Finite-Temperature Free Fermions and the Kardar-Parisi-Zhang Equation at Finite Time

We consider the system of N one-dimensional free fermions confined by a harmonic well V(x)=mω²x²/2 at finite inverse temperature β=1/T. The average density of fermions ρN(x,T) at position x is derived. For N1 and β~O(1/N), ρN(x,T) is given by a scaling function interpolating between a Gaussian at high temperature, for β1/N, and the Wigner semicircle law at low temperature, for βN^-1. In the latter regime, we unveil a scaling limit, for βω=bN^-1/3, where the fluctuations close to the edge of the support, at x~±√2N/(mω), are described by a limiting kernel Kbᶠᶠ(s,s^') that depends continuously on b and is a generalization of the Airy kernel, found in the Gaussian unitary ensemble of random matrices. Remarkably, exactly the same kernel Kbᶠᶠ(s,s^') arises in the exact solution of the Kardar-Parisi-Zhang equation in $1+1$ dimensions at finite time t, with the correspondence t=b³.