On freely indecomposable measures

Local stability of the free additive convolution

“…To show statement (i), we recall from the proof of Lemma 3.4 that Φ µα,µ β (ω α , ω β , z) = 0 is linearly S-stable at (ω α , ω β ) if 12) for some strictly positive constant c. We now show that (7.12) holds for the case µ α = µ β in the setup of (7.1). Using henceforth the shorthand F α ≡ F µα , F β ≡ F µ β , we compute…”

“…Recalling that Ω(z 0 ) /2k ≤ δ/k < 1 and using (4.10) together with (4.3), we obtain from (4.11) the estimate 12) and the analogous expansion with the rôles of the indices 1 and 2 interchanged. We therefore obtain from (2.12) and from solving the linearized equation that…”

“…A point c ∈ R is an atom of µ 1 ⊞ µ 2 , if and only if there exist a, b ∈ R such that c = a + b and µ 1 ({a}) + µ 2 ({b}) > 1; see [Theorem 7.4, [11]]. For another interesting properties of the atoms of µ 1 ⊞ µ 2 we refer the reader to [12]. The boundary behavior of the functions F µ1⊞µ2 , ω 1 and ω 2 has been studied by Belinschi [5,6,7] who proved the next two results.…”

“…We call the system (2.11) linearly S-stable at (ω 1 , ω 2 ) if 12) for some constant S. Especially, the partial Jacobian matrix, DΦ(ω 1 , ω 2 ), of (2.10) given by…”

“…Recall from (6.4) the notation f Q , where Q is an arbitrary N × N matrix. 12) implies the concentration bound…”