Universal ground state properties of free fermions in a $d$-dimensional trap

We study the ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large $N$. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large $N$. The associated edge scaling function is computed exactly and it generalizes the $1d$ result known from random matrix theory. In addition, we calculate the kernel (that characterizes any $n$-point correlation function) for large $N$ and show that, when appropriately scaled, it depends only on dimension $d$, but has otherwise universal scaling forms, both in the bulk as well as at the edges. The edge kernel, for higher $d$, generalizes the Airy kernel in one dimension, well known from random matrix theory.