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Integrable matrix theory: Level statistics.

Jasen A Scaramazza1, B Sriram Shastry2, Emil A Yuzbashyan1

  • 1Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA.

Physical Review. E
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Summary
This summary is machine-generated.

Typical integrable matrices exhibit Poisson statistics in the large N limit when the number of integrals of motion scales with N. Deviations from this behavior are rare and localized in parameter space.

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Area of Science:

  • Quantum chaos
  • Mathematical physics
  • Statistical mechanics

Background:

  • Integrable matrix ensembles are characterized by commuting partners (integrals of motion).
  • Previous work established a basis-independent construction for these matrices.

Purpose of the Study:

  • To determine the level statistics of typical integrable matrices.
  • To identify conditions leading to Poisson statistics versus level repulsion.

Main Methods:

  • Analysis of probability density functions for integrable matrix ensembles.
  • Asymptotic analysis in the N→∞ limit.
  • Numerical evidence for stationarity and ergodicity.

Main Results:

  • Poisson statistics emerge when the number of integrals of motion (n) scales as logN.
  • Level repulsion occurs when n scales slower than logN.
  • Non-Poissonian statistics are confined to measure-zero subsets of parameter space.
  • Integrable matrix ensembles demonstrate stationarity and ergodicity.

Conclusions:

  • The spectral statistics of integrable matrices are predominantly Poissonian in the large N limit.
  • Deviations from Poisson statistics are exceptional and localized.
  • These findings have implications for understanding quantum chaos and random matrix theory.