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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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Hartmann AK, Krajenbrink A, Le Doussal P. Probing the large deviations for the beta random walk in random medium. Phys Rev E 2024; 109:024122. [PMID: 38491613 DOI: 10.1103/physreve.109.024122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Accepted: 01/11/2024] [Indexed: 03/18/2024]
Abstract
We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.
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Affiliation(s)
| | - Alexandre Krajenbrink
- Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom
- Le Lab Quantique, 58 rue d'Hauteville, 75010, Paris, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'École Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
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Schawe H, Hartmann AK. Large deviations of a random walk model with emerging territories. Phys Rev E 2021; 102:062141. [PMID: 33466107 DOI: 10.1103/physreve.102.062141] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2020] [Accepted: 12/02/2020] [Indexed: 11/07/2022]
Abstract
We study an agent-based model of animals marking their territory and evading adversarial territory in one dimension with respect to the distribution of the size of the resulting territories. In particular, we use sophisticated sampling methods to determine it over a large part of territory sizes, including atypically small and large configurations, which occur with probability of less than 10^{-30}. We find hints for the validity of a large deviation principle, the shape of the rate function for the right tail of the distribution, and insight into the structure of atypical realizations.
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Affiliation(s)
- Hendrik Schawe
- Laboratoire de Physique Théorique et Modélisation, UMR-8089 CNRS, CY Cergy Paris Université, 95000 Cergy, France
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Hartmann AK, Krajenbrink A, Le Doussal P. Probing large deviations of the Kardar-Parisi-Zhang equation at short times with an importance sampling of directed polymers in random media. Phys Rev E 2020; 101:012134. [PMID: 32069556 DOI: 10.1103/physreve.101.012134] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2019] [Indexed: 11/07/2022]
Abstract
The one-point distribution of the height for the continuum Kardar-Parisi-Zhang equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. The short-time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.
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Affiliation(s)
| | - Alexandre Krajenbrink
- Laboratoire de Physique de l'École Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'École Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
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Basu U, Majumdar SN, Rosso A, Schehr G. Long-time position distribution of an active Brownian particle in two dimensions. Phys Rev E 2019; 100:062116. [PMID: 31962395 DOI: 10.1103/physreve.100.062116] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2019] [Indexed: 06/10/2023]
Abstract
We study the late-time dynamics of a single active Brownian particle in two dimensions with speed v_{0} and rotation diffusion constant D_{R}. We show that at late times t≫D_{R}^{-1}, while the position probability distribution P(x,y,t) in the x-y plane approaches a Gaussian form near its peak describing the typical diffusive fluctuations, it has non-Gaussian tails describing atypical rare fluctuations when sqrt[x^{2}+y^{2}]∼v_{0}t. In this regime, the distribution admits a large deviation form, P(x,y,t)∼exp{-tD_{R}Φ[sqrt[x^{2}+y^{2}]/(v_{0}t)]}, where we compute the rate function Φ(z) analytically and also numerically using an importance sampling method. We show that the rate function Φ(z), encoding the rare fluctuations, still carries the trace of activity even at late times. Another way of detecting activity at late times is to subject the active particle to an external harmonic potential. In this case we show that the stationary distribution P_{stat}(x,y) depends explicitly on the activity parameter D_{R}^{-1} and undergoes a crossover, as D_{R} increases, from a ring shape in the strongly active limit (D_{R}→0) to a Gaussian shape in the strongly passive limit (D_{R}→∞).
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Affiliation(s)
- Urna Basu
- Raman Research Institute, Bengaluru 560080, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alberto Rosso
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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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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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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