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The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the “scrambling” of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an asymptotic separation between the time scales of the saturation of OTOC and that of entanglement entropy in a random quantum circuit model defined on graphs with a tight bottleneck, such as tree graphs connected at the roots. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions. It also provides a more rigorous justification for an argument of arXiv:1807.04363, that black holes may be slow information scramblers. Such observations may be of fundamental importance in the understanding of the black hole information problem. The bounds we obtained for OTOC are interesting in their own right in that they generalize previous studies of OTOC on lattices to the geometries on graphs in a rigorous and general fashion.

In 2008, Christodoulou achieved a major breakthrough in the context of mathematical general relativity in being able to form trapped surfaces dynamically from initial data for the Einstein vacuum system. The results and methods which he lays out in his 600+ page manuscript has led to a flurry of activity in the last decade. I will give a rough overview of the basic ideas, describe how far theorems have come, and describe some recent progress – joint with Nikos Athanasiou – in this direction.

–Organized by Professor Shing-Tung Yau