Convex hulls of random walks: Large-deviation properties

Area of Science:

• Statistical Physics
• Probability Theory
• Computational Mathematics

Background:

• The mean perimeter and area of 2D random walks are well-understood for large times (T).
• However, the full probability distributions for these observables, P(A) and P(L), remain largely uncharacterized.
• Understanding these distributions is crucial for a complete description of random walk behavior.

Purpose of the Study:

• To numerically determine the full probability distributions P(A) and P(L) for the area and perimeter of a 2D random walk's convex hull.
• To investigate the distributions over a wide range, including rare events with probabilities as low as 10⁻³⁰⁰.
• To analyze both open random walks and closed Brownian bridges in 2D.

Main Methods:

• Employed a sophisticated large-deviation approach for numerical analysis.
• Studied distributions for probabilities down to 10⁻³⁰⁰.
• Utilized a numerical extrapolation scheme to determine rate functions for rare events.

Main Results:

• The full distributions P(A) and P(L) were computed numerically.
• For large T, universal scaling behaviors were observed as functions of A/T and L/√T, respectively.
• The rate functions for rare events show linear and square dependencies on rescaled perimeter and area.

Conclusions:

• The study provides the first comprehensive numerical results for the full distributions of random walk convex hull area and perimeter.
• Universal scaling laws govern these distributions for large time steps, independent of specific jump distributions.
• The findings offer insights into the statistical properties and rare event statistics of 2D random walks.