Large deviations in statistics of the convex hull of passive and active particles: A theoretical study

Abstract

We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and A tails behave as $\mathcal{P}\left(L\right)\sim e^{-b_N{L}^2/DT}$ and $\mathcal{P}\left(A\right)\sim e^{-c_{N}A/DT}$, while the small-$L$ and $A$ tails behave as $\mathcal{P}\left(L\right)\sim e^{-d_{N}DT/L^2}$ and $\mathcal{P}\left(A\right)\sim e^{-e_{N}DT/A}$, where $D$ is the diffusion coefficient. We calculated all of the coefficients ($b_N,c_N,d_N,e_N$) exactly. Strikingly, we find that $b_N$ and $c_N$ are independent of N, for $N\ge 3$ and $N\ge 4$, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and A tails are given by $\mathcal{P}\left(L\right)\sim e^{-T{\Psi }_N^{\text{per}}\left(L/T\right)}$ and $\mathcal{P}\left(A\right)\sim e^{-T{\Psi }_N^{\text{area}}\left(\sqrt{A}/T\right)}$ respectively. We calculate the large deviation functions ${\Psi }_N$ exactly and find that they exhibit multiple singularities. We interpret these as dynamical phase transitions of first order. We extended several of these results to dimensions $d>2$. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.

arXiv comment15 pages, 9 figures

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