  We investigate the probability distribution of Chern numbers (quantum Hall
effect integers) for a parametric version of the GUE random matrix ensemble,
which is a model for a chaotic or disordered system. The numerically-calculated
single-band Chern number statistics agree well with predictions based on an
earlier study [O. Gat and M. Wilkinson, SciPost Phys., 10, 149, (2021)] of the
statistics of the quantum adiabatic curvature, when the parametric correlation
length is small. However, contrary to an earlier conjecture, we find that the
gap Chern numbers are correlated, and that correlation is weak but
slowly-decaying. Also, the statistics of weighted sums of Chern numbers for
many bands differs markedly from predictions based upon the hypothesis that gap
Chern numbers are uncorrelated. All our results are consistent with the
universality hypothesis described in the earlier paper, including in the
previously unstudied regime of large correlation length, where the Chern
statistics is highly non-Gaussian.
