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

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, ${\mathcal{A}}_2^b(u)$, 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.

33 pages, 4 figures. The introduction slightly rewritten (and different from the published version in the journal) and misprints corrected

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• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• The periodic Schur process and free fermions at finite temperature
• Constraint Ornstein-Uhlenbeck bridges
• On the origins of Riemann-Hilbert problems in mathematics
• Non-intersecting Brownian bridges in the flat-to-flat geometry
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• Free fermions and the classical compact groups
• Non-interacting fermions in hard-edge potentials
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Quench dynamics of noninteracting fermions with a delta impurity
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Stationary time correlations for fermions after a quench in the presence of an impurity
• Decay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices