutual overlaps between samples. Parisi theory and Monte Carlo confirm this mechanism across simple and glassy phases.