Localized shocks

We study products of precursors of spatially local operators, $$ {W_x}_{{}_n}(tn)cdot cdot cdot {W}_{x_1}left({t}_1right) $$ , where W x (t) = e − iHt W x e iHt . Using chaotic spin-chain numerics and gauge/gravity duality, we show that a single precursor fills a spatial region that grows linearly in t. In a lattice system, products of such operators can be represented using tensor networks. In gauge/gravity duality, they are related to Einstein-Rosen bridges supported by localized shock waves. We find a geometrical correspondence between these two descriptions, generalizing earlier work in the spatially homogeneous case.