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

The Airy(beta) point process, a(i) (math) N-2/3 (lambda(i)-2), describes the eigenvalues lambda(i) at the edge of the Gaussian beta ensembles of random matrices for large matrix size N -> infinity. We study the probability distribution function (PDF) of linear statistics L = Sigma(i)t phi(t(-2/3 )a(i)) for large parameter t. We show the large deviation forms E-Airy, (beta)[exp(-L)] similar to exp(-t(2)Sigma[phi]) and P(L) similar to exp(-t(2)G(L/t(2))) for the cumulant generating function and the PDF. We obtain the exact rate function, or excess energy, Sigma[phi] using four apparently different methods: i) the electrostatics of a Coulomb gas, ii) a random Schrodinger problem, i.e., the stochastic Airy operator, iii) a cumulant expansion, iv) a non-local non-linear differential Painleve-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 phi, the monotonous soft walls, containing the monomials phi(x) = (u + x)(+)(gamma) 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 gamma < 3/2 (third-order free-to-pushed transition) and a perturbative deformation of the edge for gamma > 3/2 (higher-order transition). Applications are given, among them i) truncated linear statistics such as Sigma(N1)(i=1) a(i), leading to a formula for the PDF of the ground-state energy of N-1 >> 1 non-interacting fermions in a linear plus random potential, ii) (beta - 2)/r(2) interacting spinless fermions in a trap at the edge of a Fermi gas, iii) traces of large powers of random matrices. Copyright (C) EPLA, 2019