We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.

Physical review. B, Condensed matterJan 1, 199310 citations

Site-bond percolation by position-space renormalization group

Hisao Nakanishi, PETER J. REYNOLDS

Physics Letters AApr 1, 197962 citations

Percolation in one of q colors near criticality

Xiaofeng Qian, Youjin Deng, Henk W. J. Blöte

We study bond percolation in two dimensions between random site variables having one out of $q$ colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature $T$ in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several $q$ -state Potts Hamiltonians. For high $T$ , these transitions fit, irrespective of $q$ , in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$, $q$ -dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at $q=2$ . For $q<2$ the Potts critical state appears to enhance percolation, for $q>2$ it appears to suppress it. Remarkably, for $q=2$ the percolation line coincides with the only flow line extending to $T>{T}_{c}$ from the critical fixed point associated with Potts clusters.We study bond percolation in two dimensions between random site variables having one out of $q$ colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature $T$ in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several $q$-state Potts Hamiltonians. For high $T$, these transitions fit, irrespective of $q$, in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$, $q$-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at $q=2$. For $q<2$ the Potts critical state appears to enhance percolation, for $q>2$ it appears to suppress it. Remarkably, for $q=2$ the percolation line coincides with the only flow line extending to $T>{T}_{c}$ from the critical fixed point associated with Potts clusters.

Physical Review BApr 29, 200513 citations

Monte Carlo Simulations of Phase Transitions in a Two-Dimensional Random-Bond Potts Model

Ricardo Paredes V., J. Valbuena

MRS ProceedingsJan 1, 19961 citation

Critical droplets and phase transitions in two dimensions

Santo Fortunato

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

arXiv:cond-mat/0205211Physical review. B, Condensed matterJan 15, 200316 citations

We study bond percolation in two dimensions between random site variables having one out of

q

colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature

T

in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several

q

-state Potts Hamiltonians. For high

T

, these transitions fit, irrespective of

q

, in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$,

q

-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at

q=2

. For

q<2

the Potts critical state appears to enhance percolation, for

q>2

it appears to suppress it. Remarkably, for

q=2

the percolation line coincides with the only flow line extending to

T>{T}_{c}

from the critical fixed point associated with Potts clusters.

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point

{T}_{c}

of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability

{p}_{B}<1,

the corresponding site-bond clusters keep on percolating at

{T}_{c}

and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability

{p}_{c}

such that, for any

{p}_{B}>~{p}_{c},

the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for

{p}_{c}<{p}_{B}<~1

but, for

{p}_{B}{=p}_{c},

they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.