Mixing time upper bound for the uniformized Rosenthal walk on the special orthogonal groups

TL;DR

This paper establishes a polynomial-time mixing bound of order n^3 steps for a uniformized variant of the Rosenthal and Kac random walks on SO(n), using representation theory and novel combinatorial interpretations.

Contribution

It provides the first polynomial time mixing bounds for these walks on SO(n), extending to fixed angles and employing new representation-theoretic techniques.

Findings

Mixing time is O(n^3) steps in total variation distance.

Extension of results to fixed angle θ ≠ π.

Introduction of particle cascade path interpretation for Fourier coefficients.

Abstract

We prove that a uniformized variant of both the Rosenthal walk \cite{Rosenthal} and the Kac random walk \cite{Kac} on SO(n) mixes in $\cO(n^3)$ steps in total variation distance. The proof also extends easily to Rosenthal walk with fixed angle $\theta \neq \pi$. To the best of our knowledge, this is the first polynomial time bound for both walks. The techniques employed are mainly from representation theory of SO(n). But a crucial new ingredient is the interpretation of the Fourier coefficients of the character ratio as counting the number of particle cascade paths arising from the classical branching rules.