Lax random matrices from Calogero systems

Abstract

We study a class of random matrices arising from the Lax matrix structure of classical integrable systems, particularly the Calogero family of models. Our focus is the density of eigenvalues for these random matrices. The problem can be mapped to analyzing the density of eigenvalues for generalized versions of conventional random matrix ensembles, including a modified form of the log-gas. The mapping comes from the underlying integrable structure of these models. Such deep connection is confirmed by extensive Monte-Carlo simulations. Thereby we move forward not only in terms of understanding such class of random matrices arising from integrable many-body systems, but also by providing a building block for the generalized hydrodynamic description of integrable systems.

24 pages, 6 figures

References in corpus (31)

• Localization of interacting fermions at high temperature
• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Generalized HydroDynamics on an Atom Chip
• Chaos, Complexity, and Random Matrices
• Physics and Mathematics of Calogero particles
• Lecture notes on Generalised Hydrodynamics
• Generalized hydrodynamics in strongly interacting 1D Bose gases
• Onset of Random Matrix Behavior in Scrambling Systems
• Dynamics of hard rods with initial domain wall state
• A geometric viewpoint on generalized hydrodynamics
• Emergent equilibrium in many-body optical bistability
• Spectral form factors and late time quantum chaos
• Generalized Hydrodynamics in the 1D Bose gas: theory and experiments
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Kardar-Parisi-Zhang Scaling for an Integrable Lattice Landau-Lifshitz Spin Chain
• Generalized Gibbs Ensembles of the classical Toda chain
• Universal spectral form factor for many-body localization
• A short introduction to Generalized Hydrodynamics
• Random matrix ensembles associated with Lax matrices
• The GGE averaged currents of the classical Toda chain
• Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation
• Hydrodynamic Equations for the Toda Lattice
• Soliton solutions of Calogero model in harmonic potential
• Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics
• New Integrable Systems from Unitary Matrix Models
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
• Discrete integrable systems and random Lax matrices
• Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory
• Long-ranged spectral correlations in eigenstate phases
• Particle Scattering and Fusion for the Ablowitz-Ladik Chain