  The statistics of a passive tracer immersed in a suspension of active
self-propelled particles (swimmers) is derived from first principles by
considering a perturbative expansion of the tracer interaction with the
microscopic swimmer field. To first order in the swimmer density, the tracer
statistics is exactly represented as a spatial Poisson process combined with
independent swimmer-tracer scattering events, rigorously reducing the
multi-particle dynamics to two-body interactions. The Poisson representation is
valid in any dimensions and for arbitrary interaction forces and swimmer
dynamics. It provides in particular an analytical derivation of the coloured
Poisson process introduced in [K. Kanazawa et al.; Nature 579, 364 (2020)]
highlighting that such a non-Markovian process can be obtained from Markovian
dynamics by a variable transformation.
