GF（q）上稀疏随机矩阵的典型核大小和个数--一种统计物理方法
Typical kernel size and number of sparse random matrices over GF(q) - a statistical physics approach
摘要：
本文利用统计物理的方法，研究了GF（q）上一般稀疏随机矩阵在给定连通性轮廓下，在大矩阵极限下的平均个数和核大小.我们引入了一个映射的$GF(q)$矩阵到自旋系统使用的表示的循环群的q阶单位的q次复根。这种表示有利于推导的平均核大小的随机矩阵使用副本的方法，副本对称anomaly下，导致鞍点方程一般的连接分布。数值解，然后获得特定情况下的人口动力学。类似的技术也使我们能够获得一个表达式的精确和平均数的随机矩阵的任何一般的连接配置文件。我们提出了特定分布的数值结果。
英文摘要：
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.
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