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

This paper proves that flat Jackiw--Teitelboim gravity emerges as the conformal scaling limit of random flat combinatorial disks by establishing a branching exploration criterion that connects discrete row-by-row peeling processes to a continuum limit described by a harmonic Gaussian field.

Original author: Morris Ang

Published 2026-10-30

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CC BY 4.0

Original author: Morris Ang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the fabric of space itself is not smooth and continuous, but made of tiny, discrete building blocks, like a mosaic or a grid. In the realm of theoretical physics, scientists have long tried to understand how this microscopic, pixelated reality gives rise to the smooth, curved universe we experience every day. This quest often involves studying random surfaces—shapes that fluctuate and change, much like a crumpled piece of paper that is constantly being reshaped. For decades, researchers have focused on a specific type of random surface known as the Brownian map, which describes a universe with a certain kind of negative curvature, similar to the inside of a saddle. However, there is another, simpler kind of universe to consider: one that is perfectly flat, like a sheet of paper, but still random and fluctuating. This is the domain of Jackiw–Teitelboim gravity, a theory that describes two-dimensional flat space with a fluctuating boundary. While physicists have proposed what this flat universe should look like in the limit of infinite size, proving that it actually emerges from a simple, discrete model has remained a difficult challenge.

In a new study, Morris Ang from the University of California, San Diego, provides a rigorous proof that flat Jackiw–Teitelboim gravity does indeed arise as the limit of a specific random model. The researchers started with a collection of "rectilinear disks," which are shapes that look like a floor plan made entirely of rectangles joined together. These shapes are not fixed; they are generated randomly according to a specific set of rules that weigh different configurations based on their size and the number of corners they possess. The team focused on a "critical" version of this model, where the rules are tuned precisely so that the shapes are large and complex but still finite. They then imagined taking these discrete, blocky shapes and zooming out, shrinking the individual rectangles until they became invisible, effectively turning the jagged, pixelated surface into a smooth, continuous one.

The core of the discovery lies in how the researchers tracked the evolution of these shapes. Instead of looking at the whole surface at once, they used a method called a "branching exploration." Imagine standing at the edge of one of these random shapes and walking inward along a straight line. As you walk, you periodically look to your left and right to see how far you can see before hitting the edge of the shape. The distances you measure to the left and right boundaries change as you move forward. In the discrete model, these distances jump around in a specific, predictable way. The researchers showed that as the shapes get larger and the steps in the walk get smaller, these jumping distances settle into a smooth, continuous pattern governed by a specific type of random motion known as a Cauchy process. This process is distinct from the more common random walks because it allows for occasional, large jumps, which correspond to the sudden appearance of new, unexplored regions in the shape.

By carefully analyzing this movement, the team constructed a mathematical bridge between the discrete, blocky world and the smooth, continuous one. They proved that if you take a random rectilinear disk with a very long base and rescale it, the resulting shape converges to a specific continuum surface. This limit is not just any random surface; it is exactly the one described by the path-integral formulation of flat Jackiw–Teitelboim gravity. The surface is defined by a metric that is the exponential of a harmonic Gaussian field, a fancy way of saying the height of the surface fluctuates in a way that is perfectly balanced and smooth, yet random. The researchers demonstrated that the random fluctuations of the discrete shapes, when viewed from a distance, become indistinguishable from this specific continuum theory.

The proof relied on a deep structural insight: the way these shapes break apart when cut. When the researchers cut a continuous flat surface along a specific path, the remaining pieces become independent of each other, provided you know the length of the cut. This property, known as a branching Markov property, allowed them to treat the complex, jagged shapes as a collection of simpler, independent parts. They showed that the discrete models obey a similar rule, where cutting the shape reveals smaller, independent shapes that behave like the original. By verifying that the discrete models satisfy three key conditions—a specific way of breaking apart, a tightness that prevents the shapes from becoming infinitely large in a chaotic way, and a convergence of the length measurements—they established that the limit must be the flat Jackiw–Teitelboim gravity surface.

This work is significant because it confirms a long-standing conjecture about the universality of flat gravity. Just as many different types of random maps converge to the same Brownian map, this study suggests that many different models of flat, discrete surfaces will all converge to the same smooth, flat continuum theory. The researchers did not just simulate this behavior; they provided a complete mathematical proof that the convergence happens. They also identified the precise nature of the fluctuations in the final surface, describing them as a harmonic Gaussian field closely related to a concept called holomorphic multiplicative chaos. This result connects the discrete, combinatorial world of counting shapes with the continuous, geometric world of quantum gravity, offering a clear and proven pathway from the microscopic to the macroscopic in the context of flat space.

Technical Summary: JT Gravity as a Conformal Scaling Limit of Random Flat Combinatorial Disks

Problem Statement
The paper addresses the scaling limit of discrete random flat surfaces, specifically "rectilinear disks." These are topological disks equipped with a flat Riemannian metric where the boundary consists of geodesic segments meeting at angles of $\pi/2$ (convex) or $3\pi/2$ (concave). The central question is whether a specific critical ensemble of such discrete disks, known as Boltzmann rectilinear disks, converges to a continuum random surface described by flat Jackiw–Teitelboim (JT) gravity.

While the universality of the Brownian map for random planar maps (without flatness constraints) is well-established, the scaling limit for flat discrete surfaces has been a conjecture. Ferrari [Fer25, Fer26] proposed that discrete flat disks tiled by squares, triangles, or hexagons should converge to a continuum theory of flat JT gravity, where the metric is the exponential of a harmonic Gaussian field. This paper provides a rigorous proof of this convergence for a specific model and establishes a general criterion for such universality.

Methodology
The proof relies on a "branching exploration" strategy, analogous to the peeling process used in random planar maps, but adapted for flat geometry.

Discrete Exploration (Budd's Row-by-Row): The author utilizes Budd's [Bud25] row-by-row decomposition of rigid quadrangulations. By drawing horizontal and vertical rays from concave corners, a discrete disk is decomposed into rectangles. A "vertical exploration" is defined by following a vertical ray from a random point on the base, recording the lengths of horizontal chords ( $L_t, R_t$ ) as the ray progresses. This process generates a tree of "children" (unexplored sub-disks).
Continuum Exploration (Branching of JT Sectors): The author constructs a corresponding continuum exploration for flat JT gravity. This relies heavily on a companion paper [Ang] which proves a "conformal welding" identity: cutting an angle- $\pi$ JT sector along a vertical flow line yields two conditionally independent angle- $\pi/2$ JT sectors.
Stochastic Processes: The length processes $(L_t, R_t)$ in the continuum limit are identified as a Doob $h$ -transform of a pair of independent symmetric Cauchy processes, conditioned to stay in the positive quadrant and die continuously at the origin. The $h$ -function is derived from the partition function of the JT sectors.
Convergence Criterion (Theorem 1.3): The author formulates a general criterion for convergence. A discrete model converges to flat JT gravity if:
- It satisfies a branching Markov property (children are conditionally independent given their base lengths).
- The vertical exploration data (length processes and child attachment information) converges to the continuum counterpart.
- A tightness condition holds for the maximal horizontal widths of the explored regions.

Key Contributions and Results

Proof of Convergence for Boltzmann Rectilinear Disks: The primary result (Theorem 1.1) establishes that as the base length $B \to \infty$ , a Boltzmann rectilinear disk with base length $B$ , when rescaled and conformally uniformized, converges in law to the flat JT gravity surface $S$ . The convergence is in the topology of uniform convergence of conformal factors on compact subsets of the interior.
- The limiting conformal factor is described by a harmonic Gaussian field closely related to holomorphic multiplicative chaos.
- The metric on the limit surface is given by $e^{2\sigma(z)}|dz|^2$ , where $\sigma$ is a harmonic Gaussian field with specific correlations.
General Convergence Criterion (Theorem 1.3): The paper provides a robust framework for proving universality. It reduces the problem of showing a discrete flat disk model converges to JT gravity to verifying three properties: (a) the branching Markov property, (b) tightness of maximal widths, and (c) convergence of the vertical exploration data (specifically, the length process converging to the conditioned Cauchy pair).
Construction of Continuum Branching Exploration: The author explicitly constructs the branching exploration of the continuum JT surface. They show that the "children" created during the exploration are conditionally independent JT sectors, and their base lengths evolve according to the conditioned Cauchy processes. This construction mirrors the discrete row-by-row decomposition.
Partition Function Analysis: The paper computes the partition function for Boltzmann rectilinear disks (Lemma 2.1), showing it is finite and behaves asymptotically as $Z(b) \sim (2\pi b^2)^{-1}$ . It also derives the partition functions for angle- $\pi$ and angle- $\pi/2$ JT sectors, identifying the latter with the entrance density of a Cauchy excursion.

Significance and Claims
The paper claims to establish flat JT gravity as the universal scaling limit for a broad class of discrete flat disks, provided they satisfy the branching Markov property and the convergence criteria outlined.

Universality: The author expects that these properties hold for other models, such as flat triangulations or quadrangulations tiled by squares, triangles, or hexagons, thereby confirming Ferrari's conjecture that these different discretizations yield the same continuum theory.
Connection to Quantum Gravity: The work draws a parallel to the identification of the Brownian map with Liouville Quantum Gravity (LQG). While the Brownian map corresponds to $\sqrt{8/3}$ -LQG (associated with a Gaussian Free Field), the flat JT gravity limit corresponds to the harmonic part of a Gaussian free field. This connects natural discrete flat models to a continuum limit predicted by two-dimensional quantum gravity literature.
Methodological Advance: The paper introduces a "branching exploration" technique for flat surfaces that is distinct from the metric-space peeling used for the Brownian map. It leverages conformal welding and flow lines, offering a new route to studying scaling limits of flat geometries.

The author explicitly states that the main mathematical ideas and proof strategies are their own, while acknowledging that the proof of a specific proposition regarding the continuity of the developing map (Proposition 3.2) was generated using AI (GPT-6 Astra) and draws heavily on arguments from [AHS24a]. The paper does not claim to solve the full universality problem for all flat disk models but provides the "road" (Theorem 1.3) to establishing it for any model satisfying the stated criteria.

Technical Summary: JT Gravity as a Conformal Scaling Limit of Random Flat Combinatorial Disks

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