Physical review. B, Condensed matter

Directed percolation in 3+1 dimensions

Joan Adler, Jorge Berger, J. A. M. S. Duarté, Yigal Meir

Published May 1, 1988• 13 citations

Abstract

Directed percolation in four dimensions is of direct physical relevance to the world with three space and one time dimension. We present a comprehensive analysis of recently extended series for the moments of the cluster-size distribution and for the percolation probability in a ``field'' on the hypercubic lattice. We find a critical threshold, ${p}_{c}$=0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent ${\ensuremath{\Delta}}_{1}$=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.

DOI: 10.1103/physrevb.37.7529

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Physical review. B, Condensed matter

Directed percolation in 3+1 dimensions

Joan Adler, Jorge Berger, J. A. M. S. Duarté, Yigal Meir

Published May 1, 1988• 13 citations

Publisher

Abstract

{p}_{c}

=0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent

{\ensuremath{\Delta}}_{1}

=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.

DOI: 10.1103/physrevb.37.7529

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Bond and site percolation in three dimensions

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{p}_{c}

{\ensuremath{\Delta}}_{1}

=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to

L=512

, and estimate the percolation thresholds to be

{p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)

and

{p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)

. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent

{d}_{\mathrm{min}}=1.3756(3)

. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

We examine the average cluster distribution as a function of lattice probability for a very small

(L=6)

lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of

L=6

and

L=63

lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.

{E}_{p},

the percolation probability

P,

and the probability

{W}_{n}

for the appearance of

n

percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions

L.

Using the HMCSM, we calculate the existence probability

{E}_{p}

and the percolation probability

P

for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of

{E}_{p}

and

P

in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that

{E}_{p}

and

P

have universal finite-size scaling functions. This implies that the critical

{E}_{p}

is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that

{W}_{n}

for site and bond percolation on sc lattices have universal finite-size scaling functions.