Recent advances in percolation theory and its applications

Abstract

Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model.Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, dimensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive percolation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state.In the past decade, after the invention of stochastic Löwner evolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. After a short review on SLE, we will provide an overview on existence of the scaling limit and conformal invariance of the critical percolation. We will also establish a connection with the magnetic models based on the percolation properties of the Fortuin-Kasteleyn and geometric spin clusters. As an application we will discuss how percolation theory leads to the reduction of the 3D criticality in a 3D Ising model to a 2D critical behavior.Another recent application is to apply percolation theory to study the properties of natural and artificial landscapes. We will review the statistical properties of the coastlines and watersheds and their relations with percolation. Their fractal structure and compatibility with the theory of SLE will also be discussed. The present mean sea level on Earth will be shown to coincide with the critical threshold in a percolation description of the global topography.

Monodominance in tropical forests: modelling reveals emerging clusters and phase transitions

“…For variation of the dispersal radius (figure 4), finite-size scaling was performed for each dispersal radius of the identical species (1,3,5,7,10,15,20,30), meaning that, for each dispersal radius, the SM was varied (between 6000 and 10 000) for different system sizes (128 Â 128, 178 Â 178, 256 Â 256, 400 Â 400 and 512 Â 512 grid cells). Here, for system size below 400 Â 400 grid cells, 1000 simulation runs were performed and for bigger system sizes 500 simulation runs.…”

“…SM multiplied by the sum of grid cells within the DR (9450 Â 4 ¼ 37 800), is larger than that of the other species (20 Â 1256 ¼25 120). To analyse the effect of the relative total SM that is invested by all species, we systematically varied the DR of the non-monodominant species (from 1 to 30 grid cells) while keeping its SM constant (20). For each DR of the non-monodominant species, we determined the critical SM for the monodominant species that leads to the emergence of clusters.…”

String percolation threshold for elliptically bounded systems

“…The order parameter P follows the power law P ∝ (p−p c ) β , for the occupied/connected probability (p) larger or equal to the percolation threshold (p c ) with β being the critical exponent. In particular β = 5/36 in two-dimensions [9,12,14,15].…”

Multiple scaling power in liquid gallium under pressure conditions

“…29,30 This correlation length can be estimated as ξ≈14-15Å The D f dimension is determined by 10 log log 1 ⁄ ⁄ .…”