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Interplay of activity and non-reciprocity in tracer dynamics: From non-equilibrium fluctuation-dissipation to giant diffusion

Non-reciprocal interactions play a key role in shaping transport in active and passive systems, giving rise to striking nonequilibrium behavior. Here, we study the dynamics of a tracer -- active or passive -- embedded in a bath of active or passive particles, coupled through non-reciprocal interactions. Starting from the microscopic stochastic dynamics of the full system, we derive an overdamped generalized Langevin equation for the tracer, incorporating a non-Markovian memory kernel that captures bath-mediated correlations. This framework enables us to compute the tracer's velocity and displacement response, derive a generalized nonequilibrium fluctuation-dissipation relation that quantifies deviations from equilibrium behavior, and determine the mean-squared displacement (MSD). We find that while the MSD becomes asymptotically diffusive, the effective diffusivity depends non-monotonically on the degree of non-reciprocity and diverges at an intermediate value. This regime of giant diffusivity provides a generic mechanism for enhanced transport in active soft matter and has direct implications for biological systems exhibiting chase-and-run or predator-prey interactions. Our analytical predictions are supported by numerical simulations of active Brownian particles, highlighting experimentally accessible signatures of non-reciprocal interactions in soft matter.