Large Deviations of Convex Hulls of Self-Avoiding Random Walks

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 the support down to probabilities far smaller than $P=10^{-100}$ . We give an approximate closed form of the so-called large-deviation rate function $\Phi$ 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.

arXiv comment10 pages, 8 figures

References in corpus (21)

• The large deviation approach to statistical mechanics
• SLE for theoretical physicists
• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Random Convex Hulls and Extreme Value Statistics
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Sampling rare events: statistics of local sequence alignments
• Large-deviation properties of largest component for random graphs
• Spatial extent of an outbreak in animal epidemics
• Efficient implementation of the pivot algorithm for self-avoiding walks
• On large deviation properties of Erdos-Renyi random graphs
• High-precision simulation of the height distribution for the KPZ equation
• Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0
• Unravelling intermittent features in single particle trajectories by a local convex hull method
• Convex Hulls of Random Walks: Large-Deviation Properties
• Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion
• Mean perimeter and mean area of the convex hull over planar random walks
• The dimension of loop-erased random walk in 3D
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study
• Convex Hulls of Multiple Random Walks: A Large-Deviation Study
• Score distributions of gapped multiple sequence alignments down to the low-probability tail
• The Smart Kinetic Self-Avoiding Walk and Schramm-Loewner Evolution

Cited by in corpus (13)

• Long time position distribution of an active Brownian particle in two dimensions
• The convex hull of the run-and-tumble particle in a plane
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Large-deviation properties of the largest biconnected component for random graphs
• Position distribution in a generalised run and tumble process
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Ground state energy of noninteracting fermions with a random energy spectrum
• Large deviations of the length of the longest increasing subsequence of random permutations and random walks
• How many longest increasing subsequences are there?
• Large Deviations of Convex Hulls of the "True" Self-Avoiding Random Walk
• Large deviations in statistics of the convex hull of passive and active particles: A theoretical study
• Large deviations of connected components in the stochastic block model
• Large deviations of a random walk model with emerging territories