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In this work, we study the statistics of a generic noninteracting many-bodyfermionic system whose single-particle counterpart has a single-particlemobility edge (SPME). We first prove that the spectrum and the extensiveconserved quantities follow the multivariate normal distribution with avanishing standard deviation $\sim O(1/\sqrt L)$ in the thermodynamic limit,regardless of SPME. Consequently, the theorem rules out an infinite-temperatureor high-temperature many-body mobility edge (MBME) for generic noninteractingfermionic systems. Further, we also prove that the spectrum of a fermionicmany-body system with short-range interactions is qualitatively similar to thatof a noninteracting many-body system up to the third-order moment. Theseresults partially explain why neither short-range [1] nor long-rangeinteracting systems exhibit an infinite-temperature MBME.

Statistics of noninteracting many-body fermionic states: The question of a many-body mobility edge