In this section
@article{Sezik:2026:1751-8121/ae85eb,
author = {Sezik, E and Britton, C and Touma, A and Pruessner, G},
doi = {1751-8121/ae85eb},
journal = {Journal of Physics A Mathematical and Theoretical},
title = {Run, tumble and paint},
url = {http://dx.doi.org/10.1088/1751-8121/ae85eb},
volume = {59},
year = {2026}
}
TY - JOUR
AB - The visit probability, quantifying whether a particle has reached a given point for the first time by a specified time, provides access to various extreme value statistics and serves as a fundamental tool for characterising active matter models. However, previous studies have largely neglected how the visit probability depends on the internal degree of freedom driving the active particle. To address this, we calculate the ‘state-dependent’ visit probability for a run-and-tumble (RnT) particle, that is the probability that the particle first passes through (Formula presented) (Formula presented) before time (Formula presented) (Formula presented), keeping track of its internal state during first passage. This process may be thought of as the particle ‘painting’ the positions it passes through for the first time in the colour of its self-propulsion state. We perform this calculation in one dimension using Doi–Peliti field theory, by extending the tracer mechanism from previous works to incorporate such ‘polar deposition’ and demonstrate that state-dependent visit probabilities can be elegantly captured within this field-theoretic framework. We further derive the total volume covered by a right- (or left-) moving RnT particle and compare our results with known expressions for Brownian motion.
AU - Sezik,E
AU - Britton,C
AU - Touma,A
AU - Pruessner,G
DO - 1751-8121/ae85eb
PY - 2026///
SN - 1751-8113
TI - Run, tumble and paint
T2 - Journal of Physics A Mathematical and Theoretical
UR - http://dx.doi.org/10.1088/1751-8121/ae85eb
VL - 59
ER -