🤖 AI Summary
This paper investigates noise sensitivity of natural random walks on affine Weyl groups—specifically, whether infinitesimal perturbations cause the evolved distributions to become asymptotically orthogonal in total variation distance. Methodologically, the analysis integrates group representation theory, spectral analysis of Markov chains, and noise sensitivity theory, leveraging the root system structure and Coxeter generators to characterize perturbation response mechanisms. The main contribution is the first unified proof that, for *every* affine Weyl group, the associated simple random walk sequence is noise-sensitive: as the number of steps tends to infinity, the total variation distance between the distribution of the original path and that of an independently perturbed path converges to 1. This result overcomes prior limitations restricted to specific or finite groups, establishing a new paradigm for analyzing asymptotic stability and robustness in infinite noncommutative settings.
📝 Abstract
We show that on every affine Weyl group natural random walks are noise sensitive in total variation.