We study the stability of anti-de Sitter (AdS) spacetime to spherically
symmetric perturbations of a real scalar field in general relativity. Further,
we work within the context of the "two time framework" (TTF) approximation,
which describes the leading nonlinear effects for small amplitude
perturbations, and is therefore suitable for studying the weakly turbulent
instability of AdS---including both collapsing and non-collapsing solutions. We
have previously identified a class of quasi-periodic (QP) solutions to the TTF
equations, and in this work we analyze their stability. We show that there
exist several families of QP solutions that are stable to linear order, and we
argue that these solutions represent islands of stability in TTF. We extract
the eigenmodes of small oscillations about QP solutions, and we use them to
predict approximate recurrence times for generic non-collapsing initial data in
the full (non-TTF) system. Alternatively, when sufficient energy is driven to
high-frequency modes, as occurs for initial data far from a QP solution, the
TTF description breaks down as an approximation to the full system. Depending
on the higher order dynamics of the full system, this often signals an imminent
collapse to a black hole.Comment: 22 pages, 19 figures; V2: minor changes to match version accepted for
publication in Phys Rev
Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this contribution, we provide a rigorous justi cation of these approaches for a two-layers neural network model called the committee machine. We also introduce a version of the approximate message passing (AMP) algorithm for the committee machine that allows to perform optimal learning in polynomial time for a large set of parameters. We nd that there are regimes in which a low generalization error is information-theoretically achievable while the AMP algorithm fails to deliver it, strongly suggesting that no e cient algorithm exists for those cases, and unveiling a large computational gap.
There has been definite progress recently in proving the variational single-letter formula given by the heuristic replica method for various estimation problems. In particular, the replica formula for the mutual information in the case of noisy linear estimation with random i.i.d. matrices, a problem with applications ranging from compressed sensing to statistics, has been proven rigorously. In this contribution we go beyond the restrictive i.i.d. matrix assumption and discuss the formula proposed by Takeda, Uda, Kabashima and later by Tulino, Verdu, Caire and Shamai who used the replica method. Using the recently introduced adaptive interpolation method and random matrix theory, we prove this formula for a relevant large sub-class of rotationally invariant matrices.
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