Gravity without averaging

TL;DR

The paper investigates the gravity dual of a single member of a random matrix ensemble by introducing a Gaussian external-field deformation Z(σ,H0) that interpolates between fully averaged JT/dilaton gravity and a fixed Hamiltonian H0. Using exact finite‑dimensional Gaussian matrix integrals and a double-scaling limit, it maps the interpolation to deformations of JT gravity, revealing a continuous transition from wormhole-dominated, self-averaging gravity to a non-averaged theory governed by a universal Efetov S=0 saddle. The work uncovers a tearing phase where spacetime develops macroscopic holes as σ decreases, and develops a local-factorization framework by partially fixing eigenvalues of H0, linking bulk interpretations to open- and closed-string pictures. Overall, it provides a concrete, controllable route to understanding non-averaged dilaton gravity, the role of bulk couplings, and the emergence of bulk microstructure beyond JT gravity.

Abstract

We present a gravitational theory that interpolates between JT gravity, and a gravity theory with a fixed boundary Hamiltonian. For this, we consider a matrix integral with the insertion of a Gaussian with variance $σ^2$, centered around a matrix $\textsf{H}_0$. Tightening the Gaussian renders the matrix integral less random, and ultimately it collapses the ensemble to one Hamiltonian $\textsf{H}_0$. This model provides a concrete setup to study factorization, and what the gravity dual of a single member of the ensemble is. We find that as $σ^2$ is decreased, the JT gravity dilaton potential gets modified, and ultimately the gravity theory goes through a series of phase transitions, corresponding to a proliferation of extra macroscopic holes in the spacetime. Furthermore, we observe that in the Efetov model approach to random matrices, the non-averaged factorizing theory is described by one simple saddle point.

Figures (15)

• Figure 1: Phase diagram of the matrix integral \ref{['PDexternal']} and its gravitational interpretation as a function of $\sigma$. On the far right (blue region), we have $\sigma = \infty$ and our matrix model is that for JT gravity. We also added the saddle for the Efetov sigma model in the Gaussian model at $\sigma = \infty$. As we move away from $\sigma = \infty$ we enter in the orange region, where the matrix model is deformed by (ghost) brane insertions that are labelled by the eigenvalues $x_i$ of the target Hamiltonian $\textsf{H}_0$ (the different shades of orange on the small boundaries is supposed to represent that). The spectral density is given as well in this region with $\alpha_k$ given in \ref{['expCoeff']}. As $\sigma$ is decreased further in the orange region the model goes through a series of tearing phase transitions kazakov1990simple, which is manifested geometrically by very large boundaries ending on the brane. Decreasing $\sigma$ further results in the breakdown of various approximations we had made in section \ref{['sect:def']} and the model seems to enter a branched polymer phase, see section \ref{['sect:disc']}. At $\sigma = 0$ the theory is completely fixed to $H = \textsf{H}_0$. Remarkably, the Efetov model localizes on one universal saddle $S = 0$ in this regime.
• Figure 2: Spectrum $\rho(E)$ for $L=8$ and $\sigma=3,1/2,1/10$ (left to right); it transitions from the semicircle (orange) to a sum of deltas on the eigenvalues of $\textsf{H}_0$. For intermediate values of $\sigma$ there are heavy oscillations. The sharper the peaks and valleys in the spectrum, the less random the matrix integral.
• Figure 3: Spectral correlation $R(E_1,E_2)$ (top) and spectral covariance $T(E_1,E_2)$ (bottom) for $L=8$ and $\sigma = 5, 1, 1/10$ (left to right). The covariance $T(E_1,E_2)$ is only significant close to the diagonal axis, nearby eigenvalues repel; it interpolates between a ridge and the diagonal deltas. The theory factorizes, because the covariance drops to zero everywhere, except on the location of the eigenvalues of $\textsf{H}_0$ where it produces the required delta contact terms. The correlation $R(E_1,E_2)$ has a quadratic zero on the diagonal, testimony to quadratic level repulsion.
• Figure 4: Ribbon graph of quartic matrix model with coupling $\Tr(\textsf{H}_0 H)$ (left), the insertions of the external field $\textsf{H}_0$ (blue dots) are like leaves on a tree. For the gravity interpretation one must do the unitary integral (right), giving a double sum over Wick contractions or permutations of which we show two examples (orange and red). The new vertices are weighed by Weingarten functions and traces of $\textsf{H}_0$, because of the Weingarten functions the orange contraction dominates for large $L$, as explained in detail in section \ref{['sect:def']}.
• Figure 5: Discretized worldsheet for a quartic matrix integral with deformations, showing the quartic ribbon graph (blue) and the dual graph (black) which contains polygonic holes (orange) due to the deformations. The quartic interactions (black dots) are weighted by $\tau_4$ (as defined in \ref{['app:quartic']}), the extra vertices (blue dots) from de deformations are weighed by traces of $\textsf{H}_0$.