Local Semicircle Law in Random Matrix Theory

• Local Semicircle Law is a fundamental result that precisely characterizes the local behavior of eigenvalue distributions in large random matrices.
• It employs resolvent expansions, self-consistent equations, and large deviation estimates to achieve optimal convergence rates down to the eigenvalue spacing scale.
• The law underpins universality results including eigenvector delocalization and eigenvalue rigidity across various ensembles such as Wigner, sparse, and deformed matrices.

The local semicircle law is a central result in random matrix theory, describing the precise local behavior of the empirical spectral distribution of large random matrices, most notably Wigner matrices, on scales much finer than the global (macroscopic) regime. It provides quantitative bounds demonstrating that, with high probability, the Stieltjes transform of the empirical spectral distribution, the entries of the resolvent, and more generally quadratic forms in the resolvent converge to their deterministic counterparts given by Wigner’s semicircle law, all the way down to scales matching or slightly above the typical eigenvalue spacing. This convergence underpins universality of local spectral statistics, eigenvalue rigidity, and complete eigenvector delocalization in a broad variety of random matrix ensembles.

1. Definitions and Model Setup

Let $H=(h_{ij})$ be an $N\times N$ Hermitian (or real symmetric) random matrix satisfying the “Wigner matrix” conditions:

• The upper-triangular entries are independent (up to symmetry), centered, and normalized: $\mathbb{E} h_{ij} = 0$, $\mathrm{Var}(h_{ij}) = 1/N$ for $i \neq j$, $\mathrm{Var}(h_{ii}) = 2/N$ (Knowles et al., 2011).
• Higher moments are uniformly bounded, or satisfy subexponential decay (Benaych-Georges et al., 2016, Erdos et al., 2012).

For such a matrix, the resolvent (Green’s function) at spectral parameter $z = E + i\eta$ ($\eta>0$) is $G(z) = (H - z I)^{-1}$, with normalized empirical Stieltjes transform $m_N(z) = N^{-1} \mathrm{Tr}\, G(z)$. The semicircle law has density $\varrho_{sc}(x)=\frac{1}{2\pi}\sqrt{(4-x^2)_+}$ and Stieltjes transform $m_{sc}(z)$. The Stieltjes transform is the unique solution to $m_{sc}(z) + 1/m_{sc}(z) + z = 0$, $\Im m_{sc}(z)>0$.

The spectral window in which the law operates is typically $|E| \leq \Sigma$ for some fixed $\Sigma>2$, and $\eta \in [N^{-1+\epsilon}, \Sigma]$ for arbitrary small $\epsilon>0$ (Benaych-Georges et al., 2016).

2. Statement of the Local Semicircle Law

The prototypical local semicircle law has the following form. Let

$$\Psi(z) = \sqrt{ \frac{ \Im m_{sc}(z) }{ N\eta } + \frac{1}{N\eta} }$$

Then for all $z$ in the spectral domain and with very high probability,

$| m_N(z) - m_{sc}(z) | \prec \frac{1}{N\eta}$

$$\max_{i,j} |G_{ij}(z) - \delta_{ij} m_{sc}(z) | \prec \Psi(z)$$

where $\prec$ denotes stochastic domination: for any $\varepsilon, D>0$, $\mathbb{P}( |X| > N^\varepsilon Y ) \leq N^{-D}$ for large $N$ (Benaych-Georges et al., 2016, Erdos et al., 2012).

The same optimal bounds hold for a wide variety of Wigner-type ensembles, generalized Wigner matrices with inhomogeneous variances, random regular graphs, sparse Erdős–Rényi graphs (Bauerschmidt et al., 2015, Erdős et al., 2011), random band matrices with sufficient bandwidth, ensembles with additional symmetries (Alt, 2015), and certain correlated and exchangeable ensembles (Curie–Weiss type) (Fleermann et al., 2019).

3. Methods of Proof and Key Technical Ingredients

The proof of the local semicircle law proceeds by an intricate blend of resolvent expansions, large deviation estimates, self-consistent equations, and stability analysis.

1. Resolvent/Schur complement expansion: Each diagonal entry $G_{ii}(z)$ satisfies

$$G_{ii}(z) = \frac{1}{ H_{ii} - z - \sum_{k,\ell\neq i} H_{ik} G^{(i)}_{k\ell}(z) H_{\ell i} }$$

where $G^{(i)}(z)$ is the resolvent of the minor with the $i$-th row and column removed (Benaych-Georges et al., 2016).

2. Self-consistent equation: By averaging over $i$, an approximate quadratic equation for $m_N(z)$ is derived. For Wigner case,

$1 + z m_N(z) + m_N(z)^2 = O(\Psi(z))$

which is compared via stability estimates to the deterministic equation for $m_{sc}(z)$ (Benaych-Georges et al., 2016).

1. Large deviation control: Quadratic forms in the random entries are bounded uniformly in high moments by subexponential or appropriately controlled algebraic tails (Erdos et al., 2012, Benaych-Georges et al., 2016).
2. Fluctuation averaging: Due to independence and centering, averages of fluctuation errors (e.g., $\frac{1}{N}\sum_i Q_i$ for certain quadratic forms $Q_i$) are smaller by a factor $(N\eta)^{-1/2}$ than individual terms, a crucial input to descend the error from $O(\Psi(z))$ to $O((N\eta)^{-1})$ in $| m_N - m_{sc}|$ (Benaych-Georges et al., 2016, Erdos et al., 2012).
3. Multiscale bootstrapping: Control at large $\eta$ is iteratively propagated to the fine scale $\eta\sim N^{-1+\epsilon}$ by continuity in $\eta$ and careful tracking of probability estimates (Benaych-Georges et al., 2016).
4. Stability analysis: The nonvanishing derivative of the self-consistent equation ensures that the deviation $| m_N(z) - m_{sc}(z) |$ remains tightly controlled once errors in the Schur complement are small compared to the stability radius $(\kappa+\eta)^{1/2}$, where $\kappa = ||E|-2|$ (Knowles et al., 2011).
5. Isotropic law and extension: For any deterministic unit vectors $v,w$, the isotropic extension controls

$$|\langle v, (G(z) - m_{sc}(z)I ) w \rangle| \leq (\log N)^{C} \Psi(z) \|v\| \|w\|$$

with high probability, in particular under vanishing third moment or further spectral restrictions (Knowles et al., 2011).

4. Regimes, Optimality, and Extensions

The local semicircle law is optimal down to the scale $\eta \gtrsim N^{-1+\epsilon}$ (for arbitrary small $\epsilon>0$), corresponding to spectral windows containing $N\eta \sim N^{\epsilon}$ eigenvalues—barely more than one. Near the spectral edge and for more general ensembles (e.g., with nontrivial variance profiles), additional factors reflecting spectral stability or the inhomogeneity parameter $M$ are introduced (Erdos et al., 2012, Ajanki et al., 2013). The law remains valid:

• For generalized Wigner ensembles with varied variances, with the deterministic bound $1/(M\eta)$ replacing $1/(N\eta)$ (Erdos et al., 2012).
• In sparse or dependent models—Erdős–Rényi graphs, random $d$-regular graphs, Curie–Weiss ensembles, and band matrices under suitable conditions (Erdős et al., 2011, Bauerschmidt et al., 2015, Fleermann et al., 2019).
• For finite moment conditions ($4+\delta$ moments suffice; see ongoing refinement to minimal fourth moment with logarithmic corrections) (Götze et al., 2016, Götze et al., 2015, Götze et al., 2019).

In deformed Wigner models $H = W + \lambda V$ (with $V$ diagonal, independent of $W$), a local version of the deformed semicircle law holds: the spectral density converges locally to the free convolution of the semicircle law and the law of $V$ (Lee et al., 2013).

At the spectral edge, dedicated combinatorial and moment methods yield local law and rigidity of order $n^{-2/3+\epsilon}$ for GUE/GOE and Gaussian $\beta$-ensembles (Wong, 2011, Sosoe et al., 2011).

5. Applications: Universality, Rigidity, and Delocalization

The local semicircle law has several key corollaries:

• Eigenvector Delocalization: All eigenvectors are completely delocalized: with high probability, $\max_{j,k}|u_j(k)|^2 \leq C (\log N)^C / N$ (Benaych-Georges et al., 2016, Knowles et al., 2011, Götze et al., 2015, Götze et al., 2019).
• Eigenvalue Rigidity: The random eigenvalues $\lambda_j$ are close (within $O(N^{-2/3} (\min(j,N+1-j))^{-1/3} (\log N)^C)$) to their classical locations under the semicircle law; the same holds with appropriate modifications for generalized and deformed models (Benaych-Georges et al., 2016, Erdos et al., 2012, Götze et al., 2015).
• Universality of Local Spectral Statistics: Together with Green function comparison, the local law enables detailed control and universality results for $n$-point correlation functions and eigenvalue gap distributions, conditional on matching low-order moments (Benaych-Georges et al., 2016, Erdos et al., 2012, Knowles et al., 2011).

In random regular graphs and Erdős–Rényi graphs with $pN\to\infty$, analogous statements imply optimal delocalization and semicircle law down to scale $N^{-1}$ (up to logarithmic corrections) (Bauerschmidt et al., 2015, Erdős et al., 2011). For correlated Curie–Weiss-type models with slow correlation decay, the same optimal entrywise law holds under de Finetti-type assumptions (Fleermann et al., 2019).

6. Variants, Generalizations, and Limitations

The local semicircle law extends to:

• Symmetry classes with additional constraints (e.g., fourfold symmetry) (Alt, 2015).
• Generalized variance profiles and doubly stochastic matrices, yielding optimal bounds for band matrices and covariance-type (Marchenko–Pastur) models at the hard edge (Erdos et al., 2012, Ajanki et al., 2013).
• Tridiagonal models for $\beta$-ensembles via moment and resolvent expansion methods (Wong, 2011, Sosoe et al., 2011).
• Deterministic matrices, where the law quantifies deviation from the semicircle law purely in terms of deterministic stability parameters and matrix block expansions (Anderson, 2013).

Limitations are present for matrices with heavy tails lacking the required moment conditions, or ensembles with a strong mean-field component, or insufficient independence. Rigorous extensions to $\eta \sim N^{-1}$ with no logarithmic loss remain open in low-moment/no subexponential scenarios (Götze et al., 2016, Götze et al., 2019).

7. Summary Table of Core Results

Ensemble	Entrywise Bound	Averaged Bound	Scale	Authors / arXiv id
Wigner matrix	$\Psi(z)$	$(N\eta)^{-1}$	$\eta \gtrsim N^{-1+\epsilon}$	(Benaych-Georges et al., 2016)
General variance profile	$\Psi(z)$ w/ $M$	$(M\eta)^{-1}$	$\eta \gtrsim M^{-1+\epsilon}$	(Erdos et al., 2012)
$d$-regular graphs	$(N\eta)^{-1/2} + d^{-1/2}$	as above	$\eta \gtrsim d^{-1}/N$	(Bauerschmidt et al., 2015)
Sparse Erdős–Rényi	$q^{-1} + (N\eta)^{-1/2}$	as above	$q \gtrsim (\log N)^{C}$	(Erdős et al., 2011)
Deformed Wigner	$(N\eta)^{-1/2}$	$(N\eta)^{-1}$	$\eta \gtrsim N^{-1+\epsilon}$	(Lee et al., 2013)
Fourth moment only	$(N\eta)^{-1/2} \log N$	$(N\eta)^{-1} \log N$	$\eta \gtrsim (\log N)^2/N$	(Götze et al., 2019)

All error bounds hold with overwhelming or $\zeta$-high probability. $\Psi(z)$ denotes the typical control parameter $\Psi(z)=\sqrt{\Im m_{sc}(z)/N\eta+1/N\eta}$. Further details on conditions, technical hypotheses, and exceptions are found in the corresponding references.

• Isotropic semicircle law and deformation: (Knowles et al., 2011)
• Local semicircle law for random regular and sparse graphs: (Bauerschmidt et al., 2015, Erdős et al., 2011)
• General Wigner and variance profile matrices: (Benaych-Georges et al., 2016, Erdos et al., 2012, Ajanki et al., 2013)
• Local law in bulk and edge for $\beta$-ensembles: (Wong, 2011, Sosoe et al., 2011, Bao et al., 2011)
• Moment assumptions and fourth moment law: (Götze et al., 2016, Götze et al., 2015, Götze et al., 2015, Götze et al., 2019)
• Fourfold symmetry and correlated models: (Alt, 2015, Fleermann et al., 2019)
• Deterministic local law: (Anderson, 2013)
• Deformed Wigner: (Lee et al., 2013)

These results collectively constitute the modern, highly quantitative understanding of the local spectral statistics for large random matrix ensembles and their deterministic analogues.