Dynamic Coupled Semicircle Law

• Dynamic Coupled Semicircle Law is a generalization of the Wigner semicircle law to coupled random matrices, integrating microscopic SDEs with macroscopic spectral observables.
• It establishes coupled nonlinear PDEs and eigenvalue SDEs with asymmetric drift, providing explicit large deviation bounds for spectral fluctuations.
• The framework applies to high-dimensional systems in neural networks, quantum chaos, and complex systems, revealing new universality classes and holographic correspondences.

The Dynamic Coupled Semicircle Law generalizes the classical Wigner semicircle law to interacting random matrix systems, establishing a rigorous framework for describing the spectral evolution and fluctuation structure of coupled ensembles under stochastic dynamics (Chen et al., 24 Sep 2025). This theory combines microscopic stochastic differential equations (SDEs) for eigenvalues with macroscopic trace and spectral flow observables, quantifies their large deviation behavior, and introduces new universality classes and physical correspondences—especially in high-dimensional neural, quantum, and multicomponent systems.

1. Conceptual Framework and Law Statement

The classical Wigner semicircle law states that the empirical spectral density of a Hermitian $N \times N$ Wigner matrix converges, as $N \to \infty$, to the semicircle distribution:

$$\rho_{\mathrm{sc}}(\lambda) = \frac{1}{2\pi} \sqrt{4 - \lambda^2}\, \mathbb{I}_{[-2,2]}(\lambda)$$

The Dynamic Coupled Semicircle Law extends this paradigm to systems of two (or more) coupled random matrices—$H_1(t)$ and $H_2(t)$—evolving as matrix-valued Ornstein–Uhlenbeck processes with drift coupling at rates $\gamma_{12}$ and $\gamma_{21}$. The empirical spectral measures $\mu_t^{(1)}$, $\mu_t^{(2)}$ and their associated Stieltjes transforms $G_t^{(k)}(z)$ are shown to satisfy coupled nonlinear partial differential equations (generalized Burgers-type):

$$\partial_t G_t^{(k)}(z) + G_t^{(k)}(z)\partial_z G_t^{(k)}(z) + \gamma_{k\ell} \partial_z G_t^{(\ell)}(z) = 0$$

for $k=1,2$ and $k\neq \ell$, supplemented by an initial condition corresponding to the spectral measure at $t=0$. In the limit $\gamma_{12},\gamma_{21}\to 0$, these equations decouple and revert to the single-matrix semicircle law.

2. Dyson Trace Flow: Macroscopic Description

The Dyson Trace Flow ("DTF", Editor's term) isolates the collective drift and fluctuation of matrix traces as global observables for the coupled system:

$\tau_k(t) = \mathrm{Tr}\, H_k(t)$

For a single matrix, the trace follows an Ornstein–Uhlenbeck SDE:

$$d\tau_k(t) = \sqrt{2}\, dB_k(t) - \frac{1}{2} \tau_k(t) dt$$

with stationary mean decaying exponentially and a computable covariance. In the coupled scenario:

$$\begin{aligned} d\tau_1(t) &= \sqrt{2}\, dW_1(t) - \frac{1}{2}\tau_1(t) dt + \gamma \tau_2(t) dt\ d\tau_2(t) &= \sqrt{2}\, dW_2(t) - \frac{1}{2}\tau_2(t) dt + \gamma \tau_1(t) dt \end{aligned}$$

where $W_1, W_2$ are correlated Brownian motions. The joint process is stationary Gaussian, and its covariance matrix is explicit in $\gamma$ and the Brownian correlation $\rho$. These macroscopic equations provide the basis for analyzing aggregate energy transport and non-equilibrium phenomena in coupled random matrix systems.

3. Eigenvalue SDEs and Asymmetric Coupling

At the microscopic (spectral) level, under the assumption of simultaneous diagonalizability (asymptotically justified for large $N$), the eigenvalues $\lambda_i^{(1)}$ and $\lambda_i^{(2)}$ of $H_1, H_2$ evolve according to

$$\begin{aligned} d\lambda_i^{(1)} &= \frac{1}{\sqrt{N}} dW_{1,i} - \frac{1}{2} \lambda_i^{(1)} dt + \gamma_{12} \lambda_i^{(2)} dt + \frac{1}{N}\sum_{j\neq i}\frac{1}{\lambda_i^{(1)} - \lambda_j^{(1)}} dt\ d\lambda_i^{(2)} &= \frac{1}{\sqrt{N}} dW_{2,i} - \frac{1}{2} \lambda_i^{(2)} dt + \gamma_{21} \lambda_i^{(1)} dt + \frac{1}{N}\sum_{j\neq i}\frac{1}{\lambda_i^{(2)} - \lambda_j^{(2)}} dt \end{aligned}$$

The coupling manifests as linear drift terms, modifying standard Dyson Brownian motion. Repulsion is preserved via the usual $1/(\lambda_i - \lambda_j)$ term. The existence and uniqueness of strong solutions to these SDEs ensures that the eigenvalue vectors remain in the open Weyl chamber (no collisions), facilitating the subsequent analysis of spectral evolution and large deviation behavior.

4. Large Deviation Principle and Rate Functions

A central contribution is the establishment of a large deviation principle (LDP) for both traces and spectral empirical measures. For the traces, the probability that $\tau_1/N \approx x$, $\tau_2/N \approx y$ scales as

$$\mathbb{P}\left((\tau_1/N, \tau_2/N) \approx (x,y)\right) \asymp \exp\{-N^2 I(x,y)\}$$

with $I(x,y)$ a quadratic rate function (whose coefficients may be extracted from the inverse covariance computed in the Dyson Trace Flow). For the spectral measures, the LDP is characterized via a variational principle over distribution-valued trajectories, with the action determined by the kinetic (from Brownian increments) and dynamic (from drift and coupling) terms; the minimizer recovers the coupled semicircle law.

At the level of Stieltjes transforms, the dynamic equations for $G_t^{(k)}(z)$ play the role of Euler–Lagrange equations in the LDP, connecting macroscopic spectral transport to microscopic fluctuations.

5. Extensions to Nonlinear and Non-Reciprocal Systems

The theory admits extension to coupled random matrix flows with nonlinear and non-reciprocal interactions:

• Nonlinear drift: Inclusion of higher-order coupling (e.g. cubic) in the drift terms leads to new dynamical phenomena. For example, the SDEs admit bistable equilibria, and the system may exhibit hysteresis—spectral configurations depend on prior history.
• Non-reciprocal coupling: Asymmetric rates $\gamma_{12} \neq \gamma_{21}$ render the dynamical generator non-normal, introducing transient growth, sensitivity to initial conditions, and the emergence of exceptional points where eigenvalues and eigenvectors coalesce.
• Novel scaling laws: For non-reciprocal and nonlinear systems, the scaling of fluctuations and transition rates are modified, and the spectral gap may scale anomalously with $N$ or with the coupling parameters.

These generalizations increase the relevance of the Dynamic Coupled Semicircle Law to practical applications in neuroscience (interacting neural circuits), finance (multi-asset covariance models), and statistical physics.

6. Holographic Correspondence and Quantum Chaos

A foundational aspect of the work is the connection to holography and quantum chaos:

• The coupled random matrix dyamics can be mapped onto dual AdS wormhole geometries. The degree of coupling corresponds to the wormhole throat parameter, and transitions are visible in the spectral form factor, which exhibits the "dip–ramp–plateau" structure typical of chaotic systems.
• Such a holographic correspondence provides a rigorous link between random matrix theory and maximal chaos as in the Sachdev–Ye–Kitaev (SYK) models.

This establishes a dictionary for interpreting spectral transitions and energy exchange in terms of geometric features of the dual gravitational theory, with implications for black hole information and out-of-time-order correlators.

7. Applications and Significance

The Dynamic Coupled Semicircle Law offers new analytical and probabilistic tools for exploring high-dimensional, interacting systems, including but not limited to:

• Neural networks and intelligent systems: Understanding spectral stability, cross-talk, and information propagation in multilayered or interacting neural architectures.
• Quantum dynamics and statistical mechanics: Modeling open systems, energy transfer, and decoherence in coupled many-body quantum contexts.
• Complex system engineering: Quantifying risk, correlation, and extreme events in multi-agent or multi-asset domains.

The explicit derivation of coupled Burgers-type equations for spectral measures, existence and uniqueness results for eigenvalue flows, and rigorous large deviation bounds constitute a universal foundation for future work on dynamic, multi-component random matrix models (Chen et al., 24 Sep 2025).