Longest common subsequences between words of very unequal length

We consider the expected length of the longest common subsequence between two random words of lengths $n$ and $(1-\varepsilon)kn$ over $k$-symbol alphabet. It is well-known that this quantity is asymptotic to $\gamma_{k,\varepsilon} n$ for some constant $\gamma_{k,\varepsilon}$. We show that $\gamma_{k,\varepsilon}$ is of the order $1-c\varepsilon^2$ uniformly in $k$ and $\varepsilon$. In addition, for large $k$, we give evidence that $\gamma_{k,\varepsilon}$ approaches $1-\tfrac{1}{4}\varepsilon^2$, and prove a matching lower bound.
Comment: 38 pages