⚛️ high-energy theory

A semiclassical Hilbert space for random matrix theory

This paper demonstrates that the Krylov basis evolution of thermofield double states in random matrix theory provides a semiclassical Hilbert space description analogous to the double-scaled SYK model, which reduces to the Liouville Hamiltonian of JT gravity in a low-energy continuum limit, thereby suggesting a dual bulk gravity interpretation.

Original authors: Abhirup Bhattacharya, Onkar Parrikar, Vivek Singh

Published 2026-08-26

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CC BY 4.0

Original authors: Abhirup Bhattacharya, Onkar Parrikar, Vivek Singh

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 deepest corners of theoretical physics, researchers are trying to understand how the smooth, continuous fabric of space and time emerges from the chaotic, jittery behavior of quantum particles. This is the central puzzle of quantum gravity: how does the universe look like a solid, predictable stage when, at its smallest scales, it is actually a frothing sea of uncertainty. One powerful way to study this is by looking at random matrices. Imagine a giant grid of numbers, where every entry is chosen at random according to specific rules. While these grids seem chaotic, when they become infinitely large, they reveal hidden patterns and smooth shapes that resemble the geometry of space. This approach has been particularly successful in studying a specific type of quantum system known as the Sachdev-Ye-Kitaev model, which acts as a simplified laboratory for testing ideas about black holes and the holographic principle—the idea that a volume of space can be described by information stored on its boundary.

A key insight from recent work is that the behavior of these complex quantum systems can be understood by looking at a special set of directions, or a "basis," that the system naturally explores over time. Instead of tracking every single particle, physicists can focus on a sequence of states that the system visits as it evolves. This sequence, known as the Krylov basis, acts like a ladder. As the system moves up this ladder, it reveals a hidden simplicity: the complex quantum dynamics can be described by a much simpler, almost classical, set of rules. In a famous model called the double-scaled SYK model, this ladder was found to correspond to the length of a wormhole connecting two distant regions of space, suggesting that the geometry of space itself is encoded in the way quantum information spreads.

In a new study, researchers have taken this idea and applied it to a much broader class of random systems, moving beyond the specific case of the double-scaled SYK model. They asked whether this elegant connection between quantum evolution and geometric space is a unique quirk of that one model or a general feature of chaotic quantum systems. To find out, they examined a wide variety of random matrix models, which are mathematical descriptions of systems with many interacting parts. They focused on how these systems evolve when they are in a state of thermal equilibrium, a condition where the system has settled into a steady temperature. By analyzing the mathematical structure of these evolutions, they discovered that the same kind of "ladder" description appears in almost all of these models. Just as in the specific double-scaled SYK case, the researchers found that the complex quantum behavior could be mapped onto a simple, one-dimensional chain of states. This chain acts as a semiclassical "bulk" space, a hidden interior where the physics looks smooth and local, much like the space inside a black hole.

The team then investigated what happens when they zoom in on the very low-energy behavior of these systems, which corresponds to the deepest, most stable parts of the quantum state. They found that for a large class of models that modify the high-energy details of the system while keeping the low-energy behavior the same, the description on this ladder converges to a specific, well-known equation from gravity theory. This equation, known as the Liouville Hamiltonian, describes the dynamics of two-dimensional gravity, specifically the stretching and shrinking of a wormhole. This result is significant because it suggests that the emergence of a gravitational description is not a fluke of one specific model but a robust feature of a wide range of chaotic quantum systems. The researchers showed that even when they changed the "ultraviolet" details—the high-energy, short-distance rules of the system—the low-energy geometry remained unchanged and continued to look like a wormhole governed by the laws of gravity.

The study also clarified the relationship between the mathematical tools used to describe these systems and the physical concepts they represent. By using a method involving orthogonal polynomials, which are a standard tool in mathematics for organizing complex data, the researchers were able to efficiently extract the rules governing the "ladder" states. They demonstrated that the speed at which the system moves up this ladder is determined by the shape of the energy spectrum of the underlying random matrix. In particular, the way the energy levels are distributed near the edges of the spectrum dictates how the system behaves at large scales. The researchers found that if the energy spectrum has a specific shape with sharp edges, the system's evolution mimics a free particle hopping along a line, but with a subtle, exponential potential that pulls it back, creating the gravitational effect.

One of the most compelling aspects of this work is that it provides a concrete bridge between the microscopic quantum world and the macroscopic gravitational world without relying on the specific, intricate details of the double-scaled SYK model. The researchers showed that the "bulk" Hilbert space, which is the mathematical space describing the interior of the system, can be constructed for a vast array of random matrix models. This space is not just a mathematical trick; it appears to be the actual gravitational dual of the quantum system. The fact that this construction works for models with different high-energy behaviors suggests that the emergence of gravity is a universal phenomenon in these types of systems. It implies that the geometry of space, specifically the length of a wormhole, is a direct consequence of how quantum information is organized and evolves over time.

The researchers also addressed the question of entropy, which is a measure of disorder or the number of ways a system can be arranged. They showed that the semiclassical description they developed correctly reproduces the entropy of the underlying quantum system, but in a smoothed-out, coarse-grained way. This is consistent with what we expect from gravity, where the smooth geometry we see is an average over many microscopic quantum states. The study confirms that the semiclassical description is not just an approximation but a faithful representation of the system's thermodynamics. By restricting their analysis to a fixed, finite depth in the Krylov basis, they were able to capture the essential physics of the system at any given temperature or time, without needing to account for the exponentially large number of states that exist in the full quantum description.

In summary, this work extends a profound insight from a specific quantum model to a general class of random systems. It demonstrates that the emergence of a semiclassical, gravitational description is a robust feature of chaotic quantum dynamics. The researchers have shown that by looking at the right set of states—the Krylov basis—one can see the hidden geometry of space-time appearing out of the chaos of random matrices. This suggests that the connection between quantum mechanics and gravity is deeper and more universal than previously thought, offering a new perspective on how the smooth world of general relativity arises from the discrete, probabilistic world of quantum mechanics. The findings provide strong evidence that the "bulk" Hilbert space, with its discrete wormhole length, is the natural language for describing the gravitational dual of these quantum systems, even when the high-energy details of the system are varied.

Technical Summary: A Semiclassical Hilbert Space for Random Matrix Theory

Problem Statement
The paper addresses the challenge of identifying a semiclassical "bulk" Hilbert space description for random matrix theories (RMT) that mirrors the success of the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In the DSSYK context, the "chord Hilbert space"—spanned by chord-number eigenstates—provides a semiclassical description of thermofield double (TFD) states and reduces to Jackiw-Teitelboim (JT) gravity in a specific low-energy, continuum limit. The authors investigate whether this construction generalizes to broader classes of unitarily invariant random matrix models. Specifically, they ask if the Krylov basis, constructed from the maximally entangled state $|\Omega\rangle$ and the Hamiltonian $H$ , yields a similar semiclassical effective theory for general one-cut RMTs, and whether this effective theory reproduces the Liouville Hamiltonian of JT gravity in the appropriate limits.

Methodology
The authors employ a combination of random matrix theory, orthogonal polynomial theory, and asymptotic analysis to construct the effective dynamics:

Krylov Basis Construction: They define the Krylov basis $\{|n\rangle\}$ via the Gram-Schmidt orthonormalization of the sequence $\{|\Omega\rangle, H|\Omega\rangle, H^2|\Omega\rangle, \dots\}$ . In the limit of large matrix dimension $D \to \infty$ , the Hamiltonian restricted to the subspace of fixed Krylov depth $n$ becomes a tridiagonal effective Hamiltonian $H_{\text{eff}}$ with ensemble-averaged Lanczos coefficients $a_n$ and $b_n$ .
Orthogonal Polynomials: The authors establish a direct correspondence between the ensemble-averaged Lanczos coefficients and the recursion coefficients of orthogonal polynomials with respect to the spectral density $\omega(E)$ of the matrix model. Specifically, the wavefunctions $\psi_n(E) = \langle E | n \rangle$ are identified as orthonormal polynomials satisfying the recurrence relation defined by $H_{\text{eff}}$ .
Asymptotic Analysis (Riemann-Hilbert Approach): To determine the behavior of the Lanczos coefficients at large $n$ , the authors utilize the Riemann-Hilbert problem (RHP) formulation for orthogonal polynomials. They apply the Deift-Zhou steepest descent method to solve the RHP asymptotically. This allows them to derive the large- $n$ behavior of the coefficients based on the analytic structure of the spectral density (specifically the location of zeros and poles of the weight function outside the spectral support).
Continuum and Scaling Limits: The paper investigates the "triple-scaling limit" (analogous to the DSSYK limit) where the parameter $\lambda \to 0$ (related to the distance of the nearest zero of the spectral density from the cut) and the Krylov depth $n$ scales as $n \sim \lambda^{-1} \log(\lambda^{-2}) + \ell/\lambda$ . This maps the discrete Krylov index to a continuous variable $\ell$ .

Key Contributions and Results

Universal Semiclassical Description: The authors demonstrate that for a broad class of one-cut random matrix models with even, compactly supported spectral densities featuring square-root edges, the Krylov basis provides a smooth, local, semiclassical effective description. In the large- $n$ limit, the effective Hamiltonian approaches that of a free particle hopping on a one-dimensional chain ( $b_n \to 1/2$ ).
Exponential Approach and Spectral Zeros: They show that the approach of the Lanczos coefficients $b_n$ to their asymptotic value is exponential, $b_n \sim \frac{1}{2} - \alpha e^{-cn}$ . The rate $c$ is determined by the location of the nearest pair of zeros of the spectral density (away from the spectral cut $[-1, 1]$ ). This generalizes the parameter $q = e^{-\lambda}$ found in DSSYK to a general RMT context.
Robustness of JT Gravity Emergence: The central result is that for a large class of models representing ultraviolet (UV) deformations of the DSSYK spectral density—specifically those obtained via rational Christoffel deformations (adding finite numbers of zeros and poles at locations $p_i, q_j > 1$ $p_{i}, q_{j} > 1$ )—the low-energy continuum limit of the Krylov effective Hamiltonian reduces exactly to the Liouville Hamiltonian of JT gravity.
- The authors prove that adding a finite number of zeros or poles to the DSSYK spectral density does not alter the leading-order behavior of the Lanczos coefficients in the continuum limit.
- Consequently, the effective Hamiltonian takes the form $H_{\text{cont}} = -\frac{1}{\lambda^2} + \frac{1}{2}(-\partial_\ell^2 + e^{-\ell})$ , which is the JT gravity Hamiltonian where $\ell$ represents the renormalized wormhole length.
Mathematical Framework: The paper provides explicit formulas for the Lanczos coefficients of Christoffel-deformed ensembles using determinants of orthogonal polynomials (Theorem 4.2 and Corollary 4.2) and validates these results using both the RHP method and direct recursion relations.

Significance and Claims
The paper claims that the identification of the chord Hilbert space with the Krylov basis is not an artifact specific to DSSYK but a robust feature of a wide class of random matrix theories. The significance of this work lies in:

Generalization of Bulk Duals: It suggests that the semiclassical Hilbert space associated with the discretized wormhole length in DSSYK generalizes to a large class of matrix models. This implies that the "bulk" gravitational description (in terms of a discrete wormhole length) is a universal feature of the Krylov dynamics of TFD states in these theories, independent of the specific UV completion of the spectral density.
UV Independence of Low-Energy Gravity: The results indicate that the emergence of JT gravity in the low-energy limit is insensitive to the details of the spectral density in the ultraviolet (the middle of the spectrum), provided the near-edge behavior (square-root edges and the specific scaling of the nearest zero) remains unchanged.
Interpretation of the Krylov Basis: The work provides evidence supporting the interpretation of the Krylov effective theory as the bulk Hilbert space of a dual gravitational description. The confinement of TFD states within this small Hilbert space for $O(1)$ time/temperature evolution justifies the use of this semiclassical description for computing thermodynamic quantities like entropy.

The authors conclude that their construction offers a "discretization" of the geometric length basis of JT gravity for general matrix models, reinforcing the AdS/CFT-like duality between random matrix ensembles and 2D gravity.

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