 For instance, the standard deviation of the length of the longest common subsequence of random strings of length "n" is known to be proportional to the square root of "n".One complication in performing this sort of analysis is that the random variables describing whether the characters at different pairs of positions match each other are not independent of each other.For a more mathematically tractable simplification of the longest common subsequence problem, in which the allowed matches between pairs of symbols are not controlled by whether those symbols are equal to each other but instead by independent random variables with probability 1/"k" of being 1 and ("k" − 1)/"k" of being 0, it has been shown that the distribution of the longest common subsequence length is controlled by the Tracy–Widom distribution.