Metz F L, Stariolo Daniel A
Departamento de Física, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Brazil.
Departamento de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042153. doi: 10.1103/PhysRevE.92.042153. Epub 2015 Oct 28.
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P(N)(K,λ) that a large N×N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P(N)(K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N≫1 for |λ|>0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as lnN, with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.
使用复制方法，我们开发了一种解析方法来计算概率(P^{(N)}(K,\lambda))的特征函数，其中(P^{(N)}(K,\lambda))表示稀疏随机图的大(N\times N)邻接矩阵有(K)个特征值低于阈值(\lambda)的概率。该方法原则上允许确定(P^{(N)}(K,\lambda))的所有矩，从中可以完全表征典型的样本间波动。对于具有局域化特征向量的随机图模型，我们表明，对于(|\lambda|>0)，指标方差随(N\gg1)线性缩放，其与模型相关的前置因子可以精确计算。文中讨论了厄多斯 - 雷尼随机图和正则随机图的确切结果，两者都表现出前置因子作为(\lambda)的函数具有非单调行为。这些结果与旋转不变随机矩阵形成对比，在旋转不变随机矩阵中，指标方差仅按(\ln N)缩放，具有与(\lambda)无关的通用前置因子。数值对角化结果证实了我们方法的准确性，此外，还强烈支持指标波动的高斯性质。
