Abstract
We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries.
Highlights
Most rigorous works on random matrix ensembles concern either Wigner matrices with independent entries [16,23], or invariant ensembles where the correlation structure of the matrix elements is very specific
Since the existing methods to study Wigner matrices heavily rely on independence, only very few results are available on ensembles with correlated entries, see [10,11,12,19] for the Gaussian case
We mention the very recent proof of the local semicircle law and bulk universality for the adjacency matrix of the d-regular graphs [7,8] which has a completely different specific correlation
Summary
Most rigorous works on random matrix ensembles concern either Wigner matrices with independent entries [16,23] (up to the real symmetric or complex hermitian symmetry constraint), or invariant ensembles where the correlation structure of the matrix elements is very specific. The proofs rely on the key observation that the (discrete) Fourier transform H = (hφθ ) of a translation invariantly correlated self-adjoint random matrix H has independent entries up to an additional symmetry (cf Lemma 3.2 below). Our recent results [2] on the local law and bulk universality of Wigner type matrices with a general variance matrix can be applied. The stability of the QVE implies that the solutions of (1.1) and (1.2) are close, i.e., Gφφ(z) = mφ(z) + o(1), even for spectral parameters z very close to the real axis, down to the scale Im z N −1 This yields the local law for the eigenvalue density of H. The current paper in combination with [1,2] presents such a precise analysis
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
