Physics – Condensed Matter – Other Condensed Matter
Scientific paper
2008-03-14
Physics
Condensed Matter
Other Condensed Matter
16 pages, 3 figures
Scientific paper
10.1103/PhysRevE.77.061123
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.
Alamino Roberto C.
Saad David
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