This paper advocates an unconventional analytical approach to studying the fractal geometry of percolation at the threshold, which is based on the most general methods of the differential topology. Our particular interest concentrates on the Alexander-Orbach (AO) conjecture [J. Phys. (France) Lett. 43, L625 (1982)], which assigns the universal (mean-field) value $4/3$ to the spectral fractal dimension $\tilde{d}$ at the percolation threshold for all embedding Euclidean dimensions $n$ greater than one, i.e., $n>~2$. Using the topological arguments, we show that the AO conjecture might be improved for the relatively low embedding dimensions $2<~n<~5$, for which the analytical result $\tilde{d}=1.327\mathrm{}\pm 0.001$ is proposed, instead of the original AO estimate $4/3$. Meanwhile we assume that the exact value $\tilde{d}=4/3\mathrm{}$ holds for all $n>~6$, as it follows directly from the well-known mean-field theory. The improved value of $\tilde{d}\approx 1.327$ for $2<~n<~5$ is obtained from an analysis of the basic topological properties of the percolating fractal sets at the threshold of percolation. We show that these properties could be investigated fruitfully with the introduction of the concept of the fractal manifold, which might serve as an effective instrument when considering the topology of the fractal objects. Our results indicate that the proposed value of $\tilde{d}\approx 1.327$ for the spectral dimension at the percolation threshold has the fundamental topological origin related to the most general features of the fractal geometry of percolation at criticality. We argue that the constraint $2<~n<~5$ on the topological dimension $n$ of the embedding Euclidean space is the direct consequence of the famous Whitney theorem, which establishes the embedding properties of manifolds from the viewpoint of their dimensionality. A simple topological condition that identifies the threshold of percolation is obtained for $2<~n<~5$. The particular topological restrictions implied throughout the present study are discussed and the important issue of contractibility of the fractal manifolds is pointed out.

• Received 23 December 1996

DOI:https://doi.org/10.1103/PhysRevE.56.2437