On the relation between the magnitude and exponent of OTOCs

We derive an identity relating the growth exponent of early-time OTOCs, the pre-exponential factor, and a third number called "branching time". The latter is defined within the dynamical mean-field framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models; we also explicitly define "strings" in this context. As another application, we consider an SYK chain. If the coupling strength βJ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly 2 π/β.