Bulk universality for generalized Wigner matrices

Abstract

We consider N N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density .x/ D e U.x/ . We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U 2 C 6 .R/ with at most polynomially growing derivatives and .x/ Ä C e C jxj for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

Stability of the matrix Dyson equation and random matrices with correlations

“…The argument uses a Cauchy-integral formula that was also applied in the construction of the Helffer-Sjöstrand functional calculus (cf. [15]) and it already appeared in different variants in [22,27,28].…”

“…Local laws are the first step of a general three step strategy developed in [24,25,27,29] for proving universality. The second step is to add a tiny independent Gaussian component and prove universality for this slightly deformed model via analyzing the fast convergence of the Dyson Brownian motion (DBM) to local equilibrium.…”

“…The celebrated Wigner-Dyson-Mehta (WDM) universality conjecture, as formulated in the classical book of Mehta [44], asserts that the same gap statistics holds if the matrix elements are independent and have arbitrary identical distribution (they are called Wigner ensembles). The WDM conjecture has recently been proved in increasing generality in a series of papers [18,21,24,25] for both the real symmetric and complex hermitian symmetry classes via the Dyson Brownian motion. An alternative approach introducing the four-moment comparison theorem was presented in [49,50,52].…”

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

“…We apply Proposition 6.2. The cubic equation (6.3) takes the simplified form 19) where e(ω) satisfies (6.4) and ψ ∼ 1 according to (6.7). Since |m|, f ∼ 1 and m(z) is uniformly 1/3-Hölder continuous in z (cf.…”

“…(see [19] in the special case when the sums ∑ j s i j are independent of i). In particular, if (1.1) is stable under small perturbations, then g ii is close to m i and the average N −1 ∑ i m i approximates the normalized trace of the resolvent, N −1 Tr G. Being determined by N −1 Im Tr G, as Im z → 0, the empirical spectral measure of H approaches the non-random measure with density 3) as N goes to infinity, see [8,24,37].…”