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 P(L)~e^-bNL²/DT and P(A)~e^-cNA/DT, while the small-L and -A tails behave as P(L)~e^-dNDT/L² and P(A)~e^-eNDT/A, where D is the diffusion coefficient. We calculated all of the coefficients (bN,cN,dN,eN) exactly. Strikingly, we find that bN and cN are independent of N for N≥3 and N≥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 P(L)~e^-TΨNᵖᵉʳ(L/T) and P(A)~e^-TΨNᵃʳᵉᵃ(√A/T), respectively. We calculate the rate functions ΨN 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.