Low-temperature entropy in JT gravity

Mehrdad Mirbabayi,Oliver Janssen

doi:10.1007/jhep06(2021)074

Abstract

For ensembles of Hamiltonians that fall under the Dyson classification of random matrices with β ∈ {1, 2, 4}, the low-temperature mean entropy can be shown to vanish as 〈S(T)〉 ∼ κTβ + 1. A similar relation holds for Altland-Zirnbauer ensembles. JT gravity has been shown to be dual to the double-scaling limit of a β = 2 ensemble, with a classical eigenvalue density propto {e}^{S_0}sqrt{E} when 0 < E ≪ 1. We use universal results about the distribution of the smallest eigenvalues in such ensembles to calculate κ up to corrections that we argue are doubly exponentially small in S0.

Highlights

Where the expectation on the right is over an ensemble of random N ×N Hermitian matrices
We instead focus on the particular example of JT gravity and its cousins with known ensemble duals to analyze the low-temperature behavior of Fq(T ) on the matrix model side
In the case of JT, the edge region is controlled by the well-known Airy kernel of unitary ensembles, and much is known about the distribution of the smallest eigenvalue [15], as well as its distance to the eigenvalue [16, 17]

Summary

Review of JT gravity and its matrix dual

JT gravity is a simple dilaton-gravity model in two spacetime dimensions [3, 4]. Its bulk action with an appropriately rescaled negative cosmological constant is IJT. The Jacobian of the transformation H = U †diag(λ1, λ2, · · · )U is a Vandermonde determinant, giving the following partition function for the matrix eigenvalues. The Vandermonde determinant acts as a repulsive force among the eigenvalues. This manifests itself in the expectation value of the eigenvalue density ρtotal(λ) = δ(λ − λi) , i (2.6). Unless the potential is fine-tuned, the behavior near the edge of the distribution is universal in the N → ∞ limit. To focus on this region, one takes the double-scaling limit. The knowledge of ρ0 (or the closely related spectral curve) is enough to set up the matrix model genus expansion, and this was shown in [5] to match the genus expansion of JT gravity

Low-temperature free energy in matrix models

The Airy limit

Discussion

Perret-Schehr distribution

WKB approximation