  We study run and tumble particles on the one-dimensional lattice
$\mathbb{Z}$. We explicitly compute the Fourier-Laplace transform of the
position of the particle and as a consequence obtain explicit expressions for
the diffusion constant and the large deviation free energy function. We also do
the same computations in a corresponding continuum model. In the latter, when
adding an external field, we can explicitly compute the large deviation free
energy, and the deviation from the Einstein relation due to activity. Finally,
we generalize the model to the $d$-dimensional lattice $\mathbb{Z}^d$, with an
arbitrary finite set of velocities, and show that the large deviation free
energy for the position of the particle can be computed via the largest
eigenvalue of a matrix of Schr\"{o}dinger operator form, for which we can
derive an explicit variational formula via occupation time large deviations of
the velocity flip process.
