September 28, 2025Open Access

Taming Irrationality: An Invariance Principle for the Random Billiard Walk

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This analysis extends central limit theorem for random billiard walk and highlights implications for Gaussian convergence.

The rescaled trajectories converge to isotropic Brownian motion, confirming the invariance principle under various initial conditions.
For initial directions aligned with the coroot lattice, limiting behavior shows convergence to a centered Gaussian distribution.
Analysis incorporates ergodic theory to generalize existing results, applying them to a broader range of initial conditions and dependencies.
Covariance of the limiting distribution varies continuously based on the initial direction and probability factor, indicating complex dynamics.

R. T. Carpenter (2025) studied this question.

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