Stampedes I: fishnet OPE and octagon Bootstrap with nonzero bridges

Some quantities in quantum field theory are dominated by so-called leading logs 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 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.