  The lifting of the degeneracy of the states from the graphene $n$=0 Landau
level (LL) is investigated through a non-interacting tight-binding model with
random hoppings. A disorder-driven splitting of two bands and of two critical
energies is observed by means of density of states and participation ratio
calculations. The analysis of the probability densities of the states within
the $n$=0 LL provides some insights into the interplay of lattice and disorder
effects on the splitting process. An uneven spatial distribution of the wave
function amplitudes between the two graphene sublattices is found for the
states in between the two split peaks. It is shown that as the splitting is
increased (linear increasing with disorder and square root increasing with
magnetic field), the two split levels also get increasingly broadened, in such
a way that the proportion of the overlapped states keeps approximately constant
for a wide range of disorder or magnetic field variation.
