Microscopic path structure of optimally aligned random sequences

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.