Two spin systems, each internally well-connected, linked by a narrow bottleneck. Below a certain coupling strength, the bottleneck is invisible — the two blocks undergo a phase transition as if they were one. Above that threshold, the bottleneck decouples them, and each block transitions independently.
The result (arXiv:2603.20944) establishes a sharp threshold for when bottleneck connectivity matters. Two Curie-Weiss models of N spins each, connected through a restricted set of interactions. The bottleneck's width and the coupling strength jointly determine whether the system behaves as one block or two.
The counterintuitive part: the bottleneck doesn't always suppress the phase transition. For sufficiently strong coupling through even a narrow channel, the system acts as if the channel doesn't exist. The information about collective ordering passes through the bottleneck as effectively as through the bulk. Only when the coupling weakens below the threshold does the bottleneck become functionally real.
This is a phase transition IN the effect of the bottleneck. The bottleneck is always physically present. But its functional significance switches on and off as a function of parameters. There's a narrow channel between two rooms, and whether it matters depends not on the channel's width but on how urgently the rooms need to communicate.
The technical approach combines large deviations theory with phase transition analysis, deriving the threshold through the free energy landscape's geometry. The threshold is where the free energy barrier created by the bottleneck exceeds the thermal fluctuations driving the collective transition.
This matters wherever networks have chokepoints: neural circuits, supply chains, communication networks. The question isn't whether a bottleneck exists — it's whether the system's dynamics are operating in the regime where it matters.