# Periodic Airy process and equilibrium dynamics of edge fermions in a trap

### Pierre Le Doussal 1 Satya N. Majumdar 2 Gregory Schehr 2

#### Annals of Physics, 2017, 383, pp.312 - 345

We establish an exact mapping between (i) the equilibrium (imaginary time) dynamics of non-interacting fermions trapped in a harmonic potential at temperature $T=1/\beta$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU) particles constrained to return to their initial positions after time $\beta$. Exploiting the determinantal structure of the process we compute the universal correlation functions both in the bulk and at the edge of the trapped Fermi gas. The latter corresponds to the top path of the non-intersecting OU particles, and leads us to introduce and study the time-periodic Airy$_2$ process, ${\cal A}^b_2(u)$, depending on a single parameter, the period $b$. The standard Airy$_2$ process is recovered for $b=+\infty$. We discuss applications of our results to the real time quantum dynamics of trapped fermions.

• 1. LPTENS - Laboratoire de Physique Théorique de l'ENS
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques