Lattice dynamics, random walks, and nonintegral effective dimensionality

Definitions of the nonintegral effective dimensionality of recursively defined lattices (fractal lattices) may be based on scaling properties of the lattices, or on the qualitative behavior of cooperative phenomena supported by the lattices. We examine analogs of these definitions for regular (i.e., periodic) lattices supporting long-range interactions. In particular, we show how to calculate a harmonic oscillator effective dimension, a scaling dimension, and a random walk effective dimension for simple cubic lattices with a class of long-range interactions. We examine the relationship between these three dimensions for regular lattices, and conjecture a constraint on the analogs of these dimensions for fractal lattices.