Sparse Multi-Resolvent Local Law

• The paper establishes a probabilistic estimate for products of resolvents in sparse non-Hermitian matrices, quantifying deterministic scaling and fluctuations.
• It employs Hermitisation and deterministic quaternionic insertions to derive precise averaged and entrywise error bounds over varying spectral parameters.
• The analysis underpins bulk eigenvalue universality, offering a robust framework for understanding local eigenvalue statistics in sparse complex models.

The sparse multi-resolvent local law is a probabilistic estimate governing products of resolvents of Hermitised sparse complex non-Hermitian random matrices, with deterministic insertions from a specific quaternionic algebra, in the regime of slowly decaying moments. It precisely quantifies the fluctuations and deterministic scaling arising in linear and entrywise averages of these resolvent chains, crucial for proving universality of local eigenvalue statistics in the bulk for this class of sparse matrix ensembles (Osman, 5 Aug 2025).

1. Sparse Complex Matrix Model and Hermitisation

The foundational object is an $N \times N$ complex random matrix $X = (X_{ij})$ with independent entries such that $\mathbb{E} X_{ij} = \mathbb{E} X_{ij}^2 = 0$, $\mathbb{E} |X_{ij}|^2 = 1/N$, and for all integers $r > 2$,

$$\mathbb{E}|X_{ij}|^r \leq C_r N^{-1} q^{2 - r}, \quad q = N^\epsilon, \ \epsilon > 0.$$

Sparsity is realized, for example, with $X_{ij} = N^{-\epsilon/2} \xi_{ij} x_{ij}$, where $\xi_{ij}$ is Bernoulli($p$) with $p = N^{-1+\epsilon}$ and $x_{ij}$ are i.i.d.\ mean-zero complex random variables with unit variance.

For spectral analysis in the non-Hermitian case, the Hermitisation technique is employed: for $z \in \mathbb{C}$, define the $2N \times 2N$ Hermitised matrix

$$W_z = \begin{pmatrix} 0 & X-z \ (X-z)^* & 0 \end{pmatrix},$$

with resolvent $G_z(w) = (W_z - w)^{-1}$, $w \in \mathbb{C} \setminus \mathbb{R}$. In $2\times2$ block form: $$G_z(w) = \begin{pmatrix} wH_z(w) & (X-z)H_z(w) \ H_z(w)(X-z)^* & -wH_z(w) \end{pmatrix}, \qquad H_z(w) = (|X-z|^2 - w^2)^{-1}.$$

Deterministic $2N \times 2N$ matrices are drawn from the quaternionic algebra $\mathbb{H} = \mathrm{span}\{ E_+, E_-, F, F^* \}$, consisting of block-diagonal/anti-diagonal matrices built from $N \times N$ identity matrices:

• $E_+ = \mathrm{diag}(I_N, 0)$,
• $E_- = \mathrm{diag}(0, I_N)$,
• $$F = \begin{pmatrix}0 & I_N \ 0 & 0 \end{pmatrix}$$,
• $$F^* = \begin{pmatrix}0 & 0 \ I_N & 0 \end{pmatrix}$$.

2. Precise Statement of the Sparse Multi-Resolvent Local Law

Consider fixed integers $m\geq1$, $B_1,\dots,B_m \in \mathbb{H}$, spectral parameters $$w_j \in \mathsf{D}(\delta,\tau):= \{ w=E+i\eta : |E|\leq\delta|\eta|, N^{-1+\tau}\leq|\eta|\leq10 \}$$, and $z$ in the open unit disk. Define

$$a = \#\{j: B_j \in \{F, F^*\}\}, \qquad \eta = \min_j |\Im w_j|.$$

Write $G_j = G_z(w_j)$, and let $M_z(w_1,B_1,\dots,B_m)$ denote the explicit $2N \times 2N$ deterministic approximation.

The theorem provides, with probability $1 - O(N^{-D})$ for any $D>0$, two central estimates:

• Averaged Law:

$$\left\langle G_1B_1\cdots G_mB_m - M_z(w_1,B_1,\dots,B_m)B_m \right\rangle \;\prec\; \left( \frac{1}{N\eta} + \frac{1}{q} \right) \frac{1}{\eta^{m - a/2 - 1} \wedge 1 }.$$

• Entrywise Law:

$$\left| (G_1B_1\cdots G_{m+1})_{\mathfrak{x},\mathfrak{y}} - \left[ M_z(w_1,B_1,\dots,w_{m+1}) \right]_{\mathfrak{x},\mathfrak{y}} \right| \prec \left( \frac{1}{\sqrt{N\eta}} + \frac{1}{q} \right) \frac{1}{\eta^{m - a/2}}.$$

Here $\langle A \rangle = (2N)^{-1} \mathrm{Tr}\,A$, stochastic domination is denoted by $\prec$, and the normalization and deterministic scaling depend on $m$, $a$, and $\eta$.

The factors $1/q$ capture additional fluctuations due to sparsity, and the $\eta$-scaling reflects the chain length and presence of off-diagonal quaternionic insertions.

3. Key Error Bounds and Uniformity

The law asserts uniform control over all choices of $B_1,\dots,B_m \in \{E_+,E_-,F,F^*\}$ and all suitable spectral parameters. The precise error term is given by: $$\left\langle G_1B_1\cdots G_mB_m - M_z(w_1,B_1,\dots,B_m)B_m \right\rangle \prec \left( \frac{1}{N\eta} + \frac{1}{q} \right)\eta^{-(m-a/2-1)},$$

$$\max_{\mathfrak{x},\mathfrak{y}} \left| (G_1B_1\cdots G_{m+1})_{\mathfrak{x},\mathfrak{y}} - [M_z(\cdot)]_{\mathfrak{x},\mathfrak{y}} \right| \prec \left( \frac{1}{\sqrt{N\eta}} + \frac{1}{q} \right)\eta^{-(m-a/2)}.$$

These bounds are effective down to the local spectral scale $\eta \geq N^{-1+\tau}$, reflecting true local statistics in the bulk.

4. Proof Strategy and Technical Ingredients

The proof follows a three-stage "Zigzag" argument structure developed by Cipolloni–Erdős–Schröder, specialized to the moment and sparsity regime at hand:

• Global Regime: For large $\Im w$, trivial operator-norm bounds and recursive moment-estimates yield the law.
• Characteristic Flow ("Zig"): Matrix $X$ is coupled to a Gaussian reference via an Ornstein-Uhlenbeck SDE, while spectral parameters are deformed, ensuring that the deterministic approximation $M_t$ matches the drift. The local law is propagated down to small $\eta$ along the flow.
• Removal of Gaussian Component ("Zag"): A comparison argument using cumulant expansions in the sparsity parameter eliminates the Gaussian part, with each use of a higher cumulant gaining a $1/q$ factor. The method harnesses Ward identities for resolvent sums, Cauchy–Schwarz for off-diagonal smallness, and spectral decomposition to reduce long chains.

This strategy leverages the fact that, for $\epsilon > 0$, moment decay suffices to truncate cumulant expansions, controlling deviations due to sparse structure and slow tail decay (Osman, 5 Aug 2025).

5. Applications and Corollaries

The main application is the proof of bulk universality for eigenvalue statistics of sparse complex non-Hermitian matrices. By combining the multi-resolvent law with Girko’s formula and Lindeberg replacement techniques, the result yields universality under only $4+\epsilon$ moment (with truncation) or in the sparse Bernoulli-product model. Extensions include:

• Handles products of resolvents with more general deterministic insertions from $\mathbb{H}$, enabling access to questions of eigenvector overlap and correlations of multi-resolvent traces.
• Higher-moment and fluctuation bounds for resolvent chains via refinements of the cumulant expansion.

A plausible implication is that such local laws, together with Hermitisation, may serve as a template for further extensions to even sparser regimes or non-i.i.d.\ random matrix models.

6. Connections and Further Developments

The sparse multi-resolvent local law unifies and strengthens prior approaches to bulk universality in non-Hermitian random matrix theory (see also (Osman, 5 Aug 2025), Cipolloni–Erdős–Schröder, He 2023, Maltsev–Osman PTRF 2024, Tao–Vu 2008). Its mechanism is sharply tailored for slow moment decay and sparse connectivity, advancing beyond single-resolvent or Hermitian-only results. Its scope naturally extends to random matrix models governed by Bernoulli-percolation and i.i.d.\ structure, providing comprehensive entries to the universality classification in complex, sparse settings.