Two Types of Discontinuous Percolation Transitions in Cluster Merging Processes

Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have revealed that a few models exhibit a discontinuous percolation transition (DPT) in cluster merging processes. Unlike the case of continuous transitions, understanding the nature of discontinuous phase transitions requires a detailed study of the system at hand, which has not been undertaken yet for DPTs. Here we examine the cluster size distribution immediately before an abrupt increase in the order parameter of DPT models and find that DPTs induced by cluster merging kinetics can be classified into two types. Moreover, the type of DPT can be determined by the key characteristic of whether the cluster kinetic rule is homogeneous with respect to the cluster sizes. We also establish the necessary conditions for each type of DPT, which can be used effectively when the discontinuity of the order parameter is ambiguous, as in the explosive percolation model.Comment: 9 pages, 6 figure

Cluster aggregation model for discontinuous percolation transition

The evolution of the Erd\H{o}s-R\'enyi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^{\omega}$ with $0 \le \omega < 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate equation approach for such discontinuous PTs enables us to uncover the mechanism underlying the explosive PT under the Achlioptas process.Comment: 5 pages, 5 figure

Some Boas-Bellman Type Inequalities in 2-Inner Product Spaces

Some inequalities in 2-inner product spaces generalizing Bessel's result that are similar to the Boas-Bellman inequality from inner product spaces, are given. Applications for determinantal integral inequalities are also provided

Color Reflection Invariance and Monopole Condensation in QCD

We review the quantum instability of the Savvidy-Nielsen-Olesen (SNO) vacuum of the one-loop effective action of SU(2) QCD, and point out a critical defect in the calculation of the functional determinant of the gluon loop in the SNO effective action. We prove that the gauge invariance, in particular the color reflection invariance, exclude the unstable tachyonic modes from the gluon loop integral. This guarantees the stability of the magnetic condensation in QCD.Comment: 28 pages, 3 figures, JHEP styl

Monopoles and Knots in Skyrme Theory

We show that the Skyrme theory actually is a theory of monopoles which allows a new type of solitons, the topological knots made of monopole-anti-monopole pair,which is different from the well-known skyrmions. Furthermore, we derive a generalized Skyrme action from the Yang-Mills action of QCD, which we propose to be an effective action of QCD in the infra-red limit. We discuss the physical implications of our results.Comment: 4 pages. Phys. Rev. Lett. in pres

Dilaton as a Dark Matter Candidate and its Detection

Assuming that the dilaton is the dark matter of the universe, we propose an experiment to detect the relic dilaton using the electromagnetic resonant cavity, based on the dilaton-photon conversion in strong electromagnetic background. We calculate the density of the relic dilaton, and estimate the dilaton mass for which the dilaton becomes the dark matter of the universe. With this we calculate the dilaton detection power in the resonant cavity, and compare it with the axion detection power in similar resonant cavity experiment.Comment: 23 pages, 2 figure

Norm Estimates for the Difference Between Bochner's Integral and the Convex Combination of Function's Values

Norm estimates are developed between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function values taken on a division of the interval [a,b]