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

Abstract

The Airy β point process, ai ≡ N 2/3 (λi − 2), describes the eigenvalues λi at the edge of the Gaussian β ensembles of random matrices for large matrix size N → ∞. We study the probability distribution function (PDF) of linear statistics L = i tϕ(t −2/3 ai) for large parameter t. We show the large deviation forms E Airy,β [exp(−L)] ∼ exp(−t 2 Σ[ϕ]) and P (L) ∼ exp(−t 2 G(L/t 2 )) for the cumulant generating function and the PDF. We obtain the exact rate function Σ[ϕ] 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 [1][2][3][4]. 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 ϕ(x) = (u + x) γ + and the exponential ϕ(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 Σ[ϕ] exhibits a change at γ = 3/2 between a non-perturbative hard wall like regime for γ < 3/2 (third order free-to-pushed transition) and a perturbative deformation of the edge for γ > 3/2 (higher order transition). Applications are given, among them: (i) truncated linear statistics such as N 1 i=1 ai, leading to a formula for the PDF of the ground state energy of N1 1 noninteracting fermions in a linear plus random potential (ii) ∼ (β − 2)/r 2 interacting spinless fermions in a trap at the edge of a Fermi gas (iii) traces of large powers of random matrices.√ β 2 v(x), H SAO = t −2/3 H SAO with energy levels b i = t −2/3 ε i = −t −2/3 a i and obtain

Truncated Linear Statistics Associated with the Top Eigenvalues of Random Matrices

“…This behaviour is universal, meaning that it neither depends on the choice of the ensemble, nor on the choice of the function f , provided that it is monotonous. This problem has also been considered in the edge regime N K 1, but again for a class of monotonous functions [59]. Finally, truncated linear statistics have been studied for the one dimensional plasma in Ref.…”

Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls

“…Similar phase transitions of the pulled-to-pushed type have been observed in several physics models related to random matrices [13, 47], including large- N gauge theories [3, 36, 57, 64], longest increasing subsequences of random permutations [39], quantum transport fluctuations in mesoscopic conductors [12, 33, 34, 62, 63], non-intersecting Brownian motions [28, 58], entanglement measures in a bipartite system [17, 26, 27, 49], random tilings [10, 11], random landscapes [29], and the tail analysis in the KPZ problem [41]. (See also the recent popular science articles [5, 65]).…”

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“…The precise rate function of its lower-tail LDP was later proved in [72] and [14], which confirmed the prediction of existing physics literature. The four different routes of deriving the lower-tail LDP in [63], [24], [46] and [72] were later shown to be closely related in [44]. A new route has also been recently obtained in the physics work of [48] (see also [59]).…”