Ville Samuli Keränen

math-ph — Mathematical Physics

Notes on Wasserstein distance and the JT gravitational ensemble

Contributed by Ville Samuli Keränen

This note completes the study of the Boltzmann–Wasserstein (BW) distance, an optimal-transport metric on quantum theories, begun in a companion paper (arXiv:2606.01296). There the distance was evaluated between macroscopically different theories; here we take the opposite limit, two independent draws of the JT-gravity ensemble, which are identical at the classical level. The distance is computable and is controlled by the connected spectral form factor: the ramp, the double-trumpet wormhole, regulated by the plateau, the spectral discreteness. The mean is exponentially small, $\langle \mathcal{W}^2\rangle \sim S_0\, e^{-S_0}$, whereas uncorrelated (Poisson) levels would give an unsuppressed $O(1)$ distance: the self-distance is a sharp diagnostic of spectral chaos. As a metric, the BW distance renders the ensemble a point cloud: exponentially small, strongly anisotropic, its leading principal axis a coherent “breathing” deformation of the low-energy spectrum. Its effective dimension grows asymptotically as $D_{\mathrm{eff}} \simeq 0.42\, S_0^2$, so the distance self-averages, its relative width shrinking as $\mathrm{std}/\mathrm{mean} \sim 1/S_0$. At low temperatures it localizes on the Airy edge, where a single Tracy–Widom mode dominates and self-averaging fails. We test the picture with exact-diagonalization GUE and SYK simulations: in GUE the effective dimension grows logarithmically at every temperature and is predicted at finite $N$, with no free parameters, by the same covariance; unfolded SYK exhibits the predicted anisotropy and mode structure of the cloud. Together with the companion result, this fixes the resolution of the metric: the microstate rigidity computed here is the noise floor beneath a macroscopic difference between theories.