Circular law, extreme singular values and potential theory

Abstract

a b s t r a c tConsider the empirical spectral distribution of complex random n ×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

Products of Independent non-Hermitian Random Matrices

“…sub-Gaussian matrix entries. Pan and Zhou proved the circular law for any distribution ξ with finite fourth moment [25] by building on [15] and utilizing the work of Rudelson and Vershynin in [27]. In an important development, Götze and Tikhomirov showed in [14] that the expected spectral distribution Eµ Nn converges to the uniform distribution over the unit disk as n tends to infinity assuming that sup jk E|(N n ) jk | 2 φ((N n ) jk ) < ∞, where φ(x) = (ln(1 + |x|)) 19+η , η > 0.…”

“…of (22) and (23) are bounded from above by 1 | Im α| 2 . The proof of (24)(25) follows from the resolvent identitity…”

Circular law and arc law for truncation of random unitary matrix

“…Observe that the integral in (40) is equal to P (P m ∈ F ) where the underlying probability measure P has density function f (λ 1 , · · · , λ m ) as in (31) and (32), and where F := {µ ∈ M(C); ρ(µ, µ 0 ) ≥ ϵ} is a closed set in the weak convergence topology and µ 0 / ∈ F. By Lemma 2.9, there exists a constant τ = τ (ϵ) > 0 such that…”

Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices

“…Many authors have proved versions of the circular law under various assumptions on the atom variable ξ; see for instance [2,8,16,26,33,34] and references therein. We present the most general version due to Tao and Vu [34].…”