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Recently, a hybrid percolation transitions (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. In spite of considerable effort to develop the theory of HPT, it is still incomplete, particularly when the transition is induced by cluster merging dynamics. Here, we aim to develop a theoretical framework of the HPT induced by such dynamics. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters, respectively. Finite-size scaling method for the HPT is also introduced. The conventional formula of the fractal dimension in terms of the critical exponents is not valid. Neither the giant nor finite clusters are fractals but they have fractal boundaries.
We show that the change of the fluctuation spectrum near the quantum critical point (QCP) may result in the continuous change of critical exponents with temperature due to the increase in the effective dimensionality upon approach to QCP. The latter
Via a combination of molecular dynamics (MD) simulations and finite-size scaling (FSS) analysis, we study dynamic critical phenomena for the vapor-liquid transition in a three dimensional Lennard-Jones system. The phase behavior of the model, includi
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel tempering algorithm with cluster updates and an adaptive r
First order phase transitions occur discretely from one state to another, however they often display continuous behavior. To understand this nature, it is essential to probe how the emergent phase nucleates, interacts and evolves with the initial pha
The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researche