Finite-time Complexity of the Persistent Random Walk

In the persistent random walk, after each step the walker must decide whether or not to change direction. An unbiased decision leads to the simple random walk and diffusive motion, while a complete bias leads to ballistic motion. The complexity of a process, despite lacking a universal definition, is expected to be higher at the boundary between randomness and determinism, and vanish at either extreme. We study how the complexity of a persistent random walk trajectory varies with both the persistence level and the observation time. We propose that the maximal complexity corresponds to trajectories where the average number of direction-switching events scales as the square root of the observation time. We discuss the reasoning behind this conjecture along with its possible generality. We also analyze fluctuations in particle-tracking measurements of a random walker who selects the optimal persistence value so as to maximize complexity over a given observation time.