Abstract
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \mathbb{E} \left|h_{ij}\right|^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\varepsilon_n}$ with some $\varepsilon_{n} > 0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments.
Highlights
Since the pioneering work [31] of Wigner in the fifties, random matrices have played a fundamental role in modelling complex systems
We prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W N 1−εn with some εn > 0 and with a negligible mean-field component
The Wigner-Dyson-Gaudin-Mehta conjecture, formalized in [25], states that this gap distribution is universal in the sense that it depends only on the symmetry class of the matrix, but is otherwise independent of the details of the distribution of the matrix entries
Summary
Since the pioneering work [31] of Wigner in the fifties, random matrices have played a fundamental role in modelling complex systems. The Wigner-Dyson-Gaudin-Mehta conjecture was originally stated for Wigner matrices, the methods of [7,14,19] apply to certain ensembles with independent but not identically distributed entries, which retain the mean-field character of Wigner matrices They yield universality provided the variances sij ..= E|hij |2 of the matrix entries are only required to be of comparable size (but not necessarily equal): c. We extend bulk universality, proved for generalized Wigner matrices in [17], to a large class of matrix ensembles where the upper and lower bounds on the variances (1.1) are relaxed. We use c to denote a generic small positive constant
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