  We study a lattice gas of persistent walkers, in which each site is occupied
by at most one particle and the direction each particle attempts to move to
depends on its last step. We analyse the mean squared displacement (MSD) of the
particles as a function of the particle density and their persistence (the
tendency to continue moving in the same direction). For positive persistence
the MSD behaves as expected: it increases with the persistence and decreases
with the density. However, for strong anti-persistence we find two different
regimes, in which the dependence of the MSD on the density is non-monotonic.
For very strong anti-persistence there is an optimal density at which the MSD
reaches a maximum. In an intermediate regime, the MSD as a function of the
density exhibits both a minimum and a maximum, a phenomenon which has not been
observed before. We derive a mean-field theory which qualitatively explains
this behaviour.
