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Eikonal formulation of large dynamical random matrix models.

Jacek Grela1, Maciej A Nowak1,2, Wojciech Tarnowski1

  • 1Institute of Theoretical Physics, Jagiellonian University, 30-348 Cracow, Poland.

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Summary
This summary is machine-generated.

We introduce Hamilton-Jacobi dynamics for random matrix models, unifying normal and non-normal dynamics. This approach aids in calculating complex integrals for Brownian bridge dynamics.

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Area of Science:

  • Mathematical Physics
  • Statistical Mechanics

Background:

  • Dynamical random matrix models typically focus on eigenvalue trajectories.
  • Existing methods lack a unified framework for diverse matrix dynamics.

Purpose of the Study:

  • To develop a unified Hamilton-Jacobi dynamics for large random matrix models.
  • To extend the description to both normal and non-normal dynamics.

Main Methods:

  • Utilizing an analogy from optics, specifically the duality between Fermat and Huygens principles.
  • Formulating Hamilton-Jacobi equations for random matrix dynamics.

Main Results:

  • The derived equations provide a unified description for a wide range of random matrix models.
  • The formalism successfully incorporates both normal (Hermitian, unitary) and strictly non-normal dynamics.
  • Application to Brownian bridge dynamics enables calculation of Harish-Chandra-Itzykson-Zuber integral asymptotics.

Conclusions:

  • The Hamilton-Jacobi approach offers a powerful, unified framework for studying random matrix dynamics.
  • This method simplifies the analysis of complex systems and integral calculations.