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Revisiting Brownian SYK and its possible relations to de Sitter

Alexey Milekhin, Jiuci Xu

TL;DR

The paper studies Brownian double-scaled SYK (BDSSYK) and shows that energy is conserved on average after disorder averaging, despite time-dependent randomness. It solves the model in the double-scaling limit and demonstrates hyperfast scrambling, exponential decay of correlators, a bounded spectrum, and approximate higher-point factorization, drawing connections to de Sitter holography via mappings such as $J^2=1/R_{dS}$ and $\lambda=8G_N/R_{dS}$. A chord-based algebraic formulation is developed to compute correlation functions as transition amplitudes in an auxiliary Hilbert space, providing clear expressions for two- and four-point functions. The work argues that BDSSYK captures late-time, horizon-based physics of de Sitter space in a universality class sense, while discussing tensions in entropy and the precise bulk interpretation, and outlines directions for reconciling these results with full dS physics.

Abstract

We revisit Brownian Sachdev-Ye-Kitaev model and argue that it has emergent energy conservation overlooked in the literature before. We solve this model in the double-scaled regime and demonstrate hyperfast scrambling, exponential decay of correlation functions, bounded spectrum and unexpected factorization of higher-point functions. We comment on how these results are related to de Sitter holography.

Revisiting Brownian SYK and its possible relations to de Sitter

TL;DR

The paper studies Brownian double-scaled SYK (BDSSYK) and shows that energy is conserved on average after disorder averaging, despite time-dependent randomness. It solves the model in the double-scaling limit and demonstrates hyperfast scrambling, exponential decay of correlators, a bounded spectrum, and approximate higher-point factorization, drawing connections to de Sitter holography via mappings such as and . A chord-based algebraic formulation is developed to compute correlation functions as transition amplitudes in an auxiliary Hilbert space, providing clear expressions for two- and four-point functions. The work argues that BDSSYK captures late-time, horizon-based physics of de Sitter space in a universality class sense, while discussing tensions in entropy and the precise bulk interpretation, and outlines directions for reconciling these results with full dS physics.

Abstract

We revisit Brownian Sachdev-Ye-Kitaev model and argue that it has emergent energy conservation overlooked in the literature before. We solve this model in the double-scaled regime and demonstrate hyperfast scrambling, exponential decay of correlation functions, bounded spectrum and unexpected factorization of higher-point functions. We comment on how these results are related to de Sitter holography.
Paper Structure (7 sections, 66 equations, 4 figures)

This paper contains 7 sections, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Time-ordered correlation functions
  3. Out-of-time ordered 4-point function
  4. Intermezzo: energy conservation
  5. An algebraic approach towards chords in Brownian DSSYK
  6. Comparison with de Sitter
  7. Conclusion

Figures (4)

  • Figure 1: (a) Computation of the two-point function using Schwinger--Keldysh contour. Blue chords are matter chords and the black chords represent Hamiltonians. The same contour can be used to compute time-ordered four-point function. (b) The configuration $\langle V(t_1) V(t_2) W(t_3) W(t_4) \rangle$. (c) The configuration $\langle V(t_1) W(t_3) W(t_4) V(t_2) \rangle$.
  • Figure 2: Time-contour which computes the OTOC $\langle V(0) W(t) V(0) W(t) \rangle$. Chord $12$ intersects $23$, but $14$ and $34$ do not intersect anything.
  • Figure 3: Left: standard SK contour. Right: the same computation but in the doubled Hilbert space formalism.
  • Figure 4: Penrose diagram of global $d$-dimensional dS space. Two static patches are shown in shaded blue and green. Red is the cosmological horizon. Each point hides an extra $S^{d-2}$. Bold blue and green lines are "antipode" and "pode" where $S^{d-2}$ degenerates to a point. Black line is the stretched horizon.