On the Variance of the Optimal Alignments Score for Binary Random Words and an Asymmetric Scoring Function

We investigate the order of the variance of the optimal alignments score of two independent iid binary random words having the same length. The letters are equiprobable, but the scoring function is such that one letter has a larger score than the other. In this setting, we prove that the order of variance is linear in the common length. Optimal alignments constitute a generalization of longest common subsequences, they can be represented as optimal paths in a two-dimensional last passage percolation setting with dependent weights.