  Simple causal arguments put forward by Kibble and Zurek suggest that the
scaling behaviour of condensed matter at continuous transitions is related to
the familiar universality classes of the systems at quasi-equilibrium. Although
proposed 25 years ago or more, it is only in the last few years that it has
been possible to devise experiments from which scaling exponents can be
determined and in which this scenario can be tested. In previous work, an
unusually high Kibble-Zurek scaling exponent was reported for spontaneous
fluxoid production in a single isolated superconducting Nb loop, albeit with
low density. Using analytic approximations backed up by Langevin simulations,
we argue that densities as small as these are too low to be attributable to
scaling, and are conditioned by the small size of the loop. We also reflect on
the physical differences between slow quenches and small rings, and derive some
criteria for these differences, noting that recent work on slow quenches does
not adequately explain the anomalous behaviour seen here.
