In a turbulent flow carrying a passive tracer — a dye, a temperature field, a concentration gradient — the standard expectation is diffusion. Turbulence mixes, and mixing means the tracer spreads like a random walk: the mean squared displacement grows linearly with time.
Cifani, Flandoli, and Marino (arXiv:2602.21097) show that when the noise driving the flow has heavy tails — when rare, extremely large fluctuations are allowed — the tracer spreads faster than diffusion. Super-diffusively. The mean squared displacement grows faster than linearly. This happens despite the spatial complexity of the flow field, which would normally localize or slow down transport.
The surprise is in the truncation experiment. When the tails of the noise distribution are cut — even gently, by exponential tempering that preserves the shape at moderate scales — the super-diffusion vanishes. The transport reverts to ordinary, classical diffusion. Not gradually. The transition is qualitative: heavy-tailed noise produces anomalous transport; truncated noise produces normal transport.
The physical intuition is that the rare, extremely long jumps are the mechanism sustaining anomalous behavior. Spatial complexity in the flow — eddies, turbulent structures, the labyrinthine pathways of chaotic advection — acts as a homogenizer. It wants to turn everything into a random walk. And for most of the noise distribution, it succeeds. Small and medium fluctuations get scrambled by the spatial disorder. They lose their correlations, they mix, and they diffuse.
But the heavy-tailed jumps are too large for the spatial complexity to absorb. They leap over the labyrinth. A single jump can cross many turbulent structures, carrying the tracer into a region that the flow alone would never connect on that timescale. It is the escape from the maze by going over the walls.
Remove the longest jumps, and there is no escape. The spatial complexity wins. Every fluctuation stays within the labyrinth, gets mixed, and diffuses. The transport is normal.
This creates a clean dichotomy that has nothing to do with the details of the flow. It depends only on the tail of the noise distribution. Heavy tails: anomalous. Truncated tails: normal. The threshold is the existence of rare events large enough to escape spatial structure. Either they exist, and the transport is fast, or they don't, and it isn't. The complexity of the intermediate scales is irrelevant. It is the extremes that decide.