Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations

The Airy$_\beta$ point process, $a_i \equiv N^{2/3} (\lambda_i-2)$, describes the eigenvalues $\lambda_i$ at the edge of the Gaussian $\beta$ ensembles of random matrices for large matrix size $N \to \infty$. We study the probability distribution function (PDF) of linear statistics ${\sf L}= \sum_i t \varphi(t^{-2/3} a_i)$ for large parameter $t$. We show the large deviation forms $\mathbb{E}_{{\rm Airy},\beta}[\exp(-{\sf L})] \sim \exp(- t^2 \Sigma[\varphi])$ and $P({\sf L}) \sim \exp(- t^2 G(L/t^2))$ for the cumulant generating function and the PDF. We obtain the exact rate function $\Sigma[\varphi]$ using four apparently different methods (i) the electrostatics of a Coulomb gas (ii) a random Schr\"odinger problem, i.e. the stochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-linear differential Painlev\'e type equation. Each method was independently introduced to obtain the lower tail of the KPZ equation. Here we show their equivalence in a more general framework. Our results are obtained for a class of functions $\varphi$, the monotonous soft walls, containing the monomials $\varphi(x)=(u+x)_+^\gamma$ and the exponential $\varphi(x)=e^{u+x}$ and equivalently describe the response of a Coulomb gas pushed at its edge. The small $u$ behavior of the excess energy $\Sigma[\varphi]$ exhibits a change at $\gamma=3/2$ 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 $\sum_{i=1}^{N_1} a_i$, leading to a formula for the PDF of the ground state energy of $N_1 \gg 1$ noninteracting 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.