Local laws for polynomials of Wigner matrices

Abstract

We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.

Scattering in Quantum Dots via Noncommutative Rational Functions

“…However, our method works for very general noncommutative (NC) rational expressions in large matrices with i.i.d. entries (with or without Hermitian symmetry) generalizing our previous work [13] on polynomials. For convenience of the readers interested only in the concrete scattering problem, the main part of our paper focuses on this model and we defer the general theory to Appendix A.…”

“…Together with the linearization trick, this allows us to handle arbitrary polynomials in i.i.d. random matrices [13] and in the current work we extend our method to a large class of rational functions. Note that even if the building block matrices have independent entries, the linearization of their rational expressions will have dependence, but the general MDE can handle it [see (2.13)].…”

“…Since the publication of this work, various approaches and algorithms have been developed for constructing linearizations of general classes of rational functions/expressions (see, e.g., [9] or [24] for a pedagogical presentation of the subject). For reader's convenience, we provide below a simple linearization algorithm based on the method described in [13,Section A.1]. We use the following observation: for matrices A i ∈ (C x, y, y * ) mi×mi and B j ∈ (C x, y, y * ) mj ×mj+1 , m j ∈ N, the lower right m 1 × m k submatrix of the inverse of the block matrix…”

“…Using (A.9), we can construct a linearization of rational expressions using the induction on the height. In the case of a rational expression of height 0 (polynomial function), linearization can be constructed using, e.g., the algorithm from [13,Section A.1]. If q 1 ∈ (Q q 0 ) 1 and q ∈ Q q 0 ,q 1 is a rational expression of height 1, the linearization can be obtain using the following algorithm:…”

“…The entire analysis is then restricted to this set. In particular, the indicator sets χ(•) in the proof of [13,Theorem 5.1] should be replaced by χ(•) ∩ Θ.…”