Random Matrix Theory for Complexity Growth and Black Hole Interiors

TL;DR

We address operator complexity in holographic quantum theories with continuous spectra by refining Krylov complexity to a microcanonical version and developing a complexity RG that coarse-grains large K-complexities. This framework reveals a universal sequence: exponential growth during scrambling, followed by a long, linear growth phase governed by an infinite-well random-matrix description of the post-scrambling Liouvillian, with nonperturbative e^{-2S} corrections slowing descent toward saturation. We apply the formalism to JT gravity and 2d holographic CFTs, showing consistent Lanczos ascent and plateau behavior and linking linear growth to behind-the-horizon interior geometry, with empirical alignment to maximal volume and potential interpretations via interior translation symmetry. The work establishes a universal random-matrix description of chaotic operator dynamics at large complexities and proposes a boundary-oriented, computationally tractable probe of black hole interiors, broadening connections between quantum chaos, holography, and gravity.

Abstract

We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, "microcanonical" version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy $\unicode{x2014}$a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a "complexity renormalization group" framework we develop that allows us to study the effective operator dynamics for different timescales by "integrating out" large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.

Figures (10)

• Figure 1: (Top) The two figures illustrate the behavior of maximal volume slices in an eternal AdS black hole spacetime with an infalling excitation, at different boundary times $t_R=-t_L=t$. (Left) As the particle approaches the horizon, its exponential blueshift leads to backreaction on the volume due to the shockwave effect. (Right) As we approach the scrambling time, the backreaction becomes $O(1)$ and the appropriate Penrose diagram contains an extended right black hole interior. Maximal volume slices accumulate near an interior time-slice $r=r_* \sim O(R_{sch})$ over a segment that is linearly increasing in $t$, as a result of the interior isometry. (Bottom) Dependence of maximal volume as a function of time.
• Figure 2: These three schematic plots show our expectations for the refined Krylov complexity growth of a simple local operator in a maximally chaotic quantum system with fixed average energy $E$. (Top left) The Lanczos sequence. The Lanczos ascent is the linear region of coefficient index $n < S(E)$ and the Lanczos plateau is the approximately flat region $n > S(E)$. The Lanczos descent is not clearly separated from the plateau; its slope is initially exponentially small $O(e^{-S(E)})$, and has been greatly exaggerated. (Top right) The Krylov wavefunction $\phi_{E,n}(t)$ before the scrambling time (blue), at the scrambling time $t_*$ (red), and some time after $t_*$ but before $e^{S(E)}$ (purple). Before $t=t_*$, the peak of the wavefunction $n_*(t)$ decreases in height as the wavepacket spreads. At the scrambling time, the wavefunction is peaked at $n_*(t_*) = S(E)$, and after this time the peak propagates at a roughly fixed velocity with minimal spreading along the chain as it passes through the long Lanczos plateau region. (Bottom) The Krylov complexity $C_E$ for a simple local operator. $C_E$ grows exponentially for a scrambling time as the Krylov wavefunction travels along the Lanczos ascent, and then transitions to a linearly growing regime induced by the Lanczos plateau which lasts for an exponential time. Toward the end of the plateau, when the descent effects become large, the complexity saturates at an exponential value $e^{S(E)}$.
• Figure 3: Illustration of a Lanczos sequence in a chaotic system, exhibiting all universal regimes: an early linear ascent, a parametrically long plateau and a slow ultimate descent controlled by $O(e^{-2S})$ effects. An effective Liouvillian ${\cal L}_{sys}^{(k)}$ for the low K-complexity operator subspace, i.e the "system", can be constructed by truncating the sequence at a site $k$ and taking appropriate scaling limits of $k,N\to \infty$. The resulting infinitely long, "zoomed in" Krylov chains define an effective Liouvillian at different complexity RG scales. The two limits discussed in Section \ref{['sec:scalinglimits']} are shown. In the semiclassical limit, the scrambling time is infinite and the asymptotic linear ascent describes thermal dissipation and exponential K-complexity growth ---thus outside-the-horizon physics. The $N-$scaling limit contains an infinite plateau after scrambling, as a result of which the effective Liouvillian acts like a random matrix drawn from the "infinite well" ensemble (\ref{['infinitewell1']}) at $t\gg t_{scr}$. The latter is reflected in linear K-complexity growth and holographically in behind-the-horiozon physics.
• Figure 4: Illustration of the behavior of the orthogonal polynomial smearing functions $P^E_n(\omega)$ that define the Krylov elements with definite K-complexity values (\ref{['kbasis']}) in every fixed average energy sector. Here we show an example of $n=15$ (blue) and $n=45$ (orange).
• Figure 5: (Left) An example of a Lanczos sequence $b_n$ as a function of $n$, where $S$ is the microcanonical entropy and $D(E)\sim e^{2S(E)}$ the total size of the Krylov space. Our Lanczos sequence contains all the characteristic regimes discussed in Section \ref{['sec:universal-growth']}: A fast early linear ascent and a parametrically long plateau, corrected by a non-perturbatively small $O(e^{-2S})$ descent rate. (Right) A histogram of the spectrum of ${\cal L}^{(k)}_{sys}$ ---the Liouvillian projected on a complexity subspace set by the complexity RG scale $k$. $k=D(E)$ is the spectrum of the exact microscopic ${\cal L}$, $\rho(E,\omega)$ (\ref{['rhodefinition']}). When $k\sim O(1)$ the spectrum is generally not universal but has an exponential tail in systems that thermalize (the exponential is somewhat obscured by the discretization in the figure but it can be confirmed by direct diagonalization of the corresponding ${\cal L}_{sys}^{(k)}$). When $k\sim O(S)$ the spectrum develops a "belly" that approaches the inverse semi-circle distribution (\ref{['densityflow']}). This is a signature of the universal "infinite well" random matrix ensemble controlling the dynamics in the post-scrambling regime (\ref{['infinitewell1']}) which underlies the linear growth of K-complexity. For $k\sim O(e^{2S})$ the effective spectrum matches the microscopic one in the regions where the inverse semicircle would exceed $\rho(E,\omega)$, as described around eq. (\ref{['finitesize']}) ---in the example this happens near the edge of the spectrum. In the main text we extract this RG flow of the Liouvillian spectrum analytically by taking appropriate scaling limits of the spectrum of ${\cal L}_{sys}^{(k)}$.

Theorems & Definitions (1)

• Theorem 1