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 $V(x)=m{\omega }^2{x}^2\mathrm{/}2$ at finite inverse temperature $\beta = 1/T$\mathrm{.}Theaveragedensityoffermions$\rho_N(x,T)$atposition$x$ is derived. For $N\gg 1$ and $\beta \sim \mathcal{O}(1\mathrm{/}N)$, ${\rho }_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1\mathrm{/}N$, and the Wigner semi-circle law at low temperature, for $\beta \gg N^{-1}$\mathrm{.}Inthelatterregime,weunveilascalinglimit,for$\beta {\hbar \omega}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x\sim \pm \sqrt{2\mathrm{\hslash }N\mathrm{/}(m\omega )}$, are described by a limiting kernel $K_b^{\mathrm{f}\mathrm{f}}(s,s^{\mathrm{\prime }})$ 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 $K_b^{\mathrm{f}\mathrm{f}}(s,s^{\mathrm{\prime }})$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t=b^3$.