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Flat Jackiw–Teitelboim gravity emerges as the scaling limit of random flat combinatorial disks

November 1, 2026arXiv: 2610.40645v1

Researchers have shown that flat Jackiw–Teitelboim (JT) gravity appears when random flat combinatorial disks are scaled in a particular way. By studying disks whose boundaries are built from straight segments, they proved that letting the length of one side grow without bound and rescaling the surface leads to a universal random shape governed by JT gravity.

Plot of disc simulations with base length increasing, showing darker overlap regions—illustrating convergence of the rescaled surface to flat JT gravity.
Figure 1. Simulations of Boltzmann rectilinear disks with increasing base length B, illustrating that as B tends to infinity the rescaled surface converges to a continuum random surface of flat JT gravity.Ang, Morris · arXiv:2610.40645v1 · Figure 1: · CC BY 4.0

The key to the proof is a criterion expressed through branching explorations. As the distinguished side becomes long, the surface’s geometry, when mapped to a fixed domain, converges in law to the continuum random surface defined by Ferrari’s path‑integral formulation of JT gravity. The convergence is smooth for the conformal factor on every compact interior region, and the limiting field is a harmonic Gaussian field related to holomorphic multiplicative chaos.

Illustration of JT sector simulations, showing overlap patterns—highlighting the convergence of random flat combinatorial disks to flat JT gravity.
Figure 2. Simulations of angle‑π JT sectors isometrically immersed in the plane, with the Gaussian series truncated at 107 terms, depicting that random flat combinatorial disks converge to flat JT gravity in the scaling limit.Ang, Morris · arXiv:2610.40645v1 · Figure 2: · CC BY 4.0

A central part of the argument is the construction of a branching exploration in which interval lengths evolve as conditioned Cauchy processes. The unexplored parts are conditionally independent, which mirrors Budd’s row‑by‑row peeling strategy for discrete disks. Under an additional tightness requirement on the maximal horizontal widths of the discrete disks, this exploration gives a full conformal scaling limit of the Boltzmann rectilinear disks.

The study also notes that other combinatorial disk models exhibit similar first‑row recursions, suggesting broader applicability of the convergence framework, though the tightness condition remains essential for the conformal limit to hold.

1. Flat JT gravity arises as a conformal scaling limit of random flat combinatorial disks. (f1) 2. A branching‑exploration criterion establishes convergence to this limit. (f2) 3. Rescaling a Boltzmann rectilinear disk with base length tending to infinity yields a continuum random surface of flat JT gravity, with smooth convergence of conformal factors on compact interior sets. (f3) 4. The convergence criterion requires a tightness condition on the maximal horizontal widths of the discrete disks. (f4)

Source paper

JT gravity as a conformal scaling limit of random flat combinatorial disks

MathematicsMathematics
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