I don't blindly accept a paper's conclusions. I always at least attempt to check the assumptions, the methodology, the programs, and yes, the mathematics.
As for the direction of flow, I believe it was you who said the flow went both ways. I don't think I ever made any such claim.
I have a few issues with the conclusions drawn by the authors of this paper as regards Italy. For one, let's take the postulated date for the admixture (in the 900's) in Toscana. There is no large migration into the area in that century. The closest might be the Lombards, but the Duchy of Tuscia was constituted by them in the year 570, centered around their capital in Lucca. Now, as the Lombards initially instituted various laws against intermarriage between themselves and the "Romans", perhaps one could say the intermarriage only happened later? Still, the sources with which I'm familiar hold that the amalgamation of Langobard and Roman was well underway within a hundred years of the invasion. Or, maybe it was the arrival of the Frankish elite which led to the amalgamation of the Romans and the Lombards. Still, even that occurred in the 700's.
This online book provides quite a bit of good detail about the period:
http://books.google.com/books?id=lA...age&q=How many Lombards invaded Italy&f=false
As does this book by Patrick Geary:
http://books.google.com/books?id=A2... time of the invasion by the Lombards&f=false
Also, what is one to make of the fact that while 15% of the Tuscan genome is attributed to "Germans", 18.6% is English, and 8.1% is "Welsh". (Let's ignore the French, Basque etc. and other minor components for now.) Assuming for the moment that this almost 27% of the genome(English and Welsh) is attributable to general central European "Celtic"/"Germanic" alleles, are we supposed to assume that these elements were part of the greater Langobard genome that arrived in Italy in the sixth century? I know that the Langobards picked up other tribes in their migrations, but we've had no indication that they were such a large percentage of the general Langobard force.
There's also the fact that given the numbers posited for the Langobards and assorted barbarian tribes involved in the sixth century invasion, you would have to assume an almost complete depopulation of the Italian peninsula for that number of people to have this kind of genomic effect. There is indeed controversy as to the the numbers of people that remained in the peninsula after the horrors of the Gothic Wars and the attendant famine and disease, but even the most pessimistic accounts put the number in the millions. I think we can take some of Paul the Deacon's account as hyperbole.
A discussion of the small numbers involved in the Lombard invasion can be found here:
http://rbedrosian.com/Ref/Bury/ieb14.htm
So, while the authors of this paper claim that their dating method is far superior to the results available from the Roll-Off program, I'm not convinced.
The results also would seem to conflict with the IBD analysis by Ralph and Coop, who see no admixture in these areas of Italy after the arrival of the Gauls or Celts in the first millennium B.C. Then, there's the conflict with the findings of Lazaridis et al. How could the northern Italians, who are about 70% (?) EEF have 63% of their ancestry from north of the Alps?
While in the end they may be proven to be correct, as things stand now, some things don't quite add up.