We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational $α$, we show that the rate of convergence to uniformity in the quadratic Wasserstein metric (also known as the periodic $L^2$ discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field $\mathbb{Q}(α)$, such as fundamental units and special values of zeta functions.