  Recently $q$-breathers - time-periodic solutions which localize in the space
of normal modes and maximize the energy density for some mode vector $q_0$ -
were obtained for finite nonlinear lattices. We scale these solutions together
with the size of the system to arbitrarily large lattices. We generalize
previously obtained analytical estimates of the localization length of
$q$-breathers. The first finding is that the degree of localization depends
only on intensive quantities and is size independent. Secondly a critical wave
vector $k_m$ is identified, which depends on one effective nonlinearity
parameter. $q$-breathers minimize the localization length at $k_0=k_m$ and
completely delocalize in the limit $k_0 \to 0$.
