Spread complexity and the saturation of wormhole size

TL;DR

The paper addresses how Einstein-Rosen bridge (ER) volume in holographic theories should saturate in a finite-dimensional quantum Hilbert space. By constructing finite-dimensional, tridiagonal (Krylov/Lanczos) representations that reproduce the DSSYK density of states and spectral statistics, the authors connect sub-exponential chord-basis constructions to physical Krylov subspaces and analyze spread complexity as a proxy for ER volume. They show that early-time ER growth matches semiclassical expectations, while non-perturbative, finite-size effects drive late-time saturation with a plateau near $e^{S}$, whose approach depends on the chaotic universality class (Dyson/Altland-Zirnbauer). The work extends these techniques to higher dimensions, discusses the appearance of white-hole-like dynamics at late times, and links black hole entropy to finite-dimensional Hilbert spaces in quantum gravity, offering a non-perturbative framework for ER saturation and interior dynamics.

Abstract

Recent proposals equate the size of Einstein-Rosen bridges in JT gravity to spread complexity of a dual, double-scaled SYK theory (DSSYK). We show that the auxiliary ``chord basis'' of these proposals is an extrapolation from a sub-exponential part of the finite-dimensional physical Krylov basis of a spreading thermofield double state. The physical tridiagonal Hamiltonian coincides with the DSSYK approximation on the initial Krylov basis, but deviates markedly over an exponentially large part of the state space. We non-perturbatively extend the identification of ER bridge size and spread complexity to the complete Hilbert space, and show that it saturates at late times. We use methods for tridiagonalizing random Hamiltonians to study all universality classes to which large N SYK theories and JT gravities can belong. The saturation dynamics depends on the universality class, and displays ``white hole'' physics at late times where the ER bridge shrinks from maximum size to a plateau. We describe extensions of our results to higher dimensions.

Figures (8)

• Figure 1: An example of a chord diagram. The nodes correspond to Hamiltonian insertions. The edges connecting nodes pairwise correspond to Gaussian contractions between the SYK couplings. The circle indicates this diagram is computing a trace. In the DSSYK limit, these diagrams are the only ones contributing to the Hamiltonian moments (\ref{['momtr']}).
• Figure 2: The Lanczos spectrum ($a_n$ in lighter colors and $b_n$ in darker colors) for DSSYK at different values of $q=e^{-\lambda}$ given in the titles, corresponding to the evolution of the TFD at different temperatures $\beta$ given in the legend. Only the edge near $n=0$ differs between the different temperature initial states. This is a numerical computation using the leading density of states. It is transparent that the bulk of the Lanczos spectrum, namely $n\sim \mathcal{O}(L)$ that corresponds to the descent, does not depend on the temperature. It can then be approached with the RMT techniques described in the text.
• Figure 3: The leading (large-L) Lanczos spectrum for DSSYK computed analytically using the integral equation (\ref{['intdl']}) or the saddle point equations \ref{['dssyk_RMT_potential']}, valid for $n\gg 1$ (but still $n\sim\mathcal{O}(1)$ in the large-$L$ limit) (dashed lines). We also include the small $n\sim O(1)$ analytical values at various $\beta$, corresponding to the same colors as in Fig \ref{['numerical_LZ']} (dotted lines). We also show the numerical results for comparison (solid lines). The $x$-axis is scaled logarithmically to appreciate the initial (growing) part of the spectrum. Notice the dependence on the temperature.
• Figure 4: The variance of the (large-L) Lanczos spectrum for DSSYK with random initial states computed analytically using \ref{['cov']} with a polynomial approximation, valid for $n\gg 1$, plotted in solid black. We also show numerics (orange for $\sigma_{a_n}^2$ and blue for $4\sigma_{b_n}^2$) for comparison.
• Figure 5: The spectral form factor for DSSYK associated with the TFD state at temperature $T=1$ and different values of $q=e^{-\lambda}$. Different colors correspond to different quantum chaotic universality classes. We see that at a qualitative level, the spectral form factor only depends on the Dyson index $\beta$, and not on the parameter $\alpha$ characterizing the Altland-Zirnbauer class.