Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that the critical Anderson electron in [Formula: see text] dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension [Formula: see text]. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the 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 [Formula: see text] for dimension d to be accessed by the electron. We find that [Formula: see text] has a strong peak at d very close to two. In fact, our data suggest that [Formula: see text] is non-zero on the interval [Formula: see text] and may develop a discrete part ([Formula: see text]-function) at [Formula: see text] in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although [Formula: see text] is based on effective counting, of which [Formula: see text] has no a priori knowledge, [Formula: see text] is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of [Formula: see text] in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.