We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

中文翻译：

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稀疏对称随机矩阵谱密度的腔法和复制法

我们回顾了如何计算稀疏对称随机矩阵的谱密度的问题，即无向图的加权邻接矩阵。从 Edwards-Jones 公式开始，我们说明了这一研究领域的里程碑，包括 Bray 和 Rodgers 使用复制品的开创性工作。我们首先关注空腔方法，表明它可以快速为单个实例和集成级别提供正确的递归方程。我们还描述了一种替代的复制解决方案，它被证明与腔法等效。腔和副本推导都允许我们通过求解辅助概率密度函数的积分方程来获得谱密度。我们表明可以使用随机种群动力学算法求解该方程，我们提供了它的实现。在这种形式中，谱密度自然地用来自给定度数的节点的局部贡献的叠加来写，其作用被彻底阐明。本文不包含原始材料，而是对该主题进行了教学概述。它确实是针对考虑进入该领域的学生和研究人员的。详细讨论了理论工具和数值算法，突出了概念的微妙之处和实践方面。它确实是针对考虑进入该领域的学生和研究人员的。详细讨论了理论工具和数值算法，突出了概念的微妙之处和实践方面。它确实是针对考虑进入该领域的学生和研究人员的。详细讨论了理论工具和数值算法，突出了概念的微妙之处和实践方面。