We develop a first-principles approach to compute the counting statistics in the ground state of $N$ noninteracting spinless fermions in a general potential in arbitrary dimensions $d$ (central for $d>1$). In a confining potential, the Fermi gas is supported over a bounded domain. In $d=1$, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions ${\mathcal{N}}_{\mathcal{D}}$ in a domain $\mathcal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\mathcal{N}}_{\mathcal{D}}$ grows as $N^{(d-1)/d}(A_{d}logN+B_d)$ for large $N$, and obtain the explicit dependence of $A_d,B_d$ on the potential and on the size of $\mathcal{D}$ (for a spherical domain in $d>1$). This generalizes the free-fermion results for microscopic domains, given in $d=1$ by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem $\mathcal{D}$, in any dimension, supported by exact results for $d=1$.

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• Fermi gases
• Full counting statistics
• Random matrix theory