Convex hulls of random walks in higher dimensions: A large-deviation study

Area of Science:

• Statistical Mechanics
• Computational Physics
• Geometric Probability

Background:

• Understanding the geometric properties of random walks is crucial in fields like statistical mechanics and polymer physics.
• Convex hulls of random walks in higher dimensions present complex distributions that are analytically challenging to determine.
• Investigating large deviation properties requires analyzing extremely small probabilities, often beyond direct analytical calculation.

Purpose of the Study:

• To numerically determine the distribution of hypervolume and surface area for convex hulls of random walks in higher dimensions.
• To estimate large deviation properties by analyzing probabilities significantly smaller than 10^-1000.
• To investigate the scaling behavior of these distributions with walk length (T) and compare it to known 2D cases.

Main Methods:

• Numerical simulations were employed to generate random walks in dimensions d=3 and d=4.
• The hypervolume and surface area of the convex hulls of these walks were calculated.
• Statistical analysis was performed on the distributions, focusing on tail behavior and extreme probabilities.

Main Results:

• A scaling behavior of the distribution with walk length T was observed, similar to the two-dimensional case, for arbitrary dimensions.
• The tail behavior of the distributions was characterized, providing insights into large deviation properties.
• Analytical results for the means of the distributions were confirmed, and their variances for large T were calculated.

Conclusions:

• The study provides robust numerical evidence for the scaling behavior of random walk convex hull distributions in higher dimensions.
• Numerical methods are effective for exploring large deviation properties in complex systems.
• The findings contribute to a deeper understanding of the geometry of random walks and their statistical properties.