Leptokurtosis is unlikely to be an artifact of poor paternity

Simulations suggest that if ignored, small numbers of errors in paternity assignment would inflate apparent mean dispersal distances dramatically. Even a small number of errors would be enough to generate this pattern, because falsely assigned fathers are likely to be drawn at random from the population, hence be on average much further from the mother than the true father would have been. One explanation for why unrelated candidates may be assigned high probabilities of paternity either because stochasticity in Mendelian segregation means they happen to have a higher probability of paternity than the true sire. However, our simulations showed when  fathers were included in the sample we expect them to have posterior probabilities of paternity of at least 0.98. The statistical power of the marker set to identify true father thus very high.
An alternative explanation is that the true father was missing from the set of candidates, meaning that one or more unrelated candidates had high probabilities of paternity. The MCMC results suggested that as many as 40% of the fathers were missing. However, the posterior distribution largely followed the prior (figures \ref{281170} and \ref{804996}), so its likely that the data are not informative about the proportion of missing fathers. Since the number of plants in the population numbers in the thousands it is probable that a non-zero number of true fathers were missed. However, A.majus is generally restricted to the disturbed habitat along the roadsides which were surveyed multiple times, and although some plants colonise inaccessible areas, it is not likely that there can be as many missing fathers as the posterior distribution would suggest.
Bumblebee biology can also tell us something about whether such long distances are plausible. Pollen dispersal kernels on similar scales have been reported in other systems. Bumblebees are known to to forage over hundreds of metres. Bumblebee queens in particular have been reported to travel several kilometers, and are also frequent visitors to A. majus at the start of the season as they provide generous nectar rewards (personal observation). On the other hand, pollen dispersal distances are often strongly dependent on local plant density. Most previous studies have been on trees which are considerably less dense than snapdragons, so we would expect movement between neighbours on a smaller scale in Antirrhinum.