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Critical Phenomena in Sequence Matching

Richard Arratia and Michael S. Waterman
The Annals of Probability
Vol. 13, No. 4 (Nov., 1985), pp. 1236-1249
Stable URL: http://www.jstor.org/stable/2244175
Page Count: 14
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Critical Phenomena in Sequence Matching
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Abstract

We give a generalization of the result of Erdos and Renyi on the length Rn of the longest head run in the first n tosses of a coin. Consider two independent sequences, X1 X2⋯ Xm and Y1Y 2⋯ Yn. Suppose that X1, X2,⋯ are i.i.d. μ, and Y1, Y2,⋯ are i.i.d. ν, where μ and ν are possibly different distributions on a common finite alphabet S. Let $p \equiv P(X_1 = Y_1) \in (0, 1)$. The length of the longest matching consecutive subsequence is $M_{m,n} \equiv \max \{k: X_{i+r} = Y_{j+r}$ for r = 1 to k, for some 0 ≤ i ≤ m - k, 0 ≤ j ≤ n - k}. For m and n → ∞ with log(m)/log(mn) → λ ∈ (0,1), our result is that there is a constant $K \equiv K(\mu, \nu, \lambda) \in (0, 1\rbrack$ such that $P(\lim M_{m,n}/\log_{1/p}(mn) = K) = 1$. The proof uses large deviation methods. The constant K is determined from a variational formula involving the Kullback-Liebler distance or relative entropy. A simple necessary and sufficient condition for K = 1 is given. For the case m = n (λ = 1/2) and μ = ν, K = 1. The set of (μ, ν, λ) for which K = 1 has nonempty interior. The boundary of this set is the location of a phase transition. The results generalize to more than two sequences and to Markov chains. A strong law of large numbers is given for the proportion of letters within the longest matching word; the limiting proportion exhibits critical behavior, similar to that of K.

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