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 &#34;hub&#34; 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.