Avetisov V, Hovhannisyan M, Gorsky A, Nechaev S, Tamm M, Valba O
N. N. Semenov Institute of Chemical Physics of the Russian Academy of Sciences, 119991 Moscow, Russia.
Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia.
Phys Rev E. 2016 Dec;94(6-1):062313. doi: 10.1103/PhysRevE.94.062313. Epub 2016 Dec 22.
We consider the canonical ensemble of N-vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity μ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p^{-1}] almost full subgraphs (cliques) above critical fugacity, μ_{c}, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing μ leads to the formation of a multizonal support for μ>μ_{c}. Eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuates around the mean value Np. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.
我们考虑具有淬火顶点度的N顶点厄尔多斯-雷尼（ER）随机拓扑图的正则系综，且每条闭合的三条边的逸度为μ。我们声称，在临界逸度μ_c之上，大N图会完全碎片化，形成[p^{-1}]个几乎完整的子图（团），其中p是ER边形成概率。随着μ的增加，邻接矩阵的谱密度ρ(λ)的演化导致在μ>μ_c时形成多区域支撑。从中心区域到边缘区域的特征值隧穿意味着在碎片化过程中形成一个新的团。网络基态的邻接矩阵具有块对角形式，其中块中的顶点数围绕平均值Np波动。在这种情况下，整个网络的谱密度呈三角形。我们从传统随机矩阵模型的角度解释这些现象，并推测可能的物理应用。
