We study $N$ non-interacting fermions in a trap. Using random matrix theory (RMT)

we calculate various observables in the ground state, focusing on large $N$.

In the bulk, agreement with local density approximation (LDA) is recovered. At the

edge of the Fermi gas however LDA fails, while RMT leads to predictions which we

show are universal, independent of the details of the trap potential. In particular the

position of the right most fermion fluctuates according to the celebrated Tracy Widom law. 

We then extend these results to finite temperature and arbitrary space dimension,

providing new universal predictions for correlations. These could in principle be tested

in experiments using the Fermi gas microscope. A surprising connection to the stochastic growth 

Kardar-Parisi-Zhang equation also emerges from our study. 

Based on the following works:

D.S. Dean, P. Le Doussal, S.N. Majumdar and G.Schehr

    Phys. Rev. Lett. 114, 110402 (2015)

    Phys. Rev. A 94, 063622 (2016)