London South Bank University, SE1 0AA, United Kingdom.
GRAPES, rue de la belle jardinière 483/002, B-4031 Liège Angleur Sart-Tilman, Belgium.
Phys Rev E. 2016 Jun;93(6):062115. doi: 10.1103/PhysRevE.93.062115. Epub 2016 Jun 9.
Nowadays, strict finite size effects must be taken into account in condensed matter problems when treated through models based on lattices or graphs. On the other hand, the cases of directed bonds or links are known to be highly relevant in topics ranging from ferroelectrics to quotation networks. Combining these two points leads us to examine finite size random matrices. To obtain basic materials properties, the Green's function associated with the matrix has to be calculated. To obtain the first finite size correction, a perturbative scheme is hereby developed within the framework of the replica method. The averaged eigenvalue spectrum and the corresponding Green's function of Wigner random sign real symmetric N×N matrices to order 1/N are finally obtained analytically. Related simulation results are also presented. The agreement is excellent between the analytical formulas and finite size matrix numerical diagonalization results, confirming the correctness of the first-order finite size expression.
如今，在通过基于晶格或图的模型处理凝聚态问题时，必须考虑严格的有限大小效应。另一方面，已知有向键或链路的情况在从铁电体到报价网络的各种主题中非常相关。将这两点结合起来，就会导致我们研究有限大小随机矩阵。为了获得基本材料特性，必须计算与矩阵相关的格林函数。为了获得第一个有限大小修正，可以在此框架内通过复制方法发展微扰方案。最后，我们以解析方式获得了 Wigner 随机符号实对称 N×N 矩阵的阶数为 1/N 的平均特征值谱和相应的格林函数。还呈现了相关的模拟结果。解析公式和有限大小矩阵数值对角化结果之间的一致性非常好，这证实了一阶有限大小表达式的正确性。
