Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational spectra of disordered crystals, introduced many years ago by Dyson and Schmidt, are inapplicable for highly connected disordered systems. In this paper we apply random matrix theory (RMT) to calculate the averaged vibrational spectra of such systems, in the limit of infinitely large system size. At the heart of our analysis lies a calculation of the average spectrum of the product of two positive definite random matrices by means of free probability theory techniques. We also show that this problem is intimately related with quasi-hermitian random matrix theory (QHRMT), which means that the ‘hamiltonian’ matrix is hermitian with respect to a non-trivial metric. This extends ordinary hermitian matrices, for which the metric is simply the unit matrix. The analytical results we obtain for the spectrum agree well with our numerical results. The latter also exhibit oscillations at the high-frequency band edge, which fit well the Airy kernel pattern. We also compute inverse participation ratios of the corresponding amplitude eigenvectors and demonstrate that they are all extended, in contrast with conventional disordered crystals. Finally, we compute the thermodynamic properties of the system from its spectrum of vibrations. In addition to matrix model analysis, we also study the vibrational spectra of various multi-segmented disordered pendula, as concrete realizations of highly connected mechanical systems. A universal feature of the density of vibration modes, common to both pendula and the matrix model, is that it tends to a non-zero constant at vanishing frequency.

具有高连通性的无序机械系统所代表的局限性与更常见的无序晶体情况相反。晶体中的各个离子基本上会发生最近邻相互作用。相反，本文研究的系统的所有自由度相互耦合。因此，与与无序晶体相关的稀疏矩阵相反，这种系统的线性化小振荡问题涉及两个完全正定和非交换矩阵。因此，由戴森（Dyson）和施密特（Schmidt）于多年前提出的用于确定无序晶体的平均振动谱的熟悉方法不适用于高度连接的无序系统。在本文中，我们应用随机矩阵理论（RMT）来计算此类系统的平均振动谱，在无限大的系统规模的限制。我们分析的核心是通过自由概率理论技术计算两个正定随机矩阵乘积的平均谱。我们还表明，这个问题与拟埃尔米特随机矩阵理论（QHRMT），这意味着相对于非平凡指标，“哈密尔顿”矩阵是埃尔米特式的。这扩展了普通的埃尔米特矩阵，对于该矩阵而言，度量只是单位矩阵。我们获得的光谱分析结果与我们的数值结果非常吻合。后者在高频带边缘也表现出振荡，非常适合艾里核模式。我们还计算了相应振幅特征向量的逆参与比，并证明了它们都得到了扩展，这与传统的无序晶体形成了鲜明的对比。最后，我们根据其振动频谱计算系统的热力学性质。除了矩阵模型分析之外，我们还研究了各种多段无序摆的振动谱，作为高度连接的机械系统的具体实现。