Linear statistics and pushed Coulomb gas at the edge of β-random matrices: Four paths to large deviations. (22nd February 2019)

• Record Type: Journal Article
• Authors: Krajenbrink, Alexandre; Le Doussal, Pierre
• Is Part Of: Europhysics letters. Volume 125:Number 2(2019)
• Publication Date: 2019-02-22
• Subjects: 02.10.Yn; Physics -- Periodicals; Electronic journals; 530.05
• DOI: 10.1209/0295-5075/125/20009
• Languages: English
• ISSNs: 0295-5075

Abstract

Abstract: The Airy β point process, , describes the eigenvalues at the edge of the Gaussian β ensembles of random matrices for large matrix size . We study the probability distribution function (PDF) of linear statistics for large parameter t . We show the large deviation forms and for the cumulant generating function and the PDF. We obtain the exact rate function, or excess energy, using four apparently different methods: i) the electrostatics of a Coulomb gas, ii) a random Schrödinger problem, i.e., the stochastic Airy operator, iii) a cumulant expansion, iv) a non-local non-linear differential Painlevé-type equation. Each method was independently introduced previously to obtain the lower tail of the Kardar-Parisi-Zhang equation. Here we show their equivalence in a more general framework. Our results are obtained for a class of functions ϕ, the monotonous soft walls, containing the monomials and the exponential and equivalently describe the response of a Coulomb gas pushed at its edge . The small u behavior of the excess energy exhibits a change between a non-perturbative hard-wall–like regime for (third-order free-to-pushed transition) and a perturbative deformation of the edge for (higher-order transition). Applications are given, among them i) truncated linear statistics such as, leading to a formula for the PDF of the ground-state energy of non-interacting fermions in a linear plus random potential, ii) interacting spinless fermions in a trap at the edge of a Fermi gas, iii) traces of large powers of random matrices.