The shapes of random walks

Area of Science:

• Theoretical physics
• Polymer physics
• Statistical mechanics

Background:

• Characterizing the shape of random objects is crucial in various scientific fields.
• Traditional methods often face challenges with ensemble averaging in random processes.
• A need exists for analytically simple yet quantitatively accurate shape descriptors.

Purpose of the Study:

• To present a novel theoretical framework for describing the shape of random objects.
• To simplify the quantitative analysis of object asymmetry using tensor invariants.
• To reduce complications in ensemble averaging by utilizing high-dimensional spaces.

Main Methods:

• Characterizing object asymmetry via invariants of a tensor analogous to the moment-of-inertia tensor.
• Embedding random objects in high-dimensional spaces to simplify ensemble averaging.
• Developing an expansion in powers of 1/d for linear chain and ring-type random walks in d spatial dimensions.
• Deriving exact analytical expressions for infinite spatial dimensions.

Main Results:

• The first two terms of the 1/d expansion yield shape parameters that closely match computer simulations.
• The theoretical approach provides a method for accurate expressions of the probability distribution function.
• The method demonstrates remarkable agreement with simulation data for random walks.

Conclusions:

• The presented theoretical description offers an analytically simple and quantitatively accurate method for random object shape analysis.
• The high-dimensionality approach effectively simplifies complex ensemble averaging problems.
• This framework can be extended to describe the shape of other random fractal objects.