No Outliers in the Spectrum of the Product of Independent Non-Hermitian Random Matrices with Independent Entries

Abstract

We consider products of independent square random non-Hermitian matrices. More precisely, let n ≥ 2 and let X1, . . . , Xn be independent N × N random matrices with independent centered entries (either real or complex with independent real and imaginary parts) with variance N −1 . In [10] and [15] it was shown that the limit of the empirical spectral distribution of the product X1 · · · Xn is supported in the unit disk. We prove that if the entries of the matrices X1, . . . , Xn satisfy uniform subexponential decay condition, then the spectral radius of X1 · · · Xn converges to 1 almost surely as N → ∞.

Local law for the product of independent non-Hermitian random matrices with independent entries

“…The main difference compared to [7] and thus technical difficulty arises from the fact that we cannot work directly with the Stieltjes transform and have to study the concentration for its partial traces. Similar results but for different values of the resolvent parameter were obtained in [14]. Although the approach is similar to that used in [14], many important statements should be adjusted in order to obtain strong enough estimates on a set, which is sufficiently large to imply the rigidity of the singular values of X − z.…”

Outliers in the spectrum for products of independent random matrices

“…We require the following theorem, which is based on [47, Theorem 2]. A similar statement was proven in [47,Theorem 2], where the same conclusion was shown to hold almost surely rather than with overwhelming probability. However, many of the intermediate steps in [47] are proven to hold with overwhelming probability.…”

“…Remark 2.3. A version of Theorem 2.2 was proven by Nemish in [47] under the additional assumption that the atom variables ξ 1 , . .…”

“…For the fourth moment, we use that Var(ξ) ≥ 1 2 and Jensen's inequality inequality to obtain This section is devoted to the proof of Theorem 8.1. We begin with Lemma B.1 below, which is based on [47,Theorem 4]. Throughout this section, we use √ −1 for the imaginary unit and reserve i as an index.…”

“…With Lemma B.1 in hand, we are now prepared to prove Theorem 8.1. The proof below is based on a slight modification to the arguments from [47,48]. As such, in some places we will omit technical computations and only provide appropriate references and necessary changes to results from [47,48].…”