  Some quantities in quantum field theory are dominated by so-called
$\mathit{leading\,logs}$ and can be re-summed to all loop orders. In this work
we introduce a notion of $\mathit{stampede}$ which is a simple time-evolution
of a bunch of particles which start their life in a corner - on the very right
say - and $\mathit{hop}$ their way to the opposite corner - on the left -
through the repeated action of a quantum Hamiltonian. Such stampedes govern
leading logs quantities in certain quantum field theories. The leading
euclidean OPE limit of correlation functions in the fishnet theory and null
double-scaling limits of correlators in $\mathcal{N}=4$ SYM are notable
examples. As an application, we use these results to extend the beautiful
bootstrap program of Coronado [1] to all octagons functions with arbitrary
diagonal bridge length.
