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Dyson Trace Flow and Multivariate Dynamic Coupled Semicircle Law

Cong Chen, Yong Li

TL;DR

The work introduces the Dyson Trace Flow as a macroscopic stochastic descriptor for eigenvalue dynamics and develops the Multivariate Dynamic Coupled Semicircle Law to describe the evolution of spectral measures under asymmetrical coupling. It establishes well-posedness, large-deviation principles, and a dynamic Burgers-type description for the limiting spectral laws, both in binary and multivariate settings. By incorporating nonlinear and non-reciprocal couplings, the framework reveals universal phenomena such as exceptional points, phase transitions, and holographic connections to quantum gravity, while offering concrete pathways to applications in neural networks, financial networks, and quantum many-body systems. The results unify stochastic analysis, potential theory, and large deviation theory to provide exactly solvable models for interacting random matrices with broad cross-disciplinary relevance and predictive power for complex systems.

Abstract

Interacting random matrix systems are fundamental to modern theoretical physics and data science, yet a unified framework for their analysis has been lacking. This work introduces such a universal framework, built upon two novel concepts: the Dyson Trace Flow characterizing macroscopic fluctuations, and the Multivariate Dynamic Coupled Semicircle Law describing the collective spectral behavior of multiple interacting matrix processes. We establish the stochastic evolution of eigenvalues under asymmetric coupling and prove the mathematical well-posedness of the theory. A large deviation principle is derived, enabling the calculation of rare event probabilities. The framework is extended to nonlinear and non-reciprocal interactions, revealing universal phenomena including exceptional points, bistability, and novel scaling laws. A striking connection to quantum chaos is unveiled through a holographic correspondence with wormhole geometries. By generalizing classical random matrix theory, this work provides powerful tools for understanding neural networks and complex quantum dynamics.

Dyson Trace Flow and Multivariate Dynamic Coupled Semicircle Law

TL;DR

The work introduces the Dyson Trace Flow as a macroscopic stochastic descriptor for eigenvalue dynamics and develops the Multivariate Dynamic Coupled Semicircle Law to describe the evolution of spectral measures under asymmetrical coupling. It establishes well-posedness, large-deviation principles, and a dynamic Burgers-type description for the limiting spectral laws, both in binary and multivariate settings. By incorporating nonlinear and non-reciprocal couplings, the framework reveals universal phenomena such as exceptional points, phase transitions, and holographic connections to quantum gravity, while offering concrete pathways to applications in neural networks, financial networks, and quantum many-body systems. The results unify stochastic analysis, potential theory, and large deviation theory to provide exactly solvable models for interacting random matrices with broad cross-disciplinary relevance and predictive power for complex systems.

Abstract

Interacting random matrix systems are fundamental to modern theoretical physics and data science, yet a unified framework for their analysis has been lacking. This work introduces such a universal framework, built upon two novel concepts: the Dyson Trace Flow characterizing macroscopic fluctuations, and the Multivariate Dynamic Coupled Semicircle Law describing the collective spectral behavior of multiple interacting matrix processes. We establish the stochastic evolution of eigenvalues under asymmetric coupling and prove the mathematical well-posedness of the theory. A large deviation principle is derived, enabling the calculation of rare event probabilities. The framework is extended to nonlinear and non-reciprocal interactions, revealing universal phenomena including exceptional points, bistability, and novel scaling laws. A striking connection to quantum chaos is unveiled through a holographic correspondence with wormhole geometries. By generalizing classical random matrix theory, this work provides powerful tools for understanding neural networks and complex quantum dynamics.

Paper Structure

This paper contains 50 sections, 28 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Dyson Trace Flow
  3. Definition and Existence Theorem
  4. Liouville's Theorem for DBM
  5. Extension to Coupled Systems
  6. Key Properties of Dyson Trace Flow
  7. Eigenvalue Dynamics of Asymmetrically Coupled Systems
  8. Setup and Main Theorem
  9. Large-N Limit
  10. Dynamic Coupled Semicircle Law
  11. Model Setup and Assumptions
  12. Main Theorem
  13. Proof of Theorem \ref{['th:Dynamic Coupled Semicircle Law']}
  14. Proof of main results
  15. Proof of Theorem \ref{['thm:multivariate-dynamic-coupled-semicircle']}
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $H_1(t), H_2(t), \ldots, H_k(t)$ be $k$ coupled $N \times N$ real symmetric (or complex Hermitian) matrix-valued processes satisfying the following system of stochastic differential equations: where $B_p(t)$ are matrix-valued Brownian motions with the correlation structure: and $\rho_{pp} = 1$. The initial conditions $H_p(0)$ are deterministic symmetric matrices whose empirical spectral dist

Theorems & Definitions (63)

  • Theorem 1.1: Multivariate Dynamic Coupled Semicircle Law
  • Definition 1: Symmetric/Hermitian Brownian motion
  • Definition 2: Dyson Trace Flow
  • Theorem 2.1: Existence and Uniqueness of Dyson Trace Flow
  • Theorem 2.2: Liouville's Theorem for SDEs with Additive Noise
  • Corollary 2.3: Trace-Volume Consistency
  • Theorem 2.4: Coupled Dyson Trace Flow
  • Remark 1
  • Definition 3: Asymmetrically Coupled Matrix OU Processes
  • Definition 4: Eigenvalue Processes
  • ...and 53 more