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Outliers in the Single Ring Theorem


This text is about spiked models of non-Hermitian random matrices. More specifically, we consider matrices of the type \({\mathbf {A}}+{\mathbf {P}}\), where the rank of \({\mathbf {P}}\) stays bounded as the dimension goes to infinity and where the matrix \({\mathbf {A}}\) is a non-Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if \({\mathbf {P}}\) has some eigenvalues out of the maximal circle of the single ring, then \({\mathbf {A}}+{\mathbf {P}}\) has some eigenvalues (called outliers) in the neighborhood of those of \({\mathbf {P}}\), which is not the case for the eigenvalues of \({\mathbf {P}}\) in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of \({\mathbf {A}}\) around the eigenvalues of \({\mathbf {P}}\) and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such kind of fluctuations had already been shown for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of \({\mathbf {P}}\)) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980–2031, 2014) in the Hermitian case, but only for non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a result by Tao proved specifically for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.

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  1. 1.

    Recall that any matrix \({\mathbf {M}}\) in the set \( {\fancyscript{M}}_n({\mathbb {C}})\) of \(n\times n\) complex matrices is similar to a square block diagonal matrix

    $$\begin{aligned} \left( \begin{array}{lllll} {\mathbf {R}}_{p_1}(\theta _1) &{} &{} &{} (0) \\ &{} {\mathbf {R}}_{p_2}(\theta _2) &{} &{} \\ &{} &{} \ddots &{} \\ (0)&{} &{} &{} {\mathbf {R}}_{p_r}(\theta _r) \\ \end{array} \right) \quad \text { where } \quad {\mathbf {R}}_{p}(\theta ) = \left( \begin{array}{lllll} \theta &{} 1 &{} &{} (0) \\ &{} \ddots &{} \ddots &{} \\ &{} &{} \ddots &{} 1 \\ (0)&{} &{} &{} \theta \\ \end{array} \right) \in {\fancyscript{M}}_p({\mathbb {C}}), \end{aligned}$$

    which is called the Jordan Canonical Form of \({\mathbf {M}}\), unique up to the order of the diagonal blocks [21], Chap. 3].

  2. 2.

    To sort out misunderstandings: we call the multiplicity of an eigenvalue its order as a root of the characteristic polynomial, which is greater than or equal to the dimension of the associated eigenspace.

  3. 3.

    For any \(\sigma >0\), \({\fancyscript{N}}_{{\mathbb {C}}}(0, \sigma ^2)\) denotes the centered Gaussian law on \({\mathbb {C}}\) with covariance \(\frac{1}{2}\begin{pmatrix}\sigma ^2&{}0\\ 0&{}\sigma ^2\end{pmatrix}\).

  4. 4.

    Let us recall that what is here called the multiplicity of an eigenvalue its order as a root of the characteristic polynomial, which is not smaller than the dimension of the associated eigenspace.

  5. 5.

    There is actually another assumption in the Single Ring Theorem [18], but Rudelson and Vershynin recently showed in [28] that it was unnecessary. In [4], Basak Dembo also weakened the hypotheses (roughly allowing Hypothesis 3 not to hold on a small enough set, so that \(\nu \) is allowed to have some atoms). As it follows from the recent preprint [6] that the convergence of the extreme eigenvalues first established in [19] also works in this case, we could harmlessly weaken our hypotheses down to the ones of [4].

  6. 6.

    Recall the \(\beta _{i,j}\) is the number of blocks \({\mathbf {R}}_{p_{i,j}}(\theta _i)\) in the JCF of \({\mathbf {P}}\).

  7. 7.

    Recall that the notion of multiset has been defined before Lemma 5.8: a multiset is roughly a collection of elements with possible repetitions (as in tuples) but where the order of appearance is insignificant (contrarily to tuples). For example,

    $$\begin{aligned} \left\{ 1,2,2,3 \right\} _m =\left\{ 3,2,1,2 \right\} _m \ \ne \ \left\{ 1,2,3 \right\} _m. \end{aligned}$$


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We would like to thank J. Novak for discussions on Weingarten calculus.

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Correspondence to Florent Benaych-Georges.

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Benaych-Georges, F., Rochet, J. Outliers in the Single Ring Theorem. Probab. Theory Relat. Fields 165, 313–363 (2016). https://doi.org/10.1007/s00440-015-0632-x

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  • Random matrices
  • Spiked models
  • Extreme eigenvalue statistics
  • Gaussian fluctuations
  • Ginibre matrices

Mathematics Subject Classification

  • 15A52
  • 60F05