  We consider the long-time limit of out-of-time-order correlators (OTOCs) in
two classes of quantum lattice models with time evolution governed by local
unitary quantum circuits and maximal butterfly velocity $v_{B} = 1$. Using a
transfer matrix approach, we present analytic results for the long-time value
of the OTOC on and inside the light cone. First, we consider `dual-unitary'
circuits with various levels of ergodicity, including the integrable and
non-integrable kicked Ising model, where we show exponential decay away from
the light cone and relate both the decay rate and the long-time value to those
of the correlation functions. Second, we consider a class of kicked XY models
similar to the integrable kicked Ising model, again satisfying $v_{B}=1$,
highlighting that maximal butterfly velocity is not exclusive to dual-unitary
circuits.
