Linear statistics and pushed Coulomb gas at the edge of <i>β</i> -random matrices: Four paths to large deviations

Abstract

The Airyβ 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.

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