Fast Scramblers, Horizons and Expander Graphs

TL;DR

The paper argues that expander graphs offer a concrete microscopic model for horizon thermalization, producing fast scrambling with timescales that scale as τ_s ∼ β log N for sub-cell regions and transitioning to slower diffusion for larger patches. By leveraging the optical metric, it connects hyperbolic diffusion on expander-like structures to near-horizon dynamics in de Sitter and Rindler geometries, proposing a stretched-horizon expander as the edge where scrambling is most rapid. It further analyzes the limitations of probe-based models, suggests ballistic criteria for scrambling, and discusses how to derive or motivate the expander structure from holographic duals. The work provides a geometrically motivated framework tying horizon thermodynamics, quantum information scrambling, and hyperbolic geometry together, while highlighting open questions about formalizing observables and deriving the expander structure from fundamental theories.

Abstract

We propose that local quantum systems defined on expander graphs provide a simple microscopic model for thermalization on quantum horizons. Such systems are automatically fast scramblers and are motivated from the membrane paradigm by a conformal transformation to the so-called optical metric.

Figures (2)

• Figure 1: The infinite regular tree, or Bethe lattice, is the ultimate expander. Its adjacency matrix has a continuous spectrum in the interval $[-2\sqrt{k-1} , 2\sqrt{k-1}]$. The figure shows the case $k=3$, a tessellation of the two-dimensional hyperboloid.
• Figure 2: Schematic view of the effective kinetic scrambling model in the optical frame, featuring a Cayley tree picture of the stretched horizon, a hyperbolic 'Rindler atmosphere' and an asymptotic AdS region of optical depth of ${\cal O}(\beta)$. The total optical depth of the hyperbolic section $[0, z_{\rm P}]$ is ${\cal O}(\beta\log N)$. A localized classical probe injected from the asymptotic boundary ( vertical blue line) is either absorbed or reflected at $z=z_\Lambda$. If reflected (curved red line), it scrambles fast in the effective chaotic billiard of the Rindler atmosphere. If absorbed (jagged red line), it scrambles fast by diffusion on the expander graph until it reaches the bottom at $z_{\rm P}$. Once at the bottom, it scrambles slowly by diffusion on the bottom. The fast-scrambling patch measured by the $\gamma_{ij}$ metric is always about one thermal cell.