  We investigate active lattice walks: biased continuous time random walks
which perform orientational diffusion between lattice directions in one and two
spatial dimensions. We study the occupation probability of an arbitrary site on
the lattice in one and two dimensions, and derive exact results in the
continuum limit. Next, we compute the large deviation free energy function in
both one and two dimensions, which we use to compute the moments and the
cumulants of the displacements exactly at late times. Our exact results
demonstrate that the cross-correlations between the motion in the $x$ and $y$
directions in two dimensions persist in the large deviation function. We also
demonstrate that the large deviation function of an active particle with
diffusion displays two regimes, with differing diffusive behaviors. We verify
our analytic results with kinetic Monte Carlo simulations of an active lattice
walker in one and two dimensions.
