In his "History of the Yiddish Language" (1973) Max Weinreich regarded the 13th century as a turning point in the history of the Yiddish language. He identifies it as the boundary between the Early Yiddish and Old Yiddish periods. More specifically he writes about a 13th century Babylonian Renaissance characterized by a change in the Ashkenazic norms of Hebrew pronunciation. He says that the change was centered around Rothenburg and involved scholars who bore names that were previously rare or unknown among German Jews but were used by Jews in the Middle East. The name Bablyonian Renaissance comes from Weinreich's beliefs that the pronunciation norms came from Mesopotamia and that the scholars who brought them migrated from there.
A look at studies of the history of Jewish settlement in Germany shows that the 13th century was a period when settlements increased rapidly in number and geographical extant. Judging from the maps the center of this spread was the Main Valley. Rothenburg is in this area.(Maps in Michael Toch's "Jews and Peasants in Medieval Germany" 2003 and Alfred Haverkamp, ed., "Geschichte der Juden in Mittelalter" 2002).
Genomic studies have looked a uniparental markers among the Ashkenazim, These are genetic features that are inherited exclusively from one parent. Mitochondrial genes are transmitted only from mothers and genes on the Y chromosome come only from fathers. According to one study Ashkenazic Jews predominantly show mitochondrial genes that are characteristic of European populations (about 40%) and Y chromosomes genes that are characteristic of Middle Eastern ones. ("A prehistoric European ancestry amongst Ashkenazi lineages" Martin Richards et. al, in "Nature Communications" 4, article number 2543 (2013) )
Another study looked at the whole Ashkenazic genome.("Sequencing an Ashkenazi reference panel supports population targeted personal genomics and illuminates Jewish and European origins" Itsik Pe'er et al. "Nature Communications" 5, article number 4835 (2014) ). This makes it possible to infer some aspects of the history of the population. One thing that stands out is that the population looks like the result of an even admixture of a European and a Middle Eastern population (European ancestry estimated at 46-50%). The date of this admixture was between between 600 and 800 years ago.
The same study inferred a population bottleneck in the history of Ashkenaz. That is a period when the population dropped before expanding. As a result of this bottleneck modern Ashkenazim are descended from only between 250 and 420 ancestors.They lived between 25 and 32 generations ago. This would put the bottleneck at about the same time as the admixture.
If I am interpreting these data and inferences correctly, the majority of German Jews alive before 800 years ago did not contribute ancestors to the modern Ashkeanzi population. The exception was a subset comprised mostly of women who married Middle Eastern immigrants. The descendants of this admixed population expanded rapidly.
One explanation of this rapid expansion relative to the unadmixed population may be that the admixed population was disproportionately involved in the expansion of settlements. Some support for this idea comes from a study of the European settlement of Quebec. Settlers who founded new settlements had larger families with more married children and their descendants were also more apt to found new settlements, ("Deep human genealogies reveal a slective advantage to be on an expanding wave front" Claudia Moreau, et, al."Science" 25 November 2011 v.334 pp.1148-1150)
Another possible factor was suggested to me by Lou Cleveland. The Black Plague greatly reduced the German population in 1349. People of Middle Eastern origin may have had a degree of genetic resistance to this disease.