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

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 $\mathcal{P}\left(L\right)\sim e^{-T{\mathrm{\Psi }}_N^{\text{per}}(L/T)}$ and $\mathcal{P}\left(A\right)\sim e^{-T{\mathrm{\Psi }}_N^{\text{area}}(\sqrt{A}/T)}$, respectively. We calculate the rate functions ${\mathrm{\Psi }}_N$ exactly and find that they exhibit multiple singularities. We interpret these as DPTs 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.

Figure 1

Schematic diagram of the convex hull of a trajectory of a BM. To minimize the length of the trajectory $\mathcal{L}$ constrained on the convex hull area $A$ or perimeter $L$, it is more favorable to follow the dotted (straight) line than the solid line, when going between the two points marked in the figure.

Figure 2

Solid lines: Trajectories of minimal length $\mathcal{L}$, corresponding to the right tails of the area and perimeter distributions. For the area distribution, both the open and closed BMs are plotted. Schematic diagrams of the convex hulls formed by BMs of length $\mathcal{L}$, corresponding to the right tails of the area and perimeter distributions. The dotted lines are a schematic of realistic realizations that attain large but finite area (or perimeter). For the closed case, the hull is a circle for a fixed area $A$ and for the open case, the hull is a half circle for a fixed $A$ and line segment for fixed $L$.

Figure 3

Right tails of the distributions (in log scale) of the area (a) and perimeter (b) of the convex hull for an open planar BM. The blue solid line depicts the data of $\mathcal{P}\left(A\right)$ from [23] and can be seen to be in excellent agreement with the red dashed lines which denote our theoretical predictions. Parameters are $D=1/2$, and $T=1000$ (a) and and $T=1400$ (b).

Figure 4

Schematic diagram of the convex hull formed by $N$ BMs of length $\mathcal{L}$. Here $N=6, M=5$, and $\theta =\frac{2\pi }{M}$ is the angle between two BMs participating in the solution. As explained in the text, the solutions with $M=3$ and $M=4$ are the optimal solution for perimeter and area, respectively.

Figure 5

Right tail of the distribution $\mathcal{P}\left(L\right)$ of perimeter $L$ of the convex hull for a RTP for walk length $T=512$ and $\gamma =v_0=1$. The blue circles depict the numerical data from Ref. [28] and the red dashed lines denote our theoretical prediction. The yellow dot-dashed line denotes the corresponding BM result, obtained by applying the parabolic Brownian approximation (36) of the rate function $\mathrm{\Phi }\left(z\right)$. The Brownian approximation can be seen to correctly describe the near tail of the distribution, but it breaks down in the far tail.

Figure 6

The rate function ${\psi }_N\left(z\right)$ that describes the right distribution tail for (a) area $A$ and (b) perimeter $L$ for $N$ RTPs. The solid blue line depicts ${\psi }_{N\to \infty }$ while the dashed lines correspond to ${\psi }_N$ for finite values of $N=3,4,\cdots$ (from bottom to top). The dotted lines correspond to ${\tilde{\psi }}_M$ in regions where they are not optimal for any value of $N$ (again increasing with $M$ from bottom to top). The circles denote the critical points $z_M$ at which the optimal solution changes from one local minima ($M$) to another local minima ($M+1$), signaling a first-order DPT. The blue parabolic dot-dashed lines correspond to the BM approximations [for $N\ge 4$ in (a)] and are seen to give the correct asymptotic behavior at $z\ll 1$, describing the near right tail of the distribution. The vertical, gray dot-dashed lines denote the value that the area or perimeter cannot possibly exceed for any $N$, corresponding to a convex hull which is a circle of radius $v_{0}T$.

Figure 7

The shortest curve in the three-dimensional space whose convex hull has a given surface area, obtained numerically using a gradient-descent algorithm. The figure shows the projection of the trajectory onto the $xz$ plane. The $y$ components of all the points are very small on the scale of the figure.

Figure 8

Coefficients ${\tilde{b}}_M$ and ${\tilde{c}}_M$ as a function of $M$. The $M=3$ and $M=4$ are optimal solutions for perimeter and area, respectively, i.e., they minimize the functions ${\tilde{b}}_M$ and ${\tilde{c}}_M$ over integer values $M\ge 3$.

Figure 9

Right tail distribution of (a) $\mathcal{P}\left(L\right)$ and (b) $\mathcal{P}\left(A\right)$ for multiple BMs. The markers depict the data of the distributions from Ref. [24] and the dashed and dotted lines denote our theoretical predictions for the distribution tails.