Neal Madras — Page 2

arXiv1999

A pattern theorem for lattice clusters

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration ("pattern") of sites and bonds can occur in large clusters, then it occurs at least cN times in most clusters of size n, for some constant c>0. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten. The results also apply to weighted sums, and in particular we can take a$sub n$ to be the probability that the percolation cluster containing the origin consists of exactly n sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.

math.PR↗

arXiv1994

Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents $ν$ and $2Δ_4 -γ$ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation $dν= 2Δ_4 -γ$. In two dimensions, we confirm the predicted exponent $ν= 3/4$ and the hyperscaling relation; we estimate the universal ratios $\ / \ = 0.14026 \pm 0.00007$, $\ / \ = 0.43961 \pm 0.00034$ and $Ψ^* = 0.66296 \pm 0.00043$ (68\% confidence limits). In three dimensions, we estimate $ν= 0.5877 \pm 0.0006$ with a correction-to-scaling exponent $Δ_1 = 0.56 \pm 0.03$ (subjective 68\% confidence limits). This value for $ν$ agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for $Δ_1$. Earlier Monte Carlo estimates of $ν$, which were $\approx\! 0.592$, are now seen to be biased by corrections to scaling. We estimate the universal ratios $\ / \ = 0.1599 \pm 0.0002$ and $Ψ^* = 0.2471 \pm 0.0003$; since $Ψ^* > 0$, hyperscaling holds. The approach to $Ψ^*$ is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies.

hep-lat↗