TITLE: Expected edit distance
QUESTION [26 upvotes]: 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.

REPLY [2 votes]: I did some simulations myself, (5000 of each string length 1-1000) and thought I would share them here. They are plotted against Carlo's conjectured bound of $1-1/\sqrt{2}$ (in blue).
To me, the conjecture seems quite likely, since the smallest average distance I get is 0.296 at $n=1000$.