Universality and time-scale invariance for the shape of planar Lévy processes

Randon‐Furling, Julien

doi:10.1103/physreve.89.052112

Cited by 3 publications

(10 citation statements)

References 37 publications

(50 reference statements)

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“…As in [40], let us notice that the excursion probability for a random walk is the probability that the walk, pinned at 0 at times 0 and N , visits only the positive half space. This is simply given by the probability that a bridge (that is, a random walk pinned at 0 at times 0 and N ) attains its minimum at the initial step.…”

Section: Discrete-time Casementioning

confidence: 99%

“…This walk starts at some positive value and hits its minimum (0 by construction) at time i = n 1 . The probability associated with this sub-walk is just the probability for a random walk with N − (k 1 + k 2 ) steps to hit its minimum at step i (see [40] for more details). Thus interpreting the product…”

Section: Discrete-time Casementioning

confidence: 99%

“…Following [40], we shall compute the average number of such facets by computing the probability for a triangle defined by three points on the walker's path to form a facet of the convex hull (see Fig. 1).…”

Section: Discrete-time Casementioning

confidence: 99%

“…Remarkably, this was found to have a universal form: it grows logarithmically with time [40], whatever the distribution of the infinitesimal increments along the process's path -as long as these increments are independent and identically distributed (i.e. as long as the process is what is called a Lévy process).…”

Section: Introductionmentioning

confidence: 99%

“…which generalizes the two-dimensional case, where the average number of edges on the boundary of the convex hull was shown in [40] to be given by 2 ln (T /∆t). For the sake of clarity, we first examine the 3-dimensional discrete-time case.…”

Section: Introductionmentioning

confidence: 99%

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Facets on the convex hull ofd-dimensional Brownian and Lévy motion

Randon‐Furling

Wespi

2017

Phys. Rev. E

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For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d ≥ 3, we establish an exact formula for the average number of (d − 1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d = 3, a case which is of particular interest for applications in biophysics, chemistry and polymer science.We also show that the asymptotical average number of facets behaves as, where T is the total duration of the motion and ∆t is the minimum time lapse separating points that define a facet.

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Section: Discrete-time Casementioning

confidence: 99%

Section: Discrete-time Casementioning

confidence: 99%

Section: Discrete-time Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Facets on the convex hull ofd-dimensional Brownian and Lévy motion

Randon‐Furling

Wespi

2017

Phys. Rev. E

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Convex hull of a Brownian motion in confinement

2015

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We study the effect of confinement on the mean perimeter of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory. We use a minimal model where an infinite reflecting wall confines the walk to its one side. We show that the mean perimeter displays a surprising minimum with respect to the starting distance to the wall and exhibits a nonanalyticity for small distances. In addition, the mean span of the trajectory in a fixed direction θ ∈]0, π/2[, which can be shown to yield the mean perimeter by integration over θ, presents these same two characteristics. This is in striking contrast with the one dimensional case, where the mean span is an increasing analytical function. The non-monotonicity in the 2D case originates from the competition between two antagonistic effects due to the presence of the wall: reduction of the space accessible to the Brownian motion and effective repulsion. How does one characterize the territory covered by a Brownian motion in two dimensions? This question naturally arises in ecology where the trajectory of an animal during the foraging period is well approximated by a Brownian motion [1,2] and one needs to estimate the home range of the animal, i.e., the two dimensional (2D) space over which the animal moves around over a fixed period of time [3]. The most versatile and popular method to characterize the home range consists in drawing the convex hull, i.e., the minimum convex polygon enclosing the trajectory of the animal [4,5]. The size of the home range is then naturally estimated by the mean perimeter or the mean area of the convex hull.For a single planar Brownian motion of duration t and diffusion constant D, the mean perimeter L(t) = √ 16 π D t and the mean area A(t) = π D t were computed exactly in the mathematics literature quite a while back [6][7][8]. Very recently, there has been a growing number of articles both in the physics [9][10][11][12]14] and the mathematics literature [16][17][18][19] generalizing these results in various directions. In particular, adapting Cauchy's formula [22] for closed 2D convex curves to the case of random curves, a general method was recently proposed [9, 10] to compute the mean perimeter and the mean area of the convex hull of any arbitrary stochastic process in 2D. In cases where the process is isotropic in 2D, the mean perimeter and the mean area of its convex hull can be mapped onto computing the extremal statistics of the corresponding one dimensional component process [9,10]. This procedure was then successfully used to compute exactly the mean perimeter and the mean area of a number of isotropic 2D stochastic processes such as N independent Brownian motions [9, 10], random acceleration process [11], branching Brownian motion with absorption with applications to epidemic spread [12] and for anomalous diffusion processes [14].All these results pertain to isotropic stochastic processes in the unconfined 2D geometry. However, in many practical situations, the stochastic process takes plac...

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Mean perimeter of the convex hull of a random walk in a semi-infinite medium

2015

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We study various properties of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory, in the presence of an infinite reflecting wall. Recently, in a Rapid Communication [Phys. Rev. E 91, 050104(R) (2015)], we announced that the mean perimeter of the convex hull at time t, rescaled by √ Dt, is a non-monotonous function of the initial distance to the wall. In the present article, we first give all the details of the derivation of this mean rescaled perimeter, in particular its value when starting from the wall and near the wall. We then determine the physical mechanism underlying this surprising non-monotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull. Finally, we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the Brownian motion as a function of the initial distance to the wall.

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Universality and time-scale invariance for the shape of planar Lévy processes

Cited by 3 publications

References 37 publications

Facets on the convex hull ofd-dimensional Brownian and Lévy motion

Facets on the convex hull ofd-dimensional Brownian and Lévy motion

Convex hull of a Brownian motion in confinement

Mean perimeter of the convex hull of a random walk in a semi-infinite medium

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