Geometric Implementation of Hypercubic Lattices with Noninteger Dimensionality by Use of Low Lacunarity Fractal Lattices

Abstract

It is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity. These lattices are special examples of Sierpinski carpets. Their being of low lacunarity means that they are arbitrarily close to being translationally invariant. The claim is substantiated for the Ising model in $D=1+\epsilon$ dimensions, and for resistor network models with $1<D<\mathrm{}2\mathrm{}$.