Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation

Abstract

A nontrivial family of $d$-dimensional scale-invariant fractal lattices is described, on which statistical mechanics and conductivity problems are exactly solvable for every $d$. These fractals are finitely ramified but not quasi one dimensional, and hence can be used to model the important geometrical features of the percolating cluster's backbone. Critical exponents calculated for this model agree with those of "real" systems at low dimensionalities.