Using Householder Matrices to Establish Mixing Test Critical Values

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.