  We study the dimensional crossover from 2D to 1D type behavior, which takes
place in the thermal excited rippling of a graphene honeycomb lattice, when one
of the dimensions of the layer is reduced. Through a joint study, by Monte
Carlo (MC) atomistic simulations using a quasi-harmonic potential and
analytical calculations, we find that the normal-normal correlation function
does not change its power law behavior in the long wavelength limit. However
the system size dependency of the square of out of plane displacement $ <h^2>$
changes its scaling behavior when going from a layer to a nanoribbon. We show
that a new scaling law appears which corresponds to a truly 1D behavior and we
estimate the ratio of the sample dimensions where the crossover takes place as
$R_{2D \leftrightarrow 1D}\approx 1.609$. Having explored a wide number of
realistic systems sizes, we conclude that narrow ribbons present stronger
corrugations than the square graphene sheets and we discuss the implications
for the electronic properties of freestanding graphene systems.
