Journal de physique

Application of the phenomenological renormalization to percolation and lattice animals in dimension 2

Bernard Derrida, Laurent de Seze

Published Jan 1, 1982• 238 citations

Abstract

We recall the relation between finite-size scaling and the phenomenological renormalization. We calculate the exponent ν in dimension 2 for percolation and find a good agreement with the conjecture 4/3 of den Nijs. For lattice animals, we construct transfer matrices to calculate the correlation lengths and we find ν = 0.640 8 ± 0.000 3 in dimension 2.

DOI: 10.1051/jphys:01982004303047500

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Journal de physique

Application of the phenomenological renormalization to percolation and lattice animals in dimension 2

Bernard Derrida, Laurent de Seze

Published Jan 1, 1982• 238 citations

Publisher

Abstract

DOI: 10.1051/jphys:01982004303047500

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y=\frac{3}{4}

for percolation but bad agreement with its prediction

y=\frac{3}{2}

for the 4-state Potts model.

{10}^{6}

lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-

{J}^{2}

/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.