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Microscopic path structure of optimally aligned random sequences. (English) Zbl 1464.60010
This paper studies the microscopic path structure of optimally aligned random sequences, for an alignment $$\pi$$ with gaps of two sequences $$x=x_1,\cdots,x_n$$ and $$y=y_1,\cdots,y_n$$ of equal length $$n$$. The vector $$p_{\pi}(x,y)$$ of all such ratios collected in lexicographical order is the empirical distribution vector of $$\pi$$. Let $$SET(x,y)=\{p_{\pi}(x,y)\}_{\pi}$$ be the set of all empirical distribution vectors related to $$x$$ and $$y$$. For two independent random sequences $$X=X_1,\ldots, X_n$$ and $$Y=Y_1,\ldots,Y_n$$ with i.i.d. components. The convex hull of $$SET(X,Y)$$ is given by $$SET^n=\mathrm{conv}(SET(X,Y))$$. Given a scoring function $$S$$, let $$f_S$$ be some associated linear functional. It is shown that $$SET^n$$ almost surely converges to the set $$SET$$ in the topology of the Hausdorff distance as $$n$$ tends to infinity. Moreover, the empirical distribution of all optimal alignments of $$X$$ and $$Y$$ almost surely converge to a deterministic distribution when the scoring function $$S$$ is selected such that $$f_S$$ has a unique maximizer in $$SET$$. For Lebesgue-almost every scoring function $$S$$, it is proved that the empirical distributions of all optimal alignments almost surely converge to some deterministic distribution.
##### MSC:
 60C05 Combinatorial probability 60F10 Large deviations 60K35 Interacting random processes; statistical mechanics type models; percolation theory 90C25 Convex programming 60F15 Strong limit theorems
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