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We study the sequence alignment problem and its independent version, thediscrete Hammersley process with an exploration penalty. We obtain rigorousupper bounds for the number of optimality regions in both models near the softedge. At zero penalty the independent model becomes an exactly solvable modeland we identify cases for which the law of the last passage time converges to aTracy-Widom law.

Optimality Regions and Fluctuations for Bernoulli Last Passage Models