We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of [3] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. ContentsRANDOM MATRICES WITH SLOW CORRELATION DECAY 2 of the individual eigenvalues follows a universal distribution, independent of the specifics of the random matrix itself. The former is commonly called a local law, whereas the latter is known as the Wigner-Dyson-Mehta (WDM) universality conjecture, first envisioned by Wigner in the 1950's and formalized later by Dyson and Mehta in the 1960's [36]. In fact, the conjecture extends beyond the customary random matrix ensembles in probability theory and is believed to hold for any random operator in the delocalization regime of the Anderson metal-insulator phase transition. Given this profound universality conjecture for general disordered quantum systems, the ultimate goal of local spectral analysis of large random matrices is to prove the WDM conjecture for the largest possible class of matrix ensembles. In the current paper we complete this program for random matrices with a general, slow correlation decay among its matrix elements. Previous works covered only correlations with such a fast decay that, in a certain sense, they could be treated as a perturbation of the independent model. Here we present a new method that goes well beyond the perturbative regime. It relies on a novel multi-scale version of the cumulant expansion and its rigorous Feynman diagrammatic representation that can be useful for other problems as well. To put our work in context, we now explain the previous results.In the last ten years a powerful new approach, the three-step strategy has been developed to resolve WDM universality problems, see [19] for a summary. In particular, the WDM conjecture in its classical form, stated for Wigner matrices with a general distribution of the entries, has been proven with this strategy in [14,15,21]; an independent proof for the Hermitian symmetry class was given in [42]. Recent advances have crystallized that the only model dependent step in this strategy is the first one, the local law. The other two steps, the fast relaxation to equilibrium of the Dyson Brownian motion and the approximation by Gaussian divisible ensembles, have been formulated as very general "black-box" tools whose only input is the local law [17,31,32]. Thus the proof of the WDM universality, at least for mean field ensembles, is automatically reduced to obtaining a local law.Both local law and universality have first been established for Wigner matrices, which are real symmetric or complex Hermitian N × N matrices with mean-zero entries which are independent and identically distributed (i.i.d.) up to symmetry [15,16]. For Wigner matrices it has long been known that the l...

Abstract. We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of [6] that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for p-uniform hypergraphs that correspond to p-spin glass Hamiltonians acting on n distinguishable spin-1/2 particles. At the critical threshold p = n 1/2 we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory.

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