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We have recently shown that critical Anderson electron in $D=3$ dimensionseffectively occupies a spatial region of infrared (IR) scaling dimension$d_\text{IR} \approx 8/3$. Here we inquire about the dimensional substructureinvolved. We partition space into regions of equal quantum occurrenceprobability, such that points comprising a region are of similar relevance, andcalculate the IR scaling dimension $d$ of each. This allows us to infer theprobability density $p(d)$ for dimension $d$ to be accessed by electron. Wefind that $p(d)$ has a strong peak at $d$ very close to 2. In fact, our datasuggests that $p(d)$ is non-zero on the interval $[d_\text{min}, d_\text{max}]\approx [4/3,8/3]$ and may develop a discrete part ($\delta$-function) at $d=2$in infinite-volume limit. The latter invokes the possibility that combinationof quantum mechanics and pure disorder can lead to emergence of topologicaldimensions. Although $d_\text{IR}$ is based on effective counting of which$p(d)$ has no a priori knowledge, $d_\text{IR} \ge d_\text{max}$ is an exactfeature of the ensuing formalism. Possible connection of our results to recentfindings of $d_\text{IR} \approx 2$ in Dirac near-zero modes of thermal quantumchromodynamics is emphasized.

Topological Dimensions from Disorder and Quantum Mechanics?