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The recent proliferation of correlated percolation models---models where the addition of edges/vertices is no longer independent of other edges/vertices---has been motivated by the quest to find discontinuous percolation transitions. The leader in this proliferation is what is known as explosive percolation. A recent proof demonstrates that a large class of explosive percolation-type models does not, in fact, exhibit a discontinuous transition[O. Riordan and L. Warnke, Science, {bf 333}, 322 (2011)]. We, on the other hand, discuss several correlated percolation models, the $k$-core model on random graphs, and the spiral and counter-balance models in two-dimensions, all exhibiting discontinuous transitions in an effort to identify the needed ingredients for such a transition. We then construct mixtures of these models to interpolate between a continuous transition and a discontinuous transition to search for a tricritical point. Using a powerful rate equation approach, we demonstrate that a mixture of $k=2$-core and $k=3$-core vertices on the random graph exhibits a tricritical point. However, for a mixture of $k$-core and counter-balance vertices, heuristic arguments and numerics suggest that there is a line of continuous transitions as the fraction of counter-balance vertices is increased from zero with the line ending at a discontinuous transition only when all vertices are counter-balance. Our results may have potential implications for glassy systems and a recent experiment on shearing a system of frictional particles to induce what is known as jamming.
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In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random fields on the tricritical phase diagram of the spin-1 Ising model in $d=3$. We find that random fields convert first-order phase transitions into second-order, in fact more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent $beta_u$; moreover, an extreme asymmetry of the phase diagram is very striking. To accomodate this asymmetry, the second-order boundary exhibits reentrance.
Motivated by the importance of geometric information in real systems, a new model for long-range correlated percolation in link-adding networks is proposed with the connecting probability decaying with a power-law of the distance on the two-dimensional(2D) plane. By overlapping it with Achlioptas process, it serves as a gravity model which can be tuned to facilitate or inhibit the network percolation in a generic view, cover a broad range of thresholds. Moreover, it yields a set of new scaling relations. In the present work, we develop an approach to determine critical points for them by simulating the temporal evolutions of type-I, type-II and type-III links(chosen from both inter-cluster links, an intra-cluster link compared with an inter-cluster one, and both intra-cluster ones, respectively) and corresponding average lengths. Numerical results have revealed objective competition between fractions, average lengths of three types of links, verified the balance happened at critical points. The variation of decay exponents $a$ or transmission radius $R$ always shifts the temporal pace of the evolution, while the steady average lengths and the fractions of links always keep unchanged just as the values in Achlioptas process. Strategy with maximum gravity can keep steady average length, while that with minimum one can surpass it. Without the confinement of transmission range, $bar{l} to infty$ in thermodynamic limit, while $bar{l}$ does not when with it. However, both mechanisms support critical points. In two-dimensional free space, the relevance of correlated percolation in link-adding process is verified by validation of new scaling relations with various exponent $a$, which violates the scaling law of Weinribs.
Cluster concepts have been extremely useful in elucidating many problems in physics. Percolation theory provides a generic framework to study the behavior of the cluster distribution. In most cases the theory predicts a geometrical transition at the percolation threshold, characterized in the percolative phase by the presence of a spanning cluster, which becomes infinite in the thermodynamic limit. Standard percolation usually deals with the problem when the constitutive elements of the clusters are randomly distributed. However correlations cannot always be neglected. In this case correlated percolation is the appropriate theory to study such systems. The origin of correlated percolation could be dated back to 1937 when Mayer [1] proposed a theory to describe the condensation from a gas to a liquid in terms of mathematical clusters (for a review of cluster theory in simple fluids see [2]). The location for the divergence of the size of these clusters was interpreted as the condensation transition from a gas to a liquid. One of the major drawback of the theory was that the cluster number for some values of thermodynamic parameters could become negative. As a consequence the clusters did not have any physical interpretation [3]. This theory was followed by Frenkels phenomenological model [4], in which the fluid was considered as made of non interacting physical clusters with a given free energy. This model was later improved by Fisher [3], who proposed a different free energy for the clusters, now called droplets, and consequently a different scaling form for the droplet size distribution. This distribution, which depends on two geometrical parameters, has the nice feature that the mean droplet size exhibits a divergence at the liquid-gas critical point.
We describe a percolation problem on lattices (graphs, networks), with edge weights drawn from disorder distributions that allow for weights (or distances) of either sign, i.e. including negative weights. We are interested whether there are spanning paths or loops of total negative weight. This kind of percolation problem is fundamentally different from conventional percolation problems, e.g. it does not exhibit transitivity, hence no simple definition of clusters, and several spanning paths/loops might coexist in the percolation regime at the same time. Furthermore, to study this percolation problem numerically, one has to perform a non-trivial transformation of the original graph and apply sophisticated matching algorithms. Using this approach, we study the corresponding percolation transitions on large square, hexagonal and cubic lattices for two types of disorder distributions and determine the critical exponents. The results show that negative-weight percolation is in a different universality class compared to conventional bond/site percolation. On the other hand, negative-weight percolation seems to be related to the ferromagnet/spin-glass transition of random-bond Ising systems, at least in two dimensions.
Recent advances in machine learning have become increasingly popular in the applications of phase transitions and critical phenomena. By machine learning approaches, we try to identify the physical characteristics in the two-dimensional percolation model. To achieve this, we adopt Monte Carlo simulation to generate dataset at first, and then we employ several approaches to analyze the dataset. Four kinds of convolutional neural networks (CNNs), one variational autoencoder (VAE), one convolutional VAE (cVAE), one principal component analysis (PCA), and one $k$-means are used for identifying order parameter, the permeability, and the critical transition point. The former three kinds of CNNs can simulate the two order parameters and the permeability with high accuracy, and good extrapolating performance. The former two kinds of CNNs have high anti-noise ability. To validate the robustness of the former three kinds of CNNs, we also use the VAE and the cVAE to generate new percolating configurations to add perturbations into the raw configurations. We find that there is no difference by using the raw or the perturbed configurations to identify the physical characteristics, under the prerequisite of corresponding labels. In the case of lacking labels, we use unsupervised learning to detect the physical characteristics. The PCA, a classical unsupervised learning, performs well when identifying the permeability but fails to deduce order parameter. Hence, we apply the fourth kinds of CNNs with different preset thresholds, and identify a new order parameter and the critical transition point. Our findings indicate that the effectiveness of machine learning still needs to be evaluated in the applications of phase transitions and critical phenomena.
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