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

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