Large Deviations of Convex Hulls of Self-Avoiding Random Walks

TL;DR

This paper investigates the probability distributions of the volume and surface of convex hulls formed by various self-avoiding random walks using advanced large-deviation techniques, revealing detailed insights into their extreme behaviors.

Contribution

It introduces a comprehensive numerical approach to analyze large deviations in convex hull properties of self-avoiding walks, including an approximate rate function and correlation analysis.

Findings

Distributions extend to probabilities smaller than 10^{-100}

Derived an approximate large-deviation rate function

Identified correlations between volume and surface in extreme regimes

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.