Information scrambling refers to the propagation of information throughout a
quantum system. Its study not only contributes to our understanding of
thermalization but also has wide implications in quantum information and black
hole physics. Recent studies suggest that information scrambling is mediated by
collective modes called scramblons. However, a criterion for the validity of
scramblon theory in a specific model is still missing. In this work, we address
this issue by investigating the signature of the scramblon effective theory in
random spin models with all-to-all interactions. We demonstrate that, in
scenarios where the scramblon description holds, the late-time operator size
distribution can be predicted from its early-time value, requiring no free
parameters. As an illustration, we examine whether Brownian circuits exhibit a
scramblon description and obtain a positive confirmation both analytically and
numerically. We also discuss the prediction of multiple-quantum coherence when
the scramblon description is valid. Our findings provide a concrete
experimental framework for unraveling the scramblon field theory in random spin
models using quantum simulators.Comment: 6 pages, 3 figures + supplemental materia