Typical kernel size and number of sparse random matrices over GF(q) - a statistical physics approach

TL;DR

This paper applies statistical physics methods to analyze the average kernel size and number of sparse random matrices over GF(q), using a spin system mapping and replica approach to derive analytical and numerical results.

Contribution

It introduces a novel spin system mapping for GF(q) matrices and derives saddle point equations for average kernel size and matrix count using replica symmetry.

Findings

Derived saddle point equations for kernel size and matrix count.

Numerical solutions obtained for specific connectivity distributions.

Provided analytical expressions for average number of matrices with general connectivity.

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.