Self-avoiding paths on the three dimensional Sierpinski gasket
type: Item
instance of: scholarly article
title: Self-avoiding paths on the three dimensional Sierpinski gasket
zbMATH Open document ID: 0795.60036
author: Kumiko Hattori
author: Tetsuya Hattori
author: Shigeo Kusuoka
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-08-29T00:00:00Z
review text: The pre-Sierpinski gasket is the vertex lattice in \(R^ 3\) with tetrahedral cell. For each \(n\) denote by \(F_ n\) the finite tetrahedral sublattice with size \(2^ n\) times the size of a unit tetrahedron (=cell) and by \(W^{*(n)}\) the set of self-avoiding paths \(w\) on \(F_ n\) which start at a given extreme vetex and end at another given extreme vertex. Among the probability measures investigated in this paper are the measures \(\mu_ n^*[w]=Z^*_ n (\beta_ c)^{-1} \exp(-\beta_ c L(w))\) on \(W^{*(n)}\), where \(L(w)\) is the length of \(w\) and the critical value \(\beta_ c\) of the ``temperature'' is such that, as \(n \to \infty\), the partition function \(Z^*_ n (\beta)\) tends to 0 for \(\beta>\beta_ c\) and to \(\infty\) for \(\beta<\beta_ c\). The authors prove that there is a \(\lambda\) (numerically \(\lambda=2.7965\dots)\) such that \(\lambda^{-n}L\) converges in distribution to a nontrivial limit. If \(F_ n\) is scaled to unit size and the time scale suitable transformed, the image measure \(P_ n\) of \(\mu^*_ n\) is shown to converge weakly to a probability measure on a space of continuous paths on the Sierpinski gasket. Under this limit measure the paths are almost surely self-avoiding and of Hausdorff dimension greater than one. The authors also investigate the probability measures \(\tilde P_ k\) on paths of length \(k\) on the pre-Sierpinski gasket under which all self- avoiding paths of this length are equally likely. They show that the critical exponent for the maximal displacement \(\| w \|\) of such paths is \(x=\log 2/ \log \lambda\), in the sense that, as \(k \to \infty\), \(\log \| w \| \sim \log k^ x\) in probability.
reviewed by: F. Papangelou
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 82C41
zbMATH DE Number: 487016
zbMATH Keywords: pre-Sierpinski gasket
zbMATH Keywords: self-avoiding paths
zbMATH Keywords: almost surely self-avoiding
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195167053
OpenAlex ID: W2032447648
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DOI: 10.2977/PRIMS/1195167053
description: scientific article; zbMATH DE number 487016