PERCOLATION AND FINITE SIZE SCALING IN SEVEN DIMENSIONS

Lotfi Zekri

Published Jan 1, 2004• 1 citation

Abstract

Numerical investigation of critical exponents on a hypercubic lattice with Ld random sites with L up to 33 and d up to 7 showed that above the critical dimension the phase transitions in Ising model and percolation are not alike.

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PERCOLATION AND FINITE SIZE SCALING IN SEVEN DIMENSIONS

Lotfi Zekri

Published Jan 1, 2004• 1 citation

OpenAlex

Abstract

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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.

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Percolation in the two-dimensional Ising model

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The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.

arXiv:2504.18861Physical review. EJul 22, 20252 citations

The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in

d=5,6,7

dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions

d=6,7

which are larger or equal to the assumed upper critical dimension,

{d}_{u}=6

, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

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.