We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation shows a phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs - the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide a finite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.
Analytical realization of finite-size scaling for Anderson localization. Does the band of critical states exist for d>2?
基于辅助准一维系统的考虑，提出了有限尺寸标度算法的一种解析实现。将所得的解析结果与数值计算结果进行了比较，结果表明Anderson转变点分裂到临界态带中。这一结论得到了直接的数字证据的支持（Edwards and Elderless，1972; Last and Elderless，1974; Schreiber，1985; 1990）。恢复传统图像的可能性仍然存在，但需要对原始数字数据进行彻底的重新解释。
