  Strong positional correlations between particles render the diffusion of a
tracer particle in a single file anomalous and non-Markovian. While ensemble
average observables of tracer particles are nowadays well understood, little is
known about the statistics of the corresponding functionals, i.e. the
time-average observables. It even remains unclear how the non-Markovian nature
emerges from correlations between particle trajectories at different times.
Here, we first present rigorous results for fluctuations and two-tag
correlations of general bounded functionals of ergodic Markov processes with a
diagonalizable propagator. They relate the statistics of functionals on
arbitrary time-scales to the relaxation eigenspectrum. Then we study tagged
particle local times -- the time a tracer particle spends at some predefined
location along a single trajectory up to a time t. Exact results are derived
for one- and two-tag local times, which reveal how the individual particles'
histories become correlated at higher densities because each consecutive
displacement along a trajectory requires collective rearrangements. Our results
unveil the intricate meaning of projection-induced memory on a trajectory
level, invisible to ensemble-average observables, and allow for a detailed
analysis of single-file experiments probing tagged particle exploration
statistics.
