The negative-curvature JT area-tail conjecture

Let $\rho^-_{\beta_{\mathrm q}}(A)$ be the area distribution of JT gravity with negative curvature $-2/L^2$, coupled to conformal matter with $c\leq0$, and let $\nu$ be the corresponding critical exponent. Negative-curvature area-tail conjecture. There are a strictly positive constant $\kappa^-_\nu$ and a strictly increasing exponent $\chi^-_\nu$ such that

$$\ln\rho^-_{\beta_{\mathrm q}}(A)\sim-\kappa^-_\nu\left(\frac{A}{\beta_{\mathrm q}^{\nu}L}\right)^{\chi^-_\nu}\qquad(A\to+\infty),$$

with

$$\lim_{\nu\to1/2^+}\chi^-_\nu=2,\qquad\lim_{\nu\to1^-}\chi^-_\nu=+\infty.$$

The predicted tail is rapidly decaying enough to allow all cosmological constants and is motivated by hyperbolic Brownian-path behavior and the semiclassical limit.