Bulk Phase Transition and Edge Behavior in Temporally Correlated Random Matrices

Abstract

We study long-range correlated Wigner-type matrices built from row-independent stationary Gaussian sequences. For exponentially decaying (AR(1)) correlations, the bulk spectral density deforms from the semicircle law via an explicit combinatorial "hub" mechanism, yet we verify the flatness and decay hypotheses of the matrix-Dyson-equation framework (MDE), with numerical evidence supporting Tracy-Widom edge universality for every fixed $ρ<1$ of the exponential decay correlations; the degenerate limit $ρ\to1^-$ reduces to a symmetrized Volterra operator, connecting to the singular-value cascade identified in a companion BBP analysis. For power-law correlations $dt\sim t^{-γ}$, we identify $γ_c=1/2$ as the critical point for divergence of the bulk fourth-moment, while $γ=1$ marks the breakdown of the flatness condition governing the MDE edge analysis. We prove the fourth-moment transition exactly and find numerically that the self-consistent edge varies smoothly across $γ=1$, with no evidence of a kink or discontinuity.