  The emergence of big data has led to so-called convergence complexity
analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms
behave as the sample size, $n$, and/or the number of parameters, $p$, in the
underlying data set increase. This type of analysis is often quite challenging,
in part because existing results for fixed $n$ and $p$ are simply not sharp
enough to yield good asymptotic results. One of the first convergence
complexity results for an MCMC algorithm on a continuous state space is due to
Yang and Rosenthal (2019), who established a mixing time result for a Gibbs
sampler (for a simple Bayesian random effects model) that was introduced and
studied by Rosenthal (1996). The asymptotic behavior of the spectral gap of
this Gibbs sampler is, however, still unknown. We use a recently developed
simulation technique (Qin et. al., 2019) to provide substantial numerical
evidence that the gap is bounded away from 0 as $n \rightarrow \infty$. We also
establish a pair of rigorous convergence complexity results for two different
Gibbs samplers associated with a generalization of the random effects model
considered by Rosenthal (1996). Our results show that, under strong regularity
conditions, the spectral gaps of these Gibbs samplers converge to 1 as the
sample size increases.
