Replacement Principle Convergence of singular values Uniform integrability

106 Electronic Communications in Probability Theorem 4 Universality of sum and product of random matrices. For every integer n, let M n , K n , L n be n × n complex matrices such that, for some α 0, i x 7→ x −α is uniformly bounded for ν K n n ≥1 , and ν L n n ≥1 and x 7→ x α is uniformly bounded for ν M n n ≥1 , ii for almost all a.a. z ∈ C, ν K −1 n M n L −1 n −K −1 n L −1 n z converges weakly to a probability measure ν z . Then, almost surely, µ M +K X L p n n →∞ µ, where µ depends only ν z z ∈C . For M = K = L = I, the identity matrix, this statement gives the famous circular law theorem, that was established through a long sequence of partial results [ 19 , 8 , 10 , 16 , 6 , 9 , 1 , 11 , 2 , 20 , 12 , 23 , 24 ]. In this note, the steps of proof are elementary and they borrow all difficult technical statements from previously known results. Nevertheless, this theorem generalizes Theorem 1.18 in [ 24 ] in two directions. First, we have removed the assumption of uniformly bounded second moment for ν M +K X L p n , ν K −1 M L −1 and ν K −1 L −1 . Secondly, it proves the convergence of the spectral measure. The explicit form of µ in terms of ν z is quite complicated. It is given by the forthcoming equations 2-3. This expression is not very easy to handle. However, following ideas developed in [ 21 ] or using tools from free probability as in [ 22 , 14 ], it should be possible to find more elegant formulas. For nice examples of limit spectral distributions, see e.g. [ 21 ]. It is interesting to notice that we may deal with non-centered variables X i j by including the average matrix of X p n into M , and recover [ 4 ]. Finally, as in [ 13 ], it is also possible by induction to apply Theorem 4 to product of independent copies of X with the use of the forthcoming Theorem 8. 2 Proof of Theorem 2

2.1 Replacement Principle

The following is an extension of Theorem 2.1 in [ 24 ]. The idea goes back essentially to Girko. Lemma 5 Replacement principle. Let A n , B n be n × n complex random matrices. Suppose that for a.a. z ∈ C, a.s. i ν A n −z − ν B n −z tends weakly to 0, ii ln · is uniformly integrable for ν A n −z n ≥1 and ν B n −z n ≥1 . Then a.s. µ A n − µ B n tends weakly to 0. Moreover the same holds if we replace i by i’ R lnxd ν A n −z − R lnxd ν B n −z tends to 0. Proof. It is a straightforward adaptation of [ 3 , Lemma A.2]. ƒ Corollary 6. Let A n , B n , M n be n × n complex random matrices. Suppose that a.s. M n is invertible and for a.a. z ∈ C, a.s. i ν A n −zM −1 n − ν B n −zM −1 n tends weakly to 0, ii ln · is uniformly integrable for ν A n −zM −1 n n ≥1 and ν B n −zM −1 n n ≥1 . On the spectrum of sum and product of non-hermitian random matrices 107 Then a.s. µ M n A n − µ M n B n tends weakly to 0. Proof. If M n is invertible, note that Z lnxd ν M n A n −z = 1 n ln | detM n A n − z| = Z lnxd ν A n −zM −1 n + 1 n ln | det M n |. We may thus apply Lemma 5i’-ii. Indeed, in the expression R lnxd ν A n −z − R lnxd ν B n −z , the term 1 n ln | det M n | cancels. ƒ

2.2 Convergence of singular values

The following result is due to Dozier and Silverstein. Theorem 7 Convergence of singular values, [ 5 ]. Let M n n ≥1 be a sequence of n × n complex matrices such that ν M n converges weakly to a probability measure ν. Then a.s. ν X n p n+M n converges weakly to a probability measure ρ which depends only on ν. The measure ρ has an explicit characterization in terms of ν. Its exact form is not relevant here.

2.3 Uniform integrability

In order to use the replacement principle, it is necessary to prove the uniform integrability of ln · for some empirical singular values measures. This is achieved by proving that, for some β 0, x 7→ x −β + x β is uniformly bounded. Theorem 8 Uniform integrability. Let M n n ≥1 be a sequence of n × n complex matrices, and assume that x 7→ x α is uniformly bounded for ν M n n ≥1 for some α 0. Then there exists β 0 such that a.s. x 7→ x −β + x β is uniformly bounded for ν X n p n+M n n ≥1 . In the remainder of the paper, the notation n ≫ 1 means large enough n. We start with an elementary lemma. Lemma 9 Large singular values. Almost surely, for n ≫ 1, Z x 2 d ν X p n ≤ 2. Proof. We have 1 n P n i=1 s 2 i X p n = 1 n 2 trX ∗ X = 1 n 2 P 1 ≤i, j≤n |X i j | 2 , and the latter converges a.s. to 1 by the law of large number. ƒ Corollary 10. Let 0 α ≤ 2 and let M n n ≥1 be a sequence of n × n complex matrices such that x 7→ x α is uniformly bounded for ν M n n ≥1 . Then, a.s. x 7→ x α is uniformly bounded for ν X n p n+M n n ≥1 . Proof. If M , N are n × n complex matrices, from [ 15 , Theorem 3.3.16], for all 1 ≤ i, j ≤ n with 1 ≤ i + j ≤ n + 1, s i+ j −1 M + N ≤ s i M + s j N . Hence, s 2i M + N ≤ s 2i −1 M + N ≤ s i M + s i N . 108 Electronic Communications in Probability We deduce that for any non-decreasing function, f : R + → R + and t 0, Z f xd ν M +N ≤ 2 Z f 2xd ν M + 2 Z f 2xd ν N , where we have used the inequality f x + y ≤ f 2x + f 2 y. Now, in view of Lemma 9, we may apply the above inequality to f x = x α and deduce the statement. ƒ The above corollary settles the problem of the uniform integrability of ln · at +∞ for ν X p n+M . The uniform integrability at 0+ is a much more delicate matter. The next theorem is a deep result of Tao and Vu. Theorem 11 Small singular values, [ 23 , 24 ]. Let M n n ≥1 be a sequence of n ×n complex invertible matrices such that x 7→ x α is uniformly bounded for ν M n n ≥1 for some α 0. There exist c 1 , c such that a.s. for n ≫ 1, s n X n p n + M n ≥ n −c 1 . Moreover for i ≥ n 1 −γ with γ = 0.01, a.s. for n ≫ 1, s n −i X n p n + M n ≥ c i n . Proof. The first statement is Theorem 2.1 in [ 23 ] and the second is contained in [ 24 ] see the proof of Theorem 1.20 and observe that the statement of Proposition 5.1 remains unchanged if we consider a row of the matrix X n + p nM n . ƒ Proof of Theorem 8. By Corollary 10, it is sufficient to prove that x 7→ x −β is uniformly bounded for ν X p n+M and some β 0. We have lim sup n 1 n n X i=1 s −β i X p n + M ∞, By Theorem 11, we may a.s. write for n ≫ 1, 1 n n X i=1 s −β i X p n + M ≤ 1 n ⌊n 1 −γ ⌋ X i=1 n β c 1 + 1 n n X i= ⌊n 1 −γ ⌋+1 c 2  n i ‹ β ≤ n β c 1 −γ + 1 n n X i=1 c 2  n i ‹ β . This last expression is uniformly bounded if 0 β minγc 1 , 1. ƒ

2.4 End of proof of Theorem 2