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

Influence Through Mixing: Hotspots as Benchmarks for Basic Black-Hole Behaviour