一类对称随机稀疏矩阵第一本征值问题的腔方法
Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices
摘要：
利用统计力学中的腔方法，研究了大型对称随机稀疏矩阵的第一（最大）特征值及其特征向量的性质.在树近似下，这是合理的无限大的系统，结合引入拉格朗日乘子约束的特征向量的长度，本征值问题减少到一堆最优化问题的一个单一的变量的二次函数，和系数的第一和第二次项的功能作为腔场处理在腔分析。我们表明，第一本征值是确定在这样一种方式，即腔场的分布有一个有限的值的二阶矩相对于腔场的一阶系数。所开发的方法的有效性和实用性进行检查，将其应用到两个解析可解和一个简单的，但不平凡的例子，结合数值理由。
英文摘要：
A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.
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