A Black Hole Airy Tail

Abstract

In Jackiw-Teitelboim (JT) gravity, which is dual to a random matrix ensemble, the annealed entropy differs from the quenched entropy at low temperatures and goes negative. However, computing the quenched entropy in JT gravity requires a replica limit that is poorly understood. To circumvent this, we define an intermediate quantity called the semi-quenched entropy, which has the positivity properties of the quenched entropy, while requiring a much simpler replica trick. We compute this in JT gravity in different regimes using i) a bulk calculation involving wormholes corresponding to the Airy limit of the dual matrix integral and ii) a boundary calculation involving one-eigenvalue instanton saddles proposed by Hernández-Cuenca, demonstrating consistency between these two calculations in their common regime of validity. We also clarify why similar one-eigenvalue instanton saddles cannot be used to compute the quenched entropy due to a breakdown of the saddle-point approximation for the one-eigenvalue instanton in the replica limit. Our results show how to use the gravitational path integral to prove that black holes in JT gravity have isolated ground states and to study their properties.

Figures (4)

• Figure 1: Comparing semi-quenched, annealed and quenched second Rényi entropies, zoomed in near the Airy edge $\beta=\alpha e^{2S_0/3}$ (with $\gamma=1$). Semi-quenched and annealed Rényis are obtained analytically, the quenched Rényi is obtained numerically. Notice the different low temperature behavior (inset): the quenched entropy has a $\alpha^{-3}$ power-law decay which can be obtained analytically (dotted blue line), whereas the semi-quenched entropy decays exponentially. Unlike the annealed entropy, both these entropies remain positive.
• Figure 2: Since the effective potential of JT gravity is nonperturbatively unstable, one must deform the contour (solid black line) along the line of steepest ascent starting at $z=-\frac{1}{4}$ (red dot). Complex one-eigenvalue instantons at various $k=\beta/e^{S_0}$ are shown (blue dots). To pick up the one-eigenvalue instantons at a specific $k$, one must push the contour along the line of steepest descent in the instanton action (dotted blue line). An example contour deformation for $k=1$ is sketched in dashed black.
• Figure 3: A sketch of the distribution of the one-eigenvalue instanton $p(\lambda)\exp(-\beta\lambda)$ (blue), compared to the Airy edge $p(\lambda)$ (orange). (Not to scale). As we decrease the temperature, the one-eigenvalue instanton moves away from the edge (black arrow). Notice that the continuum does not give an extra contribution to $\langle Z(\beta)\rangle$.
• Figure 4: Plotting the joint distribution for two-eigenvalue instanton $p(\lambda_1,\lambda_2)e^{-\beta(\lambda_1+\lambda_2)}$. Here, we plot for $\beta=5$. As we increase $\beta$, the location of the two eigenvalues flows down the $\lambda_1=\lambda_2$ axis (black arrows). Thus, the saddle for large $\alpha$ is given by $\lambda_1\approx\lambda_2$.