  We explore the dynamics of a tracer in an active particle harmonic chain,
investigating the influence of interactions. Our analysis involves calculating
mean-squared displacements (MSD) and space-time correlations through Green's
function techniques and numerical simulations. Depending on chain
characteristics, i.e., different time scales determined by interaction
stiffness and persistence of activity, tagged-particle MSD exhibit ballistic,
diffusive, and single-file diffusion (SFD) scaling over time, with crossovers
explained by our analytic expressions. Our results reveal transitions in bulk
particle displacement distributions from an early-time bimodal to late-time
Gaussian, passing through regimes of unimodal distributions with finite support
and negative excess kurtosis and longer-tailed distributions with positive
excess kurtosis. The distributions exhibit data collapse, aligning with
ballistic, diffusive, and SFD scaling in the appropriate time regimes. However,
at much longer times, the distributions become Gaussian. Finally, we derive
expressions for steady-state static and dynamic two-point displacement
correlations, consistent with simulations and converging to equilibrium results
for small persistence. Additionally, the two-time stretch correlation extends
to longer separation at later times, while the autocorrelation for the bulk
particle shows diffusive scaling beyond the persistence time.
