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

Abstract

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 $\mathsf{L}={\sum }_{i}t\phi (t^{-2/3}{a}_i)$ for large parameter $t$. We show the large deviation forms ${\mathbb{E}}_{\mathrm{Airy},\beta }[\exp (-\mathsf{L})]\sim \exp (-t^2\Sigma [\phi ])$ and $P(\mathsf{L})\sim \exp (-t^{2}G(L/t^2))$ for the cumulant generating function and the PDF. We obtain the exact rate function $\Sigma [\phi ]$ 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 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 $\phi$, the monotonous soft walls, containing the monomials $\phi (x)=(u+x{)}_{+}^{\gamma }$ and the exponential $\phi (x)=e^{u+x}$ and equivalently describe the response of a Coulomb gas pushed at its edge. The small $u$ behavior of the excess energy $\Sigma [\phi ]$ 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.

Main text : 8 pages. Supp mat : 49 pages

References in corpus (11)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Cleaning large correlation matrices: tools from random matrix theory
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• From Random Matrices to Stochastic Operators
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Random matrices and entanglement entropy of trapped Fermi gases
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

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