  In spite of its simplicity, the central limit theorem captures one of the
most outstanding phenomena in mathematical physics, that of universality. While
this classical result is well understood it is still not very clear what
happens to this universal behaviour when the random variables become
correlated. A fruitful mathematically laboratory to investigate the rising of
new universal properties is offered by the set of eigenvalues of random
matrices. In this regard a lot of work has been done using the standard random
matrix ensembles and focusing on the distribution of extreme eigenvalues. In
this case, the distribution of the largest -- or smallest -- eigenvalue departs
from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom
distribution. One may wonder, yet again, how robust is this new universal
behaviour captured by the Tracy-Widom distribution when the correlation among
eigenvalues changes. Few answers have been provided to this poignant question
and our intention in the present work is to contribute to this interesting
unexplored territory. Thus, we study numerically the probability distribution
for the normalized largest eigenvalue of the interacting $k$-body fermionic
orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We
find a smooth transition from a slightly asymmetric Gaussian-like distribution,
for small $k/m$, to the Tracy-Widom distribution as $k/m\to 1$, where $k$ is
the rank of the interaction and $m$ is the number of fermions. Correlations at
the edge of the spectrum are stronger for small values of $k/m$, and are
independent of the number of particles considered. Our results indicate that
subtle correlations towards the edge of the spectrum distinguish the
statistical properties of the spectrum of interacting many-body systems in the
dilute limit, from those expected for the standard random matrix ensembles.
