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

## TL;DR

This paper establishes a precise connection between the equilibrium dynamics of trapped fermions and non-intersecting Ornstein-Uhlenbeck particles, introducing a new time-periodic Airy process to describe edge fluctuations.

## Contribution

It introduces and analyzes the time-periodic Airy$_2$ process, extending the classical Airy$_2$ process to finite periods, and applies this to quantum dynamics of trapped fermions.

## Key findings

- Derived universal correlation functions for bulk and edge of fermion gas
- Introduced the time-periodic Airy$_2$ process ${
m extbf{A}}^b_2(u)$
- Connected equilibrium fermion dynamics to non-intersecting OU particles

## Abstract

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.