Random walks and quadratic number fields

2025/12/03 by Borda, Bence
#11E45 #11K38 #11R42 #60G50 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.03884

Abstract

We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group ℝ/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 L² discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field ℚ(α), such as fundamental units and special values of zeta functions.

Citations

• Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of \nα\
• Irregularity of distribution in Wasserstein distance
• One-dimensional empirical measures, order statistics, and Kantorovich transport distances
• Central limit theorem and Diophantine approximations
• Special values of partial zeta functions of real quadratic fields at nonpositive integers and Euler-Maclaurin formula
• Values of zeta functions at negative integers, Dedekind sums and toric geometry

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• On homogeneous and oscillating random walks on the integers
• Random walks with badly approximable numbers