title: Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time.
abstract: We consider the system of N one-dimensional free fermions confined by a harmonic well V(x)=mω(2)x(2)/2 at finite inverse temperature β=1/T. The average density of fermions ρ(N)(x,T) at position x is derived. For N≫1 and β∼O(1/N), ρ(N)(x,T) is given by a scaling function interpolating between a Gaussian at high temperature, for β≪1/N, and the Wigner semicircle law at low temperature, for β≫N(-1). In the latter regime, we unveil a scaling limit, for βℏω=bN(-1/3), where the fluctuations close to the edge of the support, at x∼±√[2ℏN/(mω)], are described by a limiting kernel K(b)(ff)(s,s') 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)(ff)(s,s') arises in the exact solution of the Kardar-Parisi-Zhang equation in 1+1 dimensions at finite time t, with the correspondence t=b(3).
authors.affiliation: Université Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France.
authors.forename: David S
authors.initials: DS
authors.lastname: Dean
authors.affiliation: CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France.
authors.forename: Pierre
authors.initials: P
authors.lastname: Le Doussal
authors.affiliation: Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France.
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar
authors.affiliation: Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France.
authors.forename: Grégory
authors.initials: G
authors.lastname: Schehr