Category: Statistics

Summary: Characterizing how quickly 2D random-walk winding angles approach the classical Cauchy limit, and whether that rate depends on the step distribution.

For planar random walks, the winding angle around the origin is known to converge to a Cauchy law in the long-time limit. This experiment asks how fast that convergence happens in finite walks and whether the correction exponent is universal or instead depends on the underlying step distribution.

The simulation compares Gaussian, uniform, and Cauchy step laws and measures deviations from the limiting Cauchy form using winding-angle statistics such as an L-moment ratio. By fitting how those deviations decay with walk length, it turns a classical asymptotic result into a finite-time scaling problem.

That matters because many limit theorems say little about how quickly the asymptotic regime becomes visible. The experiment is designed to show whether different microscopic step rules leave a measurable signature on the approach to the same universal limit.

Method: Repeated 2D random-walk simulations with multiple step distributions, followed by finite-time scaling fits of winding-angle deviations from the Cauchy law.

What is measured: Winding-angle deviation from Cauchy, convergence-rate exponent, L-moment ratio, and dependence on step distribution.