Universality of Spectral Distributions for Wigner Matrices with Dependent Entries in High-Dimensional Probability

This monograph establishes the universality of the limiting spectral distribution (LSD) for a broad class of Wigner matrices characterized by dependent entries, relaxing the classical independence hypothesis. We consider symmetric random matrices Xn = n-1/2[xij]i,j=1n where the upper triangular entries form a martingale difference sequence or satisfy a weak dependence condition governed by a mixing coefficient. Leveraging the Stieltjes transform method combined with Stein's method for exchangeable pairs and concentration inequalities in high-dimensional probability, we prove that the Empirical Spectral Distribution (ESD) converges weakly to the Wigner Semicircle Law almost surely, provided the entries possess finite moments of sufficiently high order and the dependence decays asymptotically. Furthermore, we explore the implications of local correlations on the edge universality and the delocalization of eigenvectors, demonstrating that the macroscopic spectral properties remain invariant under significant perturbations of the independence assumption.