Gravity as a mesoscopic system

We employ a probabilistic mesoscopic description to draw conceptual and quantitative analogies between Brownian motion and late-time fluctuations of thermal correlation functions in generic chaotic systems respecting ETH. In this framework, thermal correlation functions of `simple' operators are described by stochastic processes, which are able to probe features of the microscopic theory only in a probabilistic sense. We apply this formalism to the case of semiclassical gravity in AdS$_3$, showing that wormhole contributions can be naturally identified as moments of stochastic processes. We also point out a `Matryoshka doll' recursive structure in which information is hidden in higher and higher moments, and which can be naturally justified within the stochastic framework. We then re-interpret the gravitational results from the boundary perspective, promoting the OPE data of the CFT to probability distributions. The outcome of this study shows that semiclassical gravity in AdS can be naturally interpreted as a mesoscopic description of quantum gravity, and a mesoscopic holographic duality can be framed as a moment-vs-probability-distribution duality.