I uploaded to arXiv our paper On the behavior of the leading eigenvalue of Eigen’s evolutionary matrices. This is our second joint work with Yura Semenov and Alexander Bratus, both from Moscow, on the Eigen model. The first one was devoted a special case of permutation invariant Crow-Kimura model, and this one deals with the original Eigen’s model.
We study the eigenvalue problem
where 
The matrix 
and hence defined by a unique parameter 
Among many things that we studied in this text, here is one, which is not proved, but mostly interesting from biological point of view: the critical value of the mutation rate after which the distribution of 
![\displaystyle q^*=\frac 12\sqrt[N]{\frac{\sum w_j}{\max_j\{w_j\}}}=\sqrt[N]{\frac{\lambda(0.5)}{\lambda(1)}},](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+q%5E%2A%3D%5Cfrac+12%5Csqrt%5BN%5D%7B%5Cfrac%7B%5Csum+w_j%7D%7B%5Cmax_j%5C%7Bw_j%5C%7D%7D%7D%3D%5Csqrt%5BN%5D%7B%5Cfrac%7B%5Clambda%280.5%29%7D%7B%5Clambda%281%29%7D%7D%2C&bg=ffffff&fg=333333&s=0 "\displaystyle q^*=\frac 12\sqrt[N]{\frac{\sum w_j}{\max_j\{w_j\}}}=\sqrt[N]{\frac{\lambda(0.5)}{\lambda(1)}},")
where 
The last formula gives a surprisingly nice estimates for several examples computed numerically (the examples are in the text).
