Reduced Random Walks in the Hyperbolic Plane\hspace1pt!\hspace-3.8pt?

2025/04/27 by Defant, Colin, Lee, Mitchell
#11F06 (Secondary) #20F55 (Primary) 05C81 #20F67 #60B15 #60J10 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.19367

Abstract

We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a point on the extended real line. We devote special attention to the reduced random walk in PGL₂(ℤ) (i.e., the (2,3,∞) triangle group). In this case, we provide an explicit formula for the cumulative distribution function of the limit. This formula is written in terms of the interrobang function, a new function !\hspace-3.8pt?\colon[0,1]→ℝ that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function.

Citations

• Random Combinatorial Billiards and Stoned Exclusion Processes
• Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process
• Absorbing time asymptotics in the oriented swap process
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• Random walks on weakly hyperbolic groups
• The shape of a random affine Weyl group element and random core partitions
• Matrix Representation of the Stationary Measure for the Multispecies TASEP
• The oriented swap process
• Stationary distributions of multi-type totally asymmetric exclusion processes

Related

• Random walks in the hyperbolic plane and the question mark function
• Tracking rates of random walks
• Random walks on hyperbolic groups and their Riemann surfaces
• Local limit theorems in relatively hyperbolic groups I : rough estimates
• A local limit theorem for convergent random walks on relatively hyperbolic groups