Free Cumulants and Full Eigenstate Thermalization from Boundary Scrambling

Out-of-time-order correlation functions (OTOCs) and their higher-order generalizations present important probes of quantum information dynamics and scrambling. We introduce a solvable many-body quantum model, which we term boundary scrambling, for which the full dynamics of higher-order OTOCs is analytically tractable. These dynamics support a decomposition into free cumulants and unify recent extensions of the eigenstate thermalization hypothesis with predictions from random quantum circuit models. We obtain exact expressions for (higher-order) correlations between matrix elements and show these to be stable away from the solvable point. The solvability is enabled by the identification of a higher-order Markovian influence matrix, capturing the effect of the full system on a local subsystem. These results provide insight into the emergence of random-matrix behavior from structured Floquet dynamics and show how techniques from free probability can be applied in the construction of exactly-solvable many-body models.