Critical behavior of the geometrical spin clusters and interfaces in the two-dimensional thermalized bond Ising model

Abstract

The fractal dimensions and the percolation exponents of the geometrical spin clusters of like sign at criticality, are obtained numerically for an Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff singlecluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard twodimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles, results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters, results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained, are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.

Recent advances in percolation theory and its applications

“…In 2010, Smirnov was awarded the Fields medal for the proof of conformal invariance of percolation and the planar Ising model in statistical physics. SLE has soon found many applications and turned out to describe the vorticity lines in turbulence [26,86], domain walls of spin glasses [87,88,89], the nodal lines of random wave functions [90,91], the iso-height lines of random grown surfaces [92,93,94,95,96,97], the avalanche lines in sandpile models [98] and the coastlines and watersheds on Earth [99,101,102]. Among which, SLE could provide quite unexpected connections between some features of interacting systems and ordinary uncorrelated percolation [26,90].…”

Phase transition in a stochastic geometry model with applications to statistical mechanics

“…Finally, we discuss the concept of phase transition and percolation by examining some thermodynamic quantities 25 and the moments of the probability distribution, including free energy, magnetization, kurtosis, mean, and variance. We observed that the fluctuations of most of the quantities studied for the RCM with the connection function $$$ g(x)={I}_{\left\{|x|\le 1\right\}} $$$ or the Poisson plob model in terms of temperature parameter are similar to the fluctuations in the two‐dimensional Ising model 26,27 . These fluctuations play a central role in our understanding of phase transition.…”

“…We observed that the fluctuations of most of the quantities studied for the RCM with the connection function g (x) = I {|x|≤1} or the Poisson plob model in terms of temperature parameter are similar to the fluctuations in the two-dimensional Ising model. 26,27 These fluctuations play a central role in our understanding of phase transition. Their behavior near a critical point provides important information about the underlying many-particle interactions.…”

Universality class of epidemic percolation transitions driven by random walks

“…Percolation is one of the simplest models in probability theory which provides a suitable platform to formulate and model various natural phenomena that exhibit geometric phase transitions with universal characteristics [1][2][3][4]. Percolation has been applied to describe a wide range of critical behavior such as, among many others, flow through porous media for connectivity [5,6], networks [7][8][9], magnetic models [10][11][12][13][14][15][16][17], colloids [18,19], growth models [20], topography of planets [21][22][23] and epidemic models [24]. The concept of universality lets many microscopically different physical systems exhibit the same critical behavior with quantitatively identical features, assigned by a set of critical exponents.…”