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Area of Science:

  • Statistical mechanics
  • Computational geometry
  • Probability theory

Background:

  • Studying the geometric properties of point sets generated by random walks is crucial in statistical physics.
  • Understanding the convex hull of these point sets provides insights into their spatial distribution.
  • Previous work has focused on single random walkers (n=1).

Purpose of the Study:

  • To analyze the area (A) and perimeter (L) of convex hulls formed by n independent two-dimensional random walkers.
  • To investigate the probability densities of A and L using a large-deviation approach.
  • To determine the scaling behavior and limiting distributions of these geometric quantities.

Main Methods:

  • Utilizing a large-deviation approach to calculate probability densities for area and perimeter.
  • Analyzing the behavior of these densities in the limit of large time steps (T→∞).
  • Investigating the influence of the number of walkers (n) on the system's properties.

Main Results:

  • Probability densities for area and perimeter exhibit time-independent scaling with A/T and L/sqrt[T] respectively, as T→∞.
  • The densities for perimeter (L) and the square root of area (sqrt[A]) follow Gaussian distributions.
  • The study confirms the large-deviation principle for area and perimeter, with rate functions describable by a power law for large n.

Conclusions:

  • The convex hull properties of multiple random walkers display universal scaling behaviors.
  • The large-deviation approach accurately describes the probability distributions of area and perimeter.
  • The findings extend previous results for single walkers to systems with multiple walkers.