  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.
