An upper bound on the convergence rate of a second functional in optimal sequence alignment

Consider finite sequences X[1,n] = X1,...,Xn and Y[1,n] = Y1,...,Yn of length n, consisting of i.i.d. samples of random letters from a finite alphabet, and let S and T be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in (ln(n))1/4n^3/4 for the deviation of the score relative to T of optimal alignments with gaps of X[1,n] and Y[1,n] relative to S. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun in (J. Stat. Phys. 153 (2013) 512–529).