Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userArea covered by disks in small-bounded continuum percolating systems: An application to the string percolation model
406
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions where the number of strings distributed in the surface of interest is strongly determined by the size and the energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks and the shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems.
rate research
Read More
Using the randomized algorithm method developed by Duminil-Copin, Raoufi, Tassion (2019b) we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof that the critical parameter for percolation in this model is $1/2$ when the underlying shapes for the distinct colours arise from the same distribution and extends the work of Hirsch (2015) and M{u}ller (2016). In addition we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain `transitivity condition this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.
Discontinuous transition is observed in the equilibrium cluster properties of a percolation model with suppressed cluster growth as the growth parameter g0 is tuned to the critical threshold at sufficiently low initial seed concentration rho in contrast to the previously reported results on non- equilibrium growth models. In the present model, the growth process follows all the criteria of the original percolation model except continuously updated occupation probability of the lattice sites that suppresses the growth of a cluster according to its size. As rho varied from higher values to smaller values, a line of continuous transition points encounters a coexistence region of spanning and non- spanning large clusters. At sufficiently small values of rho (less equal 0.05), the growth parameter g0 exceeds the usual percolation threshold and generates compact spanning clusters leading to discontinuous transitions.
Abrupt transitions are ubiquitous in the dynamics of complex systems. Finding precursors, i.e. early indicators of their arrival, is fundamental in many areas of science ranging from electrical engineering to climate. However, obtaining warnings of an approaching transition well in advance remains an elusive task. Here we show that a functional network, constructed from spatial correlations of the systems time series, experiences a percolation transition way before the actual system reaches a bifurcation point due to the collective phenomena leading to the global change. Concepts from percolation theory are then used to introduce early warning precursors that anticipate the systems tipping point. We illustrate the generality and versatility of our percolation-based framework with model systems experiencing different types of bifurcations and with Sea Surface Temperature time series associated to El Nino phenomenon.
The ranges of transmission of the mobiles in a Mobile Ad-hoc Network are not uniform in reality. They are affected by the temperature fluctuation in air, obstruction due to the solid objects, even the humidity difference in the environment, etc. How the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask. Here a new model of percolation phenomena, with an additional source of disorder, has been introduced for a theoretical understanding of this problem. As in ordinary percolation, sites of a square lattice are occupied randomly with the probability $p$. Each occupied site is then assigned a circular disc of random value $R$ for its radius. A bond is defined to be occupied if and only if the radii $R_1$ and $R_2$ of the discs centered at the ends satisfy certain pre-defined condition. In a very general formulation, one divides the $R_1 - R_2$ plane into two regions by an arbitrary closed curve. One defines that a point within one region represents an occupied bond, otherwise it is a vacant bond. Study of three different rules under this general formulation, indicates that the percolation threshold is always larger and varies continuously. This threshold has two limiting values, one is $p_c$(sq), the percolation threshold for the ordinary site percolation on the square lattice and the other being unity. The variation of the thresholds are characterized by exponents, which are not known in the literature. In a special case, all lattice sites are occupied by discs of random radii $R in {0,R_0}$ and a percolation transition is observed with $R_0$ as the control variable, similar to the site occupation probability.
We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci anyons in the presence of competing interactions. Combining high-order series expansions around three exactly solvable points and exact diagonalizations, we find that the non-Abelian doubled Fibonacci topological phase is separated from two nontopological phases by different second-order quantum critical points, the positions of which are computed accurately. These trivial phases are separated by a first-order transition occurring at a fourth exactly solvable point where the ground-state manifold is infinitely many degenerate. The evaluation of critical exponents suggests unusual universality classes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
