Self-avoiding paths on the three dimensional Sierpinski gasket
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.
scientific article; zbMATH DE number 487016