Long-Range Correlated Random Matrices

Abstract: Motivated by the importance ascribed to correlations in random matrices used to model phenomena in various scientific disciplines, we report how algebraic correlations between matrix elements affect the eigenvalue statistics and spectral density of random matrices. These correlations, introduced through a long-range correlated percolation model, decay as a power law $\propto r^{-2H}$, with exponent $H > 0$. As $H$ varies, both the eigenvalue distribution and excess kurtosis undergo qualitative changes. At the threshold $H_c = 3/4$, characterized by emergent Gaussian statistics, a sign change in excess kurtosis marks a transition from a fat-tailed generalized $t$-distribution to one that gradually approaches the standard semicircle law for $H \gg H_c$. Our analytical results, based on scaling analysis and supported by extensive numerical simulations, provide clear predictions and uncover novel spectral regimes in random matrix theory. Our results connect techniques from statistical physics, percolation theory, and random matrix analysis, offering a new perspective on universality in correlated ensembles.

• The paper introduces a novel random matrix ensemble with power-law decaying correlations, identifying spectral phase transitions at H = 1/4 and H = 3/4.
• It employs analytical derivations and numerical verifications to detail how variance and excess kurtosis scale with matrix size and correlation strength.
• The study establishes distinct universality classes with implications for quantum chaos, complex networks, and related physical systems.

Spectral Regimes and Phase Transitions in Long-Range Correlated Random Matrices

Introduction and Motivation

The statistical properties of random matrices have profound implications across statistical physics, quantum chaos, neuroscience, complex networks, and financial systems. While standard Random Matrix Theory (RMT) typically assumes independent or short-range correlated entries, numerous systems of practical interest display spatially structured, long-range correlations among their constituent degrees of freedom. This work develops and analyzes a non-invariant, physically motivated random matrix ensemble where matrix entries possess tunable algebraic correlations decaying as a power law with exponent $H$, studying both analytical predictions and systematic numerical verifications (2604.22447).

Model Construction and Analytical Predictions

The investigated ensemble is generated by thresholding a correlated Gaussian field on a square lattice, leading to binary, zero-mean symmetric matrices $\mathcal{M}$ with spatial correlations $C(r) \sim r^{-2H}$ among the entries. This procedure ensures the ensemble incorporates both geometric structure and spatial correlations, distinguishing it from conventional invariant random matrix ensembles or those with solely heavy-tailed entry distributions.

The central analytical advancement is the derivation of how the spectral properties of the ensemble (e.g., bulk density, excess kurtosis, and tail behavior) depend explicitly on the exponent $H$ controlling correlation decay. The main results delineate three spectral regimes separated by critical values of $H$:

1. Subcritical regime ($0 \leq H < 1/4$): The eigenvalue distribution is a family of generalized $t$-distributions with $\text{f} \le 4$, producing divergent excess kurtosis in the thermodynamic limit ($L \rightarrow \infty$).
2. Intermediate regime ($1/4 < H < 3/4$): The exponent $\mathcal{M}$0 increases monotonically with $\mathcal{M}$1 and the eigenvalue distribution's tails are less pronounced; the excess kurtosis is positive but finite.
3. Critical point ($\mathcal{M}$2): The system undergoes a qualitative transition to Gaussian spectral statistics, with $\mathcal{M}$3, and the excess kurtosis vanishes.
4. Supercritical regime ($\mathcal{M}$4): The effect of correlations vanishes, and the spectral density transitions to the bounded semicircle law characteristic of standard RMT.

Explicitly, the family of one-point eigenvalue distributions is given by:

$\mathcal{M}$5

with the $\mathcal{M}$6-dependent exponent

$\mathcal{M}$7

Scaling Results and Numerical Verification

Theoretical moment analysis yields the scaling of the variance and excess kurtosis (EK) of the eigenvalue distribution as functions of matrix size $\mathcal{M}$8 and correlation exponent $\mathcal{M}$9. For $C(r) \sim r^{-2H}$0, the variance correction due to correlations vanishes as $C(r) \sim r^{-2H}$1, while the EK displays a transition from divergence ($C(r) \sim r^{-2H}$2) to finite values ($C(r) \sim r^{-2H}$3), marking heavier spectral tails in the former regime.

Numerical experiments robustly corroborate these predictions.

Figure 1: Simulated eigenvalue variance as a function of $C(r) \sim r^{-2H}$4 for multiple $C(r) \sim r^{-2H}$5, verifying theoretical scaling $C(r) \sim r^{-2H}$6.

The measured excess kurtosis for a range of $C(r) \sim r^{-2H}$7 values exactly follows analytical expressions. At $C(r) \sim r^{-2H}$8, EK exhibits logarithmic scaling with $C(r) \sim r^{-2H}$9, in direct agreement with analytical predictions.

Figure 2: Excess kurtosis of eigenvalues as a function of $H$0 for different $H$1, showing the crossover from divergent to finite regime.

The entire eigenvalue probability density function (PDF) shows strong finite-size collapse onto predicted analytical forms across matrix sizes and correlation exponents. At $H$2, the distribution converges to Gaussian, confirming the identified spectral phase transition. For $H$3, numerical distributions converge to the semicircle law, as expected from standard RMT.

Figure 3: Empirical PDF $H$4 for scaled eigenvalues across $H$5; the crossover from heavy-tailed to Gaussian to semicircle distributions is clearly observed.

Universality, Spectral Rigidity, and Broader Implications

The study establishes distinct universality classes for random matrices with spatial, algebraically correlated entries and highlights both the mechanism and scaling of the breakdown of standard RMT universality due to geometric correlations. In contrast to invariant or superstatistical deformations that insert heavy tails directly into the entry distribution, here the emergence of heavy-tailed, non-Gaussian spectra is a consequence of real-space correlation geometry.

The authors further emphasize that, beyond one-point statistics, the spectral two-point correlators and eigenvalue rigidity properties present additional structure. An identified rigidity threshold at $H$6 distinguishes the sublinear spectral-form-factor ramp and stretched-exponential gap distribution from Dyson-like statistics, with the latter fully recovered only for $H$7 in the bulk.

The theoretical framework outlined in this paper has ramifications for a variety of structured physical systems: for instance, deviations from RMT due to long-range correlations in quantum chaotic spin chains, EEG/brain connectivity networks, photonic and electronic lattice systems, and many complex networks where spatial effects are non-negligible.

Conclusion

This work provides a systematic theoretical and numerical analysis of random matrices with tunable long-range power-law correlations in their elements. The critical finding is the existence of two sharp phase transitions in the eigenvalue distribution—first in the excess kurtosis at $H$8, and then to Gaussian bulk statistics at $H$9, followed by crossover to the semicircle law for very weak correlations. The results sharpen understanding of universality in RMT for spatially structured systems and set a concrete baseline for interpreting non-RMT spectral phenomena in correlated physical and complex network models. Future research directions include detailed investigation of higher-order spectral correlations, the thermodynamic behavior of the extreme eigenvalues, and applications in statistically correlated, high-dimensional data settings.