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Mean area of the convex hull of a run and tumble particle in two dimensions

We investigate the statistics of the convex hull for a single run-and-tumble particle in two dimensions. Run-and-tumble particle, also known as persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles $n$ or (ii) the total observation time $t$ is kept fixed. Benchmarking the results on perimeter, we study the statistical properties of the area of the convex hull for RTP. Exploiting the connections to extreme value statistics, we obtain exact analytical expressions for the mean area for both ensembles. For fixed-$t$ ensemble, we show that the mean possesses a scaling form in $γt$ (with $γ$ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits crossover from $\sim t^3$ scaling at small times $\left( t \ll γ^{-1} \right)$ to $\sim t$ scaling at large times $\left( t \gg γ^{-1} \right)$. On the other hand, for fixed-$n$ ensemble, the mean expectedly grows linearly with $n$ for $n \gg 1$. All our analytical findings are supported with numerical simulations.