Conditionally Extended Validity of Perturbation Theory: Persistence of AdS Stability Islands

Approximating nonlinear dynamics with a truncated perturbative expan- sion may be accurate for a while, but it in general breaks down at a long time scale that is one over the small expansion parameter. There are interesting occasions in which such breakdown does not happen. We provide a mathematically general and precise definition of those occasions, in which we prove that the validity of truncated theory trivially extends to the long time scale. This enables us to utilize numerical results, which are only obtainable within finite times, to legitimately predict the dynamic when the expansion parameter goes to zero, thus the long time scale goes to infinity. In particular, this shows that existing non-collapsing solutions in the AdS (in)stability problem persist to the zero-amplitude limit, opposing the conjecture by Dias, Horowitz, Marolf and Santos that predicts a shrinkage to measure-zero [1]. We also point out why the persistence of collapsing solutions is harder to prove, and how the recent interesting progress by Bizon, Maliborski and Rostoworowski is not there yet [2].