The fine grained energy spectrum of quantum chaotic systems is widely
believed to be described by random matrix statistics. A basic scale in such a
system is the energy range over which this behavior persists. We define the
corresponding time scale by the time at which the linearly growing ramp region
in the spectral form factor begins. We call this time tramp. The
purpose of this paper is to study this scale in many-body quantum systems that
display strong chaos, sometimes called scrambling systems. We focus on randomly
coupled qubit systems, both local and k-local (all-to-all interactions) and
the Sachdev--Ye--Kitaev (SYK) model. Using numerical results for Hamiltonian
systems and analytic estimates for random quantum circuits we find the
following results. For geometrically local systems with a conservation law we
find tramp is determined by the diffusion time across the system,
order N2 for a 1D chain of N qubits. This is analogous to the behavior
found for local one-body chaotic systems. For a k-local system with
conservation law the time is order logN but with a different prefactor and
a different mechanism than the scrambling time. In the absence of any
conservation laws, as in a generic random quantum circuit, we find tramp∼logN, independent of connectivity.Comment: 61+20 pages, minor errors corrected, and significant edits in Section