The hyperbolic-lattice Schwarzian-limit conjecture

Let $L$ be the curvature length scale and $\ell_0$ the lattice edge length for regular or semi-regular hyperbolic tessellations, and let $n$ be the polygon length. Hyperbolic-lattice Schwarzian conjecture. The Schwarzian effective description of negative-curvature JT gravity is the continuum limit of the corresponding SOP lattice models, obtained by

$$L\propto\ell_0\to0,\qquad n\to\infty,\qquad nL\ fixed,$$

with $nL$ proportional to the macroscopic length $\ell$; in this limit the critical exponent is $\nu=1$ and the Hausdorff dimension $1/\nu$ is one. The construction is proposed as an effective, rather than microscopic, description because regular hyperbolic tessellations have a fixed area scale.