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...

We consider the nonlinear equation − 1 m = z + Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In [?] we qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16c], we present a complete stability analysis of the equation for any z ∈ H, including the vicinity of the singularities.

We consider the local eigenvalue distribution of large self-adjoint N × N random matrices H = H * with centered independent entries. In contrast to previous works the matrix of variances s ij = E|h ij | 2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper [1]. We show that as N grows, the resolvent, G(z) = (H − z) −1 , converges to a diagonal matrix, diag(m(z)), where m(z) = (m 1 (z), . . . , m N (z)) solves the vector equation −1/m i (z) = z + j s ij m j (z) that has been analyzed in [1]. We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.

Active systems can produce a far greater variety of ordered patterns than conventional equilibrium systems. In particular, transitions between disorder and either polar- or nematically ordered phases have been predicted and observed in two-dimensional active systems. However, coexistence between phases of different types of order has not been reported. We demonstrate the emergence of dynamic coexistence of ordered states with fluctuating nematic and polar symmetry in an actomyosin motility assay. Combining experiments with agent-based simulations, we identify sufficiently weak interactions that lack a clear alignment symmetry as a prerequisite for coexistence. Thus, the symmetry of macroscopic order becomes an emergent and dynamic property of the active system. These results provide a pathway by which living systems can express different types of order by using identical building blocks.