Abstract
In this note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed (i.i.d.) entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere.
Highlights
We endow the space of probability measures on C with the topology of weak convergence: a sequence of probability measuresn 1 converges weakly to μ is for any bounded continuous function f : C → R, f dμn − f dμ converges to 0 as n goes to infinity
We will say that a measurable function f : C → R is uniformly bounded forn 1 if lim sup |f |dμn < ∞
The above definitions will be used for probability measures on R+ = [0, ∞) and functions f : R+ → R
Summary
For two sequences of probability measures (μn)n 1, (μ′n)n 1, we will use μn − μ′n n→∞ 0, or say that μn − μ′n tends weakly to 0, if f dμn − f dμ′n converges to 0 for any bounded continuous function f . We will say that a measurable function f : C → R is uniformly bounded for (μn)n 1 if lim sup |f |dμn < ∞. It is known that almost surely (a.s.) for n large enough, X is invertible (see the forthcoming Theorem 11) and μX−1Y is a well defined random probability measure on C.
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