  Understanding when evolutionary algorithms are efficient or not, and how they
efficiently solve problems, is one of the central research tasks in
evolutionary computation. In this work, we make progress in understanding the
interplay between parent and offspring population size of the $(\mu,\lambda)$
EA. Previous works, roughly speaking, indicate that for $\lambda \ge
(1+\varepsilon) e \mu$, this EA easily optimizes the OneMax function, whereas
an offspring population size $\lambda \le (1 -\varepsilon) e \mu$ leads to an
exponential runtime.
  Motivated also by the observation that in the efficient regime the
$(\mu,\lambda)$ EA loses its ability to escape local optima, we take a closer
look into this phase transition. Among other results, we show that when $\mu
\le n^{1/2 - c}$ for any constant $c > 0$, then for any $\lambda \le e \mu$ we
have a super-polynomial runtime. However, if $\mu \ge n^{2/3 + c}$, then for
any $\lambda \ge e \mu$, the runtime is polynomial. For the latter result we
observe that the $(\mu,\lambda)$ EA profits from better individuals also
because these, by creating slightly worse offspring, stabilize slightly
sub-optimal sub-populations. While these first results close to the phase
transition do not yet give a complete picture, they indicate that the boundary
between efficient and super-polynomial is not just the line $\lambda = e \mu$,
and that the reasons for efficiency or not are more complex than what was known
so far.
