  In quantum many-body systems, interactions play a crucial role in the
emergence of information scrambling. When particles interact throughout the
system, the entanglement between them can lead to a rapid and chaotic spreading
of quantum information, typically probed by the growth in operator size in the
Heisenberg picture. In this study, we explore whether the operator undergoes
scrambling when particles interact solely through a single impurity in generic
spatial dimensions, focusing on fermion systems with spatial and temporal
random hoppings. By connecting the dynamics of the operator to the symmetric
exclusion process with a source term, we demonstrate the presence of an
escape-to-scrambling transition when tuning the interaction strength for
fermions in three dimensions. As a comparison, systems in lower dimensions are
proven to scramble at arbitrarily weak interactions unless the hopping becomes
sufficiently long-ranged. Our predictions are validated using both a Brownian
circuit with a single Majorana fermion per site and a solvable Brownian SYK
model with a large local Hilbert space dimension. This suggests the
universality of the theoretical picture for free fermion systems with spatial
and temporal randomness.
