  The super-cooled $N3$ model exhibits an increasingly slow dynamics as density
approaches the model's random closest packing density. Here, we present a
direct measurement of the dynamical correlation function $G_4(r,t)$, showing
the emergence of a growing length scale $\xi_4$ across which the dynamics is
correlated. The correlation length measured, up to 120 lattice sites, power-law
diverges as the density approaches $\rho_t$, the density at which the fluid
phase of the model is predicted to terminate. It is shown that the four-point
susceptibility, often used as an agent to estimate $\xi_4$, does not depend
simply on the latter. Rather, it depends strongly on the short-range behavior
of $G_4(r,t)$. Consequently, $\chi_4$ peaks before $\xi_4$ reaches its maximal
value. The two quantities should therefore be studied independently.
