Korea Adv Inst Sci & Technol, Daejeon, South Korea..

Schnelli, Kevin

KTH. IST Austria, Klosterneuburg, Austria..

2018 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 171, no 1-2, p. 543-616Article in journal (Refereed) Published

Abstract [en]

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the ErdAs-R,nyi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the ErdAs-R,nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue when p is much larger than wth a deterministic shift of order (Np)(-1)..

Place, publisher, year, edition, pages

Springer Berlin/Heidelberg, 2018

Keywords

Local law, Sparse random matrices, Erdos-Renyi graph

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.

2017 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 3, p. 947-990Article in journal (Refereed) Published

Abstract [en]

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.