I have a reference you might be interested in:
Waterman, M.S., L. Gordon, and R. Arratia. Phase transitions in
sequence matches and nucleic acid structure. PNAS 84: 1239-1243, 1987.
They use the Erdos-Renyi result of the length of the longest run of
heads in successive coins flips to calculate the probability of
long base-paired sequences. The phase transition they refer to is
that the minimum energy, and thus the alignment selected, changes
from a linear to a logarithmic relationship with the mismatch penalty.
Center for Complex Systems
Florida Atlantic University
tomh at bambi.ccs.fau.edu