Central limit theorem and large deviations for run and tumble particles on a lattice

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.