The edit or Levenshtein distance between two strings is the minimum number of single symbol insertions, deletions and substitutions to transform one string into another. For example $$\operatorname{E}(01010,00100)=2.$$

Let $E_n$ be a random variable giving the edit distance between two random binary strings of length $n$.

What bounds can be found for

$$\lim_{n \to \infty} \frac{\mathbb{E}(E_n)}{n} = c\;?$$

According to my non-extensive simulations, $c \approx 0.288$.

There is related work by Luecker on the expected length of the longest common subsequence which has lower and upper bounds of $0.788071n$ and $0.826280n$. This was a line of work originally started by Chvatal and Sankoff. However the longest common subsequence length and $n$ minus the edit distance need not be similar.