Local single ring theorem on optimal scale

Abstract

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U (N ). Let Σ be a non-negative deterministic N by N matrix. The single ring theorem [26] asserts that the empirical eigenvalue distribution of the matrix X := U ΣV * converges weakly, in the limit of large N , to a deterministic measure which is supported on a single ring centered at the origin in C. Within the bulk regime, i.e., in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N −1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N ).Remark 1.3. In Theorem 1.2, U and V may be both Haar distributed on U (N ) or on O(N ). Remark 1.4. In its original form Theorem 1.2 was proved by Guionnet, Krishnapur and Zeitouni in [26] under a further assumption on the smallest singular value of the matrix X − z, z ∈ C. This hard-to-check condition was removed by Rudelson and Vershynin in [34] (c.f., Theorem 2.6 below), which yields Theorem 1.2.Remark 1.5. The measure ρ σ was first computed in [27]. It has a direct interpretation in free probability theory, in fact it is the Brown measure of the free product of a Haar unitary and an element σ on a noncommutative probability space; see [27] for more details. * We will often use the convention that capital letters indicate random matrices and the corresponding small letters indicate their limiting objects.

Local inhomogeneous circular law

“…Then the corresponding Dyson equations are much simpler, in fact they consist of only two scalar equations and they are characterized by a vector of parameters (of the singular values of T ) instead of an entire matrix of parameters S. The vanishing third moment condition in [31] is necessary to compare the general distribution with the Gaussian case via a moment matching argument. We also mention the recent proof of the local single ring theorem on optimal scale in the bulk [9]. This concerns another prominent non-Hermitian random matrix ensemble that consists of matrices of the form U ΣV , where U , V are two independent Haar distributed unitaries and Σ is deterministic (may be assumed to be diagonal).…”

Universality of the least singular value for the sum of random matrices

“…with overwhelming probability for sufficiently large N not depending on t. Further, by Lemma A.2 of [16], there exists c > 0 such that inf z∈D I Imw X ≥ c for large enough N , independent of t. This implies the desired claim for G(z, t) at any fixed t. Observe there is an implicit dependence ofw X on t. This estimate may then be transferred to G ii , using the resolvent identity, and made uniform in t, using a standard stochastic continuity argument, as in [33,Theorem 3.16]. This completes the proof.…”

“…The model M is a natural interpretation of the notion of the sum of two generic random matrices and exhibits strong correlations between its entries, unlike the matrices studied in [34,73]. It was previously studied in [16], where its spectrum was controlled on scale N −1+ε . The Hermitian version of this model, H = V * XV + U * Y U , has attracted significant interest.…”

“…While the essential technical content is unchanged, this somewhat streamlines the proofs. Further, we use [16] to prove a strong law at small energies, paralleling the use of [12][13][14][15] in [33] to establish a strong law in the bulk of the spectrum. We also comment on an interesting difference between the real and complex cases which does not arise in the Hermitian model.…”

“…The first assumption is to prevent the y i from accumulating around any point E ∈ R. This is illustrated by Proposition C.1. The second and third are required to apply [16,Theorem 4.4] to control the Green functions of (a modification of) M , as is done in Section 5. The remaining assumptions are required in Subsection 5.2.…”