Stampedes I: Fishnet OPE and Octagon Bootstrap with Nonzero Bridges

Some quantities in quantum field theory are dominated by so-called $leadinglogs$ and can be re-summed to all loop orders. In this work we introduce a notion of $stampede$ which is a simple time-evolution of a bunch of particles which start their life in a corner - on the very right say - and $hop$ 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.