We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ
 L
 ≤ 2πk
 B
 T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

A bound on chaos

The authors study in detail the quantum mechanical model of $N$ Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large $N$ limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.

Remarks on the Sachdev-Ye-Kitaev model

We use holography to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the t = 0 slice, creating a shock wave. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to the firewall controversy.

/pdf/black-holes-and-the-butterfly-effect-4v604bigor.pdf

Black holes and the butterfly effect

We argue that the following three statements cannot all be true: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. Perhaps the most conservative resolution is that the infalling observer burns up at the horizon. Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic distances from the horizon.

Black Holes: Complementarity or Firewalls?

We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly $AdS_2$ spacetimes. The asymptotic symmetries of $AdS_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the $AdS_2$ geometry and they are explicitly broken by the small deformation away from $AdS_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of out of time order correlators that is indicative of chaos. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.

Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $\lambda_L \le 2 \pi k_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

Douglas Stanford

Papers

A bound on chaos

Remarks on the Sachdev-Ye-Kitaev model

Black holes and the butterfly effect

Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space

A bound on chaos