Fast Scramblers, Democratic Walks and Information Fields

TL;DR

This paper introduces solvable democratic random walks on complete graphs to model information spreading in highly interconnected systems, revealing conditions for fast scrambling and implications for black hole physics.

Contribution

It presents a new class of exactly solvable democratic walks and links their properties to information scrambling and black hole dynamics.

Findings

Hierarchies where characteristic time saturates fast scrambling conjecture

Democratic walks accurately model information spreading in fully connected systems

Stability of a subsystem implies global scrambling in democratic systems

Abstract

We study a family of weighted random walks on complete graphs. These `democratic walks' turn out to be explicitly solvable, and we find the hierarchy window for which the characteristic time scale saturates the so-called fast scrambling conjecture. We show that these democratic walks describe well the properties of information spreading in systems in which every degree of freedom interacts with every other degree of freedom, such as Matrix or infinite range models. The argument is based on the analysis of suitably defined `Information fields' ($\mathcal{I}$), which are shown to evolve stochastically towards stationarity due to unitarity of the microscopic model. The model implies that in democratic systems, stabilization of one subsystem is equivalent to global scrambling. We use these results to study scrambling of infalling perturbations in black hole backgrounds, and argue that the near horizon running coupling constants are connected to entanglement evolution of single particle perturbations in democratic systems.