Deformed Single Ring Theorem

• Deformed single ring theorem is a framework that characterizes both global and local eigenvalue distributions of non-Hermitian matrices with deterministic deformations.
• It employs advanced techniques like free probability, Hermitization, and analytic subordination to rigorously define the annular support and rotational invariance of the limiting spectral measure.
• The theorem establishes convergence to a Brown measure with optimal local law results, showcasing precise spectral behavior down to the microscopic scale.

The deformed single ring theorem characterizes the limiting and local eigenvalue distributions of random non-Hermitian matrices of the form $X = U \Sigma V^* + A$, where $U$ and $V$ are independent Haar-distributed unitary (or orthogonal) matrices, $\Sigma$ is a deterministic nonnegative diagonal matrix, and $A$ is a general deterministic deformation. This theorem merges techniques from free probability, random matrix theory, and operator algebras to describe both the global spectral measure (Brown measure) and the local spectral statistics in the large dimension limit, encompassing results for both pure and deformed models. The framework subsumes the classical single ring theorem and rigorously establishes the annular support and rotational invariance of the limiting law, as well as the behavior and stability of spectral outliers under finite- and full-rank deformations.

1. Model and General Formulation

Consider, for each integer $N$, deterministic matrices $A_N$, diagonal nonnegative $\Sigma_N$ with $\|\Sigma_N\| \le M$, and independent Haar unitary matrices $U_N, V_N$. The central object is the random matrix

$Y_N = U_N \Sigma_N V_N^* + A_N.$

The empirical eigenvalue law is given by

$$\mu_{Y_N} = \frac{1}{N} \sum_{i=1}^N \delta_{\lambda_i(Y_N)}.$$

Under appropriate control on the operator norms and *-moment convergence of $A_N$ and $\Sigma_N$, there exists a limit $(\mathcal{A}, \tau)$ where the empirical distributions converge in *-distribution to $a \in \mathcal{A}$, $\sigma \in \mathcal{A}_+$, and, via asymptotic freeness, $U_N \Sigma_N V_N^* \rightarrow T$, an $R$-diagonal operator free from $a$. The main probabilistic input stems from the unitaries; the deformations $A_N$ and values of $\Sigma_N$ are deterministic (Ho et al., 2022, Ho et al., 15 Feb 2025).

2. Limiting Law: Brown Measure and Single-Ring Structure

For $A_N \equiv 0$, the classical single ring theorem asserts that $\mu_{Y_N}$ converges to a deterministic, radially symmetric "single-ring" measure $\mu_\Sigma$ supported in the annulus

$\{z \in \mathbb{C} : r_- < |z| < r_+\},$

where $r_+^2 = \int x^2 d\mu_\sigma(x)$ and $r_-^{-2} = 2\int x^{-2} d\mu_\sigma(x)$, with the convention $r_-=0$ if the integral diverges (Bao et al., 2016). The measure $\mu_\Sigma$ is explicitly given in terms of the Haagerup–Larsen formula: $$d\mu_\Sigma(w) = \frac{\rho_\Sigma(r)}{2\pi} dr d\theta,\quad \text{with } \rho_\Sigma(r) = \frac{1}{2\pi r} \partial_r \Phi(r),$$ where $\Phi$ involves the free additive convolution of symmetrized singular-value measures.

For general deformations ($A_N \neq 0$), the limiting law is described by the Brown measure of $y = T + a$, defined as the unique probability measure $\mu_{y}$ with logarithmic potential

$$\tau[\log|y-\lambda|] = \int_\mathbb{C} \log|z-\lambda| \, \mu_{y}(dz).$$

The Brown measure $\mu_{T+a}$ has support given by the set where the equation for analytic subordination functions admits solutions, and its density in the interior is obtained as the push-forward of the free additive convolution of the symmetrized singular-value laws of $A_N$ and $T$ (Ho et al., 2022, Ho et al., 15 Feb 2025).

3. Local Law and Optimal Scale Results

Within the bulk regime (compact subsets strictly inside the limiting annulus or Brown measure support), the convergence of the empirical spectral measure holds down to the optimal microscopic scale $N^{-1/2+\varepsilon}$: $$\left| \frac{1}{N} \sum_{j=1}^N f_{w_0}(\lambda_j(Y_N)) - \int_\mathbb{C} f_{w_0}(w) d\mu_{T+a}(w) \right| = O(N^{-1+2\varepsilon})\|\Delta f\|_{L^1},$$ for rescaled test functions $$f_{w_0}(w) := N^{2\varepsilon} f(N^{\varepsilon}(w-w_0))$$. This result demonstrates that local spectral statistics coincide with the deterministic limiting Brown measure down to the eigenvalue spacing scale in the bulk, and macroscopic convergence holds at the rate $O(N^{-1})$ (Bao et al., 2016, Ho et al., 2022).

4. Conditions, Assumptions, and Generality

The primary conditions are:

• $\|\Sigma_N\| \le M$ uniformly in $N$.
• The empirical law of the singular values of $\Sigma_N$ converges weakly to a compactly supported measure $\mu_\sigma$ not concentrated at zero.
• For $A_N$, uniform norm bounds and convergence of all *-moments in a tracial $W^*$-probability space.

For the undeformed model ($A_N \equiv 0$), the only restriction is non-degeneracy of the singular-value law and compactness of support, no further regularity such as density or spectral gap is required (Bao et al., 2016). In the general deformed case, invertibility up to slow-growing factors of $\|\Sigma_N^{-1}\|\le N^\alpha$ and mild technical conditions on Cauchy transforms—which are always met for Hermitian or unitary $A_N$—ensure applicability without further assumptions (Ho et al., 2022).

5. Techniques: Hermitization, Free Probability, and Subordination

The proof methodology is fundamentally based on Girko’s Hermitization: studying the $2N \times 2N$ Hermitian matrix

$$H_N(\lambda) = \begin{pmatrix} 0 & Y_N - \lambda \ (Y_N - \lambda)^* & 0 \end{pmatrix}$$

whose spectrum encodes the singular values of $Y_N-\lambda$. The logarithmic potential of the empirical law is linked to the normalized trace of $\log|H_N(\lambda)|$, and Green’s function and Stieltjes transforms are used to connect resolvents to spectral density (Bao et al., 2016, Ho et al., 2022).

A central step is the use of free additive convolution and the corresponding analytic subordination framework: subordination functions describe the Cauchy transforms of the limiting measures. The convergence analysis combines large-$\eta$ concentration, recursive moment bounds, fluctuation averaging, and a bootstrapping argument to handle the small-$\eta$ (microscopic) regime. The explicit control of resolvent entries and the integration via Stieltjes inversion enables precise local law statements down to $N^{-1/2+\varepsilon}$ (Bao et al., 2016).

6. Outlier Eigenvalues and Deformation Effects

For deformations of bounded rank, finite rank perturbations $P_n$ yield outlier eigenvalues of $A_n + P_n$ located at the eigenvalues of $P_n$ outside the outer circle or inside the inner disk of the single ring, with fluctuations characterized by the Jordan canonical form and correlated Ginibre spectra (Benaych-Georges et al., 2013).

The full rank deformed single ring theorem, as established in Ho–Yin–Zhong, generalizes these results: in the domain exterior to the ring, outlier eigenvalues persist if associated spikes of $A_n$ are present, while no outliers occur in the interior region. This holds provided the required spectral separation and well-conditioning on $A_n$ (Ho et al., 15 Feb 2025). The precise location and stability of outliers are determined using a Schur complement and resolvent expansion, building on the Weingarten calculus for Haar unitaries.

7. Corollaries, Special Cases, and Extensions

Notable corollaries include simplification in the case of Hermitian or unitary $A_N$, where minimal assumptions fully guarantee convergence to the Brown measure. For pure unitary limits, the limiting Brown measure reduces to the single ring law, while for non-trivial deformations, the annulus may thicken or collapse depending on the distribution of singular values.

Special models, such as diagonal $\Sigma_N$ with two unequal masses or scalar multiples, make explicit the dependence of the inner and outer radii on the singular value moments. Jordan block deformations illustrate regularization phenomena, corroborating and extending prior work on spectral universality in non-Hermitian random matrices (Ho et al., 2022, Benaych-Georges et al., 2013).

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References:

• Ho & Zhong, "Deformed single ring theorems" (Ho et al., 2022)
• Ho, Yin, & Zhong, "Outlier eigenvalues for full rank deformed single ring random matrices" (Ho et al., 15 Feb 2025)
• Bao, Erdős, Schnelli, "Local single ring theorem on optimal scale" (Bao et al., 2016)
• Benaych-Georges & Rochet, "Outliers in the Single Ring Theorem" (Benaych-Georges et al., 2013)