  Considering the optimal alignment of two i.i.d. random sequences of length
$n$, we show that when the scoring function is chosen randomly, 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 is interesting because it 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 yields furthermore a robust alternative to optimal alignment
scores for testing the relatedness of genetic sequences.
