Using Householder Matrices to Establish Mixing Test Critical Values

TL;DR

This paper introduces a method for establishing critical values in mixing tests by using Monte Carlo simulations of unistochastic matrices derived from Householder matrices, aiding in the classification of stirring protocols.

Contribution

It proposes a novel approach to determine critical values for mixing tests using Monte Carlo distributions from unistochastic matrices based on Householder matrices.

Findings

Monte Carlo distributions effectively establish critical values

Unistochastic matrices derived from Householder matrices model mixing properties

Method improves hypothesis testing accuracy for mixing protocols

Abstract

A measure-preserving dynamical system can be approximated by a Markov shift with a bistochastic matrix. This leads to using empirical stochastic matrices to measure and estimate properties of stirring protocols. Specifically, the second largest eigenvalue can be used to statistically decide if a stirring protocol is weak-mixing, ergodic, or nonergodic. Such hypothesis tests require appropriate probability distributions. In this paper, we propose using Monte Carlo empirical probability distributions from unistochastic matrices to establish critical values. These unistochastic matrices arise from randomly constructed Householder matrices.