Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

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Abstract

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.
Original languageEnglish
Article number061123
Pages (from-to)1-12
Number of pages12
JournalPhysical Review E
Volume77
Issue number6
DOIs
Publication statusPublished - 17 Jun 2008

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Copyright of the American Physical Society.

Keywords

  • random matrices
  • Galois fields
  • statistical mechanics
  • replica theory

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