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Spectral statistics of permutation matrices.

Idan Oren1, Uzy Smilansky

  • 1Department of Physics of Complex Systems, Weizmann Institute of Science, , Rehovot 76100, Israel.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|December 18, 2013
PubMed
Summary
This summary is machine-generated.

We computed spectral properties of large permutation matrices, linking them to number theory

Keywords:
permutation matricesrandom matrix theoryspectral statistics

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Area of Science:

  • Mathematics
  • Number Theory
  • Statistical Mechanics

Background:

  • Permutation matrices are fundamental objects in combinatorics and group theory.
  • Spectral properties of random matrices are crucial in understanding complex systems.

Purpose of the Study:

  • To compute the mean two-point spectral form factor and spectral number variance for large permutation matrices.
  • To establish a connection between spectral properties and number-theoretic functions.

Main Methods:

  • Utilized generalized divisor functions from number theory.
  • Applied classical number theory results to derive asymptotic expansions.
  • Analyzed the two-point correlation function of permutation matrices.

Main Results:

  • Derived expressions for the mean two-point spectral form factor and spectral number variance.
  • Identified leading and next-to-leading terms in the asymptotic expansion.
  • Expressed the two-point correlation function using generalized divisor functions.

Conclusions:

  • Provided a new perspective on the spectral properties of permutation matrices.
  • Improved upon existing results in the asymptotic analysis of these properties.
  • Demonstrated the utility of number-theoretic tools in analyzing spectral statistics.