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The Airy$_\beta$ point process, $a_i \equiv N^{2/3} (\lambda_i-2)$, describesthe eigenvalues $\lambda_i$ at the edge of the Gaussian $\beta$ ensembles ofrandom matrices for large matrix size $N \to \infty$. We study the probabilitydistribution function (PDF) of linear statistics ${\sf L}= \sum_i t\varphi(t^{-2/3} a_i)$ for large parameter $t$. We show the large deviationforms $\mathbb{E}_{{\rm Airy},\beta}[\exp(-{\sf L})] \sim \exp(- t^2\Sigma[\varphi])$ and $P({\sf L}) \sim \exp(- t^2 G(L/t^2))$ for the cumulantgenerating function and the PDF. We obtain the exact rate function$\Sigma[\varphi]$ using four apparently different methods (i) theelectrostatics of a Coulomb gas (ii) a random Schr\"odinger problem, i.e. thestochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-lineardifferential Painlev\'e type equation. Each method was independently introducedto obtain the lower tail of the KPZ equation. Here we show their equivalence ina 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 $\varphi(x)=e^{u+x}$ andequivalently describe the response of a Coulomb gas pushed at its edge. Thesmall $u$ 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 theedge for $\gamma>3/2$ (higher order transition). Applications are given, amongthem: (i) truncated linear statistics such as $\sum_{i=1}^{N_1} a_i$, leadingto 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) tracesof large powers of random matrices.

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