Nakerst Goran, Denisov Sergey, Haque Masudul
Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany.
NordSTAR - Nordic Center for Sustainable and Trustworthy AI Research, Pilestredet 52, N-0166, Oslo, Norway.
Phys Rev E. 2023 Jul;108(1-1):014102. doi: 10.1103/PhysRevE.108.014102.
The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter φ, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with D vertices (number of system states) and 2φD edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as ∼φ, while its variance is ∼sqrt[φ]. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of D. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, φ=const, and find that starting already at φ⩾2, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.
一个复杂多态系统的演化通常被解释为一个连续时间马尔可夫过程。为了对这类系统的弛豫动力学进行建模，我们引入了一组随机稀疏矩阵，它们可作为马尔可夫演化的生成器。稀疏性由参数φ控制，φ是生成器矩阵每行每列的非零元素数量。因此，该集合的一个成员由一个具有D个顶点（系统状态数）和2φD条边且权重随机分布的有向正则图的拉普拉斯矩阵来表征。我们研究了稀疏性对生成器谱的影响。结果表明，稀疏性会缩小非稀疏随机生成器所特有的大谱隙。我们证明了特征值分布的一阶矩按 ∼φ缩放，而其方差为 ∼sqrt[φ]。通过使用极值理论，我们展示了谱边缘的形状是如何由相应权重分布的尾部决定的，并阐明了谱隙作为D的函数的行为。最后，我们分析了超稀疏生成器（φ = const）的复杂间距比统计量，发现从φ⩾2开始，生成器的谱就表现出吉尼贝里正交系综所特有的普遍性质。
