Abstract
We have recently shown that critical Anderson electron in D=3 dimensions effectively occupies a spatial region of infrared (IR) scaling dimension dIR ≈ 8/3. Here we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probability, such that points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density p(d) for dimension d to be accessed by electron. We find that p(d) has a strong peak at d very close to 2. In fact, our data suggests that p(d) is non-zero on the interval [dmin, dmax] ≈ [4/3,8/3] and may develop a discrete part (δ-function) at d=2 in infinite-volume limit. The latter invokes the possibility that combination of quantum mechanics and pure disorder can lead to emergence of topological dimensions. Although dIR is based on effective counting of which p(d) has no a priori knowledge, dIR ≥ dmax is an exact feature of the ensuing formalism. Possible connection of our results to recent findings of dIR ≈ 2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.