  Effective theories are being developed for fields outside black holes, often
with an unusual open-system feel due to the influence of large number of
degrees of freedom that lie out of reach beyond the horizon. What is often
difficult when interpreting such theories is the absence of comparisons to
simpler systems that share these features. We propose here such a simple model,
involving a single external scalar field that mixes in a limited region of
space with a `hotspot' containing a large number of hot internal degrees of
freedom. Since the model is at heart gaussian it can be solved explicitly, and
we do so for the mode functions and correlation functions for the external
field once the hotspot fields are traced out. We compare with calculations that
work perturbatively in the mixing parameter, and by doing so can precisely
identify its domain of validity. We also show how renormalization-group EFT
methods can allow some perturbative contributions to be resummed beyond leading
order, verifying the result using the exact expression.
