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Singular-potential random-matrix model arising in mean-field glassy systems.

Gernot Akemann1, Dario Villamaina2, Pierpaolo Vivo3

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Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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This study introduces a new random matrix model with a single pole potential, finding applications in glassy systems. The research reveals a two-interval spectral density for mean-field glassy systems, confirmed by simulations.

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Area of Science:

  • Statistical Mechanics
  • Quantum Chaos
  • Number Theory

Background:

  • Random matrix theory is crucial in understanding complex systems.
  • Previous work explored potentials with first- and second-order poles in quantum chaos and number theory.
  • Mean-field glassy systems present unique challenges in statistical mechanics.

Purpose of the Study:

  • To investigate an invariant random matrix ensemble with a distorted Gaussian potential featuring a single pole.
  • To apply this model to mean-field glassy systems.
  • To derive and solve the loop equation in the planar limit for this class of potentials.

Main Methods:

  • Derivation and solution of the loop equation in the planar limit.
  • Analysis of potentials with arbitrary order single poles.
  • Imposition of the traceless matrices constraint.
  • Consideration of both repulsive and attractive poles.
  • Detailed analysis of a second-order pole for a zero-trace model.

Main Results:

  • The macroscopic spectral density for mean-field glassy systems is supported on two disconnected intervals.
  • Edge points of the spectral density are determined by the traceless matrices constraint.
  • A one-cut solution for unbounded potentials with attractive poles is ruled out by the traceless condition.
  • The standard semicircle is recovered in the limit of a vanishing pole.
  • Results are valid for orthogonal, unitary, and symplectic invariant matrices in the planar limit.

Conclusions:

  • The developed random matrix model provides a novel framework for studying mean-field glassy systems.
  • The two-interval spectral density is a key characteristic of these systems under the considered potential.
  • The findings are robust, as confirmed by numerical simulations and independent analytical calculations.