Random walks and quadratic number fields

TL;DR

This paper explores the connection between random walks on the circle group involving quadratic irrationals and the arithmetic properties of quadratic number fields, revealing how algebraic invariants influence convergence rates.

Contribution

It introduces a new link between random walk convergence and algebraic invariants of quadratic fields, bridging probability and number theory.

Findings

Convergence rate depends on fundamental units of quadratic fields

Special zeta function values influence the random walk behavior

Arithmetic invariants govern the uniformity convergence in Wasserstein metric

Abstract

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 $\alpha$, 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 $\mathbb{Q}(\alpha)$, such as fundamental units and special values of zeta functions.