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We develop a first-principle approach to compute the counting statistics inthe ground-state of $N$ noninteracting spinless fermions in a general potentialin arbitrary dimensions $d$ (central for $d>1$). In a confining potential, theFermi gas is supported over a bounded domain. In $d=1$, for specificpotentials, this system is related to standard random matrix ensembles. Westudy the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ ina domain $\cal{D}$ of macroscopic size in the bulk of the support. We show thatthe variance of ${\cal N}_{\cal D}$ grows as $N^{(d-1)/d} (A_d \log N + B_d)$for large $N$, and obtain the explicit dependence of $A_d, B_d$ on thepotential and on the size of ${\cal D}$ (for a spherical domain in $d>1$). Thisgeneralizes the free-fermion results for microscopic domains, given in $d=1$ bythe Dyson-Mehta asymptotics from random matrix theory. This leads us toconjecture similar asymptotics for the entanglement entropy of the subsystem$\cal{D}$, in any dimension, supported by exact results for $d=1$.

Counting statistics for non-interacting fermions in a $d$-dimensional potential