Considering optimal alignments of two i.i.d. random sequences of length $n$, we show that for Lebesgue-almost all scoring functions, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges to a unique limiting distribution as $n$ tends to infinity. This result helps understanding the microscopic path structure of a special type of last-passage percolation problem with correlated weights, an area of long-standing open problems. Characterizing the microscopic path structure also yields robust alternatives to the use of optimal alignment scores alone for testing the homology of genetic sequences.
"Microscopic path structure of optimally aligned random sequences." Bernoulli 26 (1) 1 - 30, February 2020. https://doi.org/10.3150/18-BEJ1053