Borel-Cantelli Lemma and Derandomization
I don't think they mean derandomization in the traditional sense.  Try looking at the application of the BC lemma in this paper for an example of what they are talking about: http://www.cs.bu.edu/~reyzin/hash.html.

They say "asymptotic" because most BB separations apply to concepts like one-way functions, which are defined asymptotically.  Their result is instead a "concrete" bound that applies to all values of the security parameters, not just sufficiently large values.