Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Critical dynamics below the percolation threshold

S. Jain, Eduardo Lage

Published Dec 7, 1993• 2 citations
Publisher

Abstract

The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.

DOI
10.1088/0305-4470/26/23/004

Related papers

Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.

Journal of Physics A Mathematical and General•Mar 23, 1995•5 citations

Monte Carlo Renormalization Group of Dilute 2D Ising Dynamics

Vladimir V. Prudnikov, O. N. Markov

The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.

Europhysics Letters (EPL)•Jan 20, 1995•16 citations

Breakdown of dynamic scaling at the percolation threshold

S. Jain

The critical dynamics of the two-dimensional Ising model at the bond percolation threshold is investigated by Monte Carlo simulations on a 64*64 lattice. Conventional dynamic scaling breaks down at low temperatures: the logarithm of the relaxation time depends quadratically upon the logarithm of the thermal correlation length. The coefficients of the quadratic and linear terms are 0.51 and 3.25, respectively. The results are compared with recent experimental and analytic work.

Journal of Physics A Mathematical and General•Feb 1, 1986•23 citations

The breakdown of dynamic scaling of the bond-diluted Ising model at the percolation threshold for hypercubic lattices

J M Nunes da Silva, E J S Laga

The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p .

Journal of Physics C Solid State Physics•May 20, 1987•7 citations

Non-universality in the dynamics at the percolation threshold

S. Jain

The dynamical behavior of the two-dimensional Ising model at the site percolation threshold on L*L lattices with L<or=64 is studied by Monte Carlo simulations. The results are consistent with the recently proposed singular dynamic scaling. The new dynamical exponents are estimated to be A s =0.62+or-0.12 and B s =3.92+or-0.89. Comparison with bond dilution strongly suggests that universality is violated for this model.

Journal of Physics A Mathematical and General•Aug 1, 1986•19 citations

Dimensionality dependence in the singular dynamic scaling in the dilute Ising model

Bharat B. Biswal, Debashish Chowdhury

The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=${\mathit{Ax}}^{2}$+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.

Physical Review A•Apr 1, 1991•7 citations
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p .
The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=Ax2{\mathit{Ax}}^{2}Ax2+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.