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Single-file systems, in which particles diffuse in narrow channels while notovertaking each other, is a fundamental model for the tracer subdiffusionobserved in confined geometries, such as in zeolites or carbon nanotubes.Twenty years ago, the mean squared displacement of a tracer was determined atlarge times, for any diffusive single-file system. Since then, for a generalsingle-file system, even the determination of the fourth cumulant, which probesthe deviation from Gaussianity, has remained an open question. Here, we fillthis gap and provide an explicit formula for the fourth cumulant of anarbitrary single-file system. Our approach also allows us to quantify theperturbation induced by the tracer on its environment, encoded in thecorrelation profiles. These explicit results constitute a first step towardsobtaining a closed equation for the correlation profiles for arbitrarysingle-file systems.

Tracer diffusion beyond Gaussian behavior: explicit results for general single-file systems