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Mukherjee S, Smith NR. Large deviations in statistics of the convex hull of passive and active particles: A theoretical study. Phys Rev E 2024; 109:044120. [PMID: 38755832 DOI: 10.1103/physreve.109.044120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2023] [Accepted: 03/08/2024] [Indexed: 05/18/2024]
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 P(L)∼e^{-b_{N}L^{2}/DT} and P(A)∼e^{-c_{N}A/DT}, while the small-L and -A tails behave as P(L)∼e^{-d_{N}DT/L^{2}} and P(A)∼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≥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^{-TΨ_{N}^{per}(L/T)} and P(A)∼e^{-TΨ_{N}^{area}(sqrt[A]/T)}, respectively. We calculate the rate functions Ψ_{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.
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Affiliation(s)
- Soheli Mukherjee
- Department of Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000, Israel
| | - Naftali R Smith
- Department of Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000, Israel
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Cygan W, Sandrić N, Šebek S. Convex hulls of stable random walks. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp826] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Wojciech Cygan
- Technische Universität Dresden, Faculty of Mathematics, Institute of Mathematical Stochastics, Zellescher Weg 25, 01069 Dresden, Germany
| | - Nikola Sandrić
- Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
| | - Stjepan Šebek
- Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
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Alexander Y, Eldan R. Cone points of Brownian motion in arbitrary dimension. ANN PROBAB 2019. [DOI: 10.1214/19-aop1335] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Schawe H, Hartmann AK, Majumdar SN. Large deviations of convex hulls of self-avoiding random walks. Phys Rev E 2018; 97:062159. [PMID: 30011525 DOI: 10.1103/physreve.97.062159] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2018] [Indexed: 06/08/2023]
Abstract
A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely, the classical self-avoiding walk, the smart-kinetic self-avoiding walk, and the loop-erased random walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than P=10^{-100}. We give an approximate closed form of the so-called large-deviation rate function Φ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further, we show correlations between the two observables also in the limits of atypical large or small values.
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Affiliation(s)
- Hendrik Schawe
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alexander K Hartmann
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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McRedmond J, Wade AR. The convex hull of a planar random walk: perimeter, diameter, and shape. ELECTRON J PROBAB 2018. [DOI: 10.1214/18-ejp257] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Schawe H, Hartmann AK, Majumdar SN. Convex hulls of random walks in higher dimensions: A large-deviation study. Phys Rev E 2017; 96:062101. [PMID: 29347304 DOI: 10.1103/physreve.96.062101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2017] [Indexed: 06/07/2023]
Abstract
The distribution of the hypervolume V and surface ∂V of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than P=10^{-1000} to estimate large deviation properties. For arbitrary dimensions and large walk lengths T, we suggest a scaling behavior of the distribution with the length of the walk T similar to the two-dimensional case and behavior of the distributions in the tails. We underpin both with numerical data in d=3 and d=4 dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large T.
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Affiliation(s)
- Hendrik Schawe
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alexander K Hartmann
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Tikhomirov K, Youssef P. When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull? ANN PROBAB 2017. [DOI: 10.1214/15-aop1079] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Randon-Furling J, Wespi F. Facets on the convex hull of d-dimensional Brownian and Lévy motion. Phys Rev E 2017; 95:032129. [PMID: 28415327 DOI: 10.1103/physreve.95.032129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2017] [Indexed: 06/07/2023]
Abstract
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 〈F_{T}^{(d)}〉∼2[ln(T/Δt)]^{d-1}, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.
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Affiliation(s)
- Julien Randon-Furling
- SAMM (EA 4543), Université Paris-1 Panthéon-Sorbonne, Centre Pierre Mendès-France, 90 rue de Tolbiac, 75013 Paris, France
| | - Florian Wespi
- IMSV, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
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Dewenter T, Claussen G, Hartmann AK, Majumdar SN. Convex hulls of multiple random walks: A large-deviation study. Phys Rev E 2016; 94:052120. [PMID: 27967062 DOI: 10.1103/physreve.94.052120] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2016] [Indexed: 11/07/2022]
Abstract
We study the polygons governing the convex hull of a point set created by the steps of n independent two-dimensional random walkers. Each such walk consists of T discrete time steps, where x and y increments are independent and identically distributed Gaussian. We analyze area A and perimeter L of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below 10^{-900}. We find that the densities exhibit in the limit T→∞ a time-independent scaling behavior as a function of A/T and L/sqrt[T], respectively. As in the case of one walker (n=1), the densities follow Gaussian distributions for L and sqrt[A], respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for T→∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n→∞ as found in the n=1 case. We also investigated the behavior of the averages as a function of the number of walks n and found good agreement with the predicted behavior.
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Affiliation(s)
- Timo Dewenter
- Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany
| | - Gunnar Claussen
- Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany.,Fachbereich Ingenieurwissenschaften, Jade Hochschule Wilhelmshaven/Oldenburg/Elsfleth, D-26389 Wilhelmshaven, Germany
| | | | - Satya N Majumdar
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université de Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
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Chupeau M, Bénichou O, Majumdar SN. Mean perimeter of the convex hull of a random walk in a semi-infinite medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022145. [PMID: 26382381 DOI: 10.1103/physreve.92.022145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2015] [Indexed: 06/05/2023]
Abstract
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 [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 nonmonotonous function of the initial distance to the wall. In this 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 nonmonotonicity 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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Affiliation(s)
- Marie Chupeau
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Satya N Majumdar
- CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, 91405 Orsay Cedex, France
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Chupeau M, Bénichou O, Majumdar SN. Convex hull of a Brownian motion in confinement. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:050104. [PMID: 26066101 DOI: 10.1103/physreve.91.050104] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2014] [Indexed: 06/04/2023]
Abstract
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 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 to the one-dimensional case, where the mean span is an increasing analytical function. The nonmonotonicity in the two-dimensional 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.
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Affiliation(s)
- Marie Chupeau
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
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