  I show that random distributions of vortex-antivortex pairs (rather than of
individual vortices) lead to scaling of typical winding numbers W trapped
inside a loop of circumference C with the square root of C when the expected
winding numbers are large. Such scaling is consistent with the Kibble-Zurek
mechanism (KZM). By contrast, distribution of individual vortices with randomly
assigned topological charges would result in the dispersion of W scaling with
the square root of the area inside C. Scaling of the dispersion of W and of the
probability of detection of non-zero W with C can be also studied for loops so
small that non-zero windings are rare. In this case I show a doubling of the
scaling of dispersion with C when compared to the scaling of dispersion in the
large W regime. Moreover, probability of trapping of a non-zero W becomes, in
this case, proportional to the area subtended by C (hence, to the square of
circumference). This quadruples, as compared with large winding numbers regime,
the exponent in the power law dependence of the frequency of trapping of W=+1
or W=-1 on C. Such change of the power law exponent by a FACTOR OF FOUR implies
quadrupling of the scaling of the frequency of winding number trapping with the
quench rate, and is of key importance for experimental tests of KZM.
