⚛️ high-energy theory

Cardy random matrix theory

This paper introduces and numerically validates a "Cardy random matrix model" that, by enforcing $\mathbb{S}$ -modular invariance through energy coarse-graining, successfully reproduces the Cardy density of states at macroscopic scales while exhibiting Gaussian Unitary Ensemble statistics at mesoscopic scales, offering a computational pathway toward solving the matrix dual of 3D quantum gravity.

Original authors: Marco Ambrosini, Alexandre Belin, Jan de Boer, Julian Sonner

Published 2026-09-29

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CC BY 4.0

Original authors: Marco Ambrosini, Alexandre Belin, Jan de Boer, Julian Sonner

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to understand the universe, physicists often look for the most fundamental building blocks of reality. For decades, a powerful idea has been that the chaotic, complex behavior of quantum systems—like the jumbled energy levels inside an atom or a black hole—can be described by the same statistical rules that govern random numbers. This field, known as random matrix theory, suggests that if you look closely enough at a chaotic system, its energy levels repel each other in a very specific, predictable way, much like how people instinctively avoid standing too close to strangers in a crowded room. However, describing gravity in three dimensions, particularly the kind of gravity that exists in a universe with a negative curvature known as anti-de Sitter space, has been a stubborn puzzle. Unlike simpler models of gravity, this three-dimensional version requires a theory that respects a deep symmetry called modular invariance, a rule that ensures the physics looks the same whether you view the system from a certain angle or flip it inside out. For a long time, scientists could not build a computer model that respected this rule while also behaving like a chaotic quantum system, leaving a gap between the mathematics of gravity and the statistics of chaos.

A team of researchers has now constructed a new type of mathematical model that bridges this gap, effectively creating a "Cardy random matrix theory." They started with a simple premise: if they could force a collection of random numbers to obey the strict rules of modular invariance, would the resulting pattern of numbers naturally look like the energy spectrum of a chaotic universe? To test this, they did not try to solve the complex equations exactly. Instead, they created a simulation where the numbers were allowed to violate the rules slightly, but only by a tiny, controlled amount. They organized these numbers into groups, or "bins," and applied a mathematical pressure that pushed the number of items in each bin to match a specific target. This target was derived from a famous formula known as the Cardy formula, which predicts how the number of possible states in a system should grow as the energy increases. The researchers found that when they ran their simulation, the numbers arranged themselves perfectly to match this target growth rate, confirming that the rules of modular invariance are strong enough to force the system into the correct shape.

The most surprising discovery, however, was what happened when they looked at the arrangement of the numbers inside those groups. While the overall count of numbers in each bin was rigidly fixed by the modular rules, the individual numbers within a single bin were not static. They moved around, pushing against one another in a way that is characteristic of chaotic systems. The researchers observed that the spacing between these neighboring numbers followed the exact statistical pattern predicted by random matrix theory, a behavior known as level repulsion. This revealed a hidden hierarchy of scales in their model. At the largest scale, the system is frozen and ordered, obeying the strict constraints of gravity. But at a smaller scale, inside each group, the system becomes fluid and chaotic, behaving exactly like a random matrix. This transition from order to chaos happens at a specific threshold, which the researchers identified as a "Thouless time," a concept that marks the moment when a system begins to show its chaotic nature.

To verify these findings, the team used powerful computer simulations to generate millions of random configurations and measured how the numbers distributed themselves. They compared two different scenarios: one where a specific "vacuum" state was present to act as a seed for the growth, and one where it was absent. When the seed was present, the simulation produced the expected exponential growth of states, matching the Cardy formula with high precision. When the seed was removed, the growth stopped, and the system remained bounded, proving that the chaotic tower of states is directly sourced by the vacuum. The simulations also showed that the system displays the correct "spectral form factor," a mathematical tool used to measure how correlations in a system evolve over time. The results showed a linear ramp, a signature of chaos, appearing only after a specific time delay determined by the size of the bins used in the simulation.

This work represents a significant step forward in understanding the quantum nature of gravity. By showing that a simple set of constraints can generate both the correct large-scale density of states and the correct small-scale chaotic statistics, the researchers have provided a concrete example of how gravity and chaos can coexist in a single mathematical framework. Their model does not claim to be the final theory of everything, but it offers a working blueprint for how to build such a theory on a computer. It demonstrates that the rigid laws of modular invariance do not suppress chaos; rather, they set the stage for it to emerge at the right scale. This insight helps resolve a long-standing tension in theoretical physics, suggesting that the universe can be both highly ordered in its global structure and wildly chaotic in its local details, with the transition between the two governed by the very same rules that define the geometry of space and time.

Technical Summary: Cardy Random Matrix Theory

Problem Statement
The paper addresses the challenge of formulating a dual to pure 3D quantum gravity (AdS $_3$ ) as an ensemble average over Conformal Field Theory (CFT) data. While the duality between Jackiw-Teitelboim (JT) gravity and matrix integrals is well-established, the 3D case is more complex due to the presence of both a spectrum and Operator Product Expansion (OPE) coefficients, subject to stringent consistency constraints like crossing symmetry and modular invariance. Previous work introduced a "constraint $^2$ " random tensor-matrix model where the probability distribution is proportional to $\exp(-V^2)$ , with $V$ vanishing on data satisfying bootstrap constraints. However, the scaling limits required to extract physical behavior from this model, particularly how to enforce modular invariance while recovering random matrix theory (RMT) statistics, remained unclear. The authors aim to construct and analyze a simplified "Cardy Random Matrix Theory" (Cardy RMT) that implements S-modular invariance as the sole constraint to understand these scaling limits and the emergence of the Cardy density of states.

Methodology
The authors construct a single-matrix model where the eigenvalues $h_i$ represent the scaling dimensions of primary operators in a 2D CFT. The methodology involves three main steps:

Constraint Regularization: The S-modular invariance condition, $Z(\tau) = Z(-1/\tau)$ , is recast in terms of Virasoro characters. To make this suitable for a matrix potential, the authors regularize the constraint by coarse-graining the spectrum into bins of fixed width $\epsilon$ in Liouville momentum space. This converts the ill-defined equality between a discrete sum of delta functions and a smooth kernel into a condition on the number of eigenvalues $N_{P,\epsilon}$ in each bin.
Constraint $^2$ Potential: Following the "maximal ignorance" prescription, the authors soften the hard constraint into a Gaussian penalty. The partition function takes the form:
$Z = \int \prod_j dh_j \, \Delta(h)^2 \exp\left[ -\frac{1}{4a\epsilon^2} \sum_P \left( N_{P,\epsilon} - \epsilon \sum_i S_{PP_i}(1) \right)^2 \right]$
Here, $\Delta(h)$ is the Vandermonde determinant, $a$ is a parameter controlling the tolerance of the constraint, and $S_{PP'}(1)$ is the modular S-kernel. The potential drives the coarse-grained density of states toward the Cardy profile $\rho_{\text{Cardy}} \sim e^{2\pi Q P}$ .
Scaling Limits and Analysis: The authors investigate the double-scaling limit where $N, E \to \infty$ $N, E \to \infty$ and $a, \epsilon \to 0$ $a, ϵ \to 0$ , while keeping specific combinations fixed (e.g., $t = aN$ $t = a N$ , $G = \epsilon E$ $G = ϵ E$ ). They employ:
- Monte Carlo Simulations: Markov Chain Monte Carlo (MCMC) methods are used to sample the matrix model for finite $N$ and UV cutoffs, testing both "clipped" (spectrum stops at cutoff) and "completed" (frozen modular tail) boundary conditions.
- Analytical Saddle-Point Analysis: Deriving saddle-point equations to determine the macroscopic density and the "Cardy transition" (the point where the vacuum becomes the preferred light state).
- Topological Recursion: Using the cut-pinching approximation, the authors reduce the high-genus spectral curve (arising from multi-cut bin structures) to a genus-1 (and effectively genus-0) problem to compute connected correlators and the spectral form factor.

Key Contributions and Results

Emergence of the Cardy Profile: The numerical and analytical results confirm that the constraint $^2$ potential successfully drives the coarse-grained density of states to the Cardy profile above the black hole threshold, provided the vacuum (identity operator) is occupied. Without the vacuum source, the spectrum remains bounded and does not exhibit exponential growth.
Hierarchy of Scales: The model exhibits a distinct separation of scales:
- Macroscopic Scale ( $> \epsilon$ ): The potential rigidly fixes the bin occupations to the Cardy target. The density appears smooth and non-fluctuating.
- Mesoscopic Scale ( $\sim \epsilon$ ): Inside each bin, the Vandermonde repulsion rearranges eigenvalues. The model displays Gaussian Unitary Ensemble (GUE) statistics, including Wigner-Dyson level spacing and a linear ramp in the connected spectral form factor (SFF).
- Microscopic Scale ( $\sim e^{-S}$ ): Individual levels are resolved, and the SFF saturates to a plateau.
The Cardy Transition: Analytical study of the potential minima reveals a "Cardy transition." For a fixed number of states $N$ , there is a critical value $N_c$ (scaling exponentially with the cutoff $E$ ) above which the configuration with an occupied vacuum has a lower potential energy than configurations with no light states or other light states. This suggests a mechanism for the dynamical selection of the vacuum in a full theory.
Thouless Time and RMT Onset: The width of the coarse-graining window $\epsilon$ (or the energy width $\delta$ ) sets the effective Thouless time $t_{\text{Th}} \sim 1/\delta$ . The model behaves as a continuous theory (decaying SFF $\propto t^{-3}$ ) at times $t < t_{\text{Th}}$ and transitions to RMT behavior (linear ramp) at $t > t_{\text{Th}}$ . This identifies the parameter regime where the model describes a chaotic CFT.
Boundary Conditions: The paper compares "clipped" and "completed" boundary conditions at the UV cutoff. "Completed" conditions, which include the frozen modular tail of the light spectrum, remove edge artifacts (Gibbs ringing) and allow the Cardy profile to match the spectrum over a wider energy range.

Significance and Claims
The authors claim that this work provides a "first step towards solving the matrix/tensor model dual to 3D quantum gravity on a computer." By isolating the spectral sector and regularizing modular invariance, they demonstrate that:

It is possible to construct a matrix model where modular invariance is an output rather than an input, naturally generating the Cardy density of states.
The model successfully reproduces the expected hierarchy of scales in chaotic CFTs: a rigid macroscopic bootstrap constraint coexisting with microscopic random matrix statistics.
The "constraint $^2$ " approach, when combined with appropriate scaling limits, offers a viable pathway to understanding the non-perturbative structure of 3D gravity duals, specifically how the vacuum and heavy states organize themselves to satisfy consistency conditions.

The paper remains modest regarding the full 3D gravity dual, noting that the current model is a simplified "toy" (single matrix, scalar states, no OPE coefficients). However, it establishes the technical framework and scaling limits necessary to extend these results to the full tensor model involving OPE coefficients and spin quantization.

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