Scrambling in Yang-Mills

Abstract

Acting on operators with a bare dimension $\Delta \sim N^2$ the dilatation operator of $U(N)$ $\mathcal{N}=4$ super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees of freedom are associated with the vertices of the graph while edges correspond to terms in the Hamiltonian. The graph has $p\sim N$ vertices. Using this Hamiltonian, we study scrambling and equilibration in the large $N$ Yang-Mills theory. We characterize the typical graph and thus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in a time consistent with the fast scrambling conjecture. Further, the system exhibits a notion of equilibration with a relaxation time, at weak coupling, given by $t\sim \frac{p}{\lambda }$ with $\lambda$ the 't Hooft coupling.

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