  Self-similarity and long-range correlations are the remarkable features of
the Earth's surface topography. Here we develop an approach based on
percolation theory to study the geometrical features of Earth. Our analysis is
based on high-resolution, 1 arc min, ETOPO1 global relief records.We find some
evidence for abrupt transitions that occurred during the evolution of the
Earth's relief network, indicative of a continental/cluster aggregation. We
apply finite-size-scaling analysis based on a coarse-graining procedure to show
that the observed transition is most likely discontinuous. Furthermore, we
study the percolation on two-dimensional fractional Brownian motion surfaces
with Hurst exponent $H$ as a model of long-range correlated topography, which
suggests that the long-range correlations may play a key role in the observed
discontinuity on Earth. Our framework presented here provides a theoretical
model to better understand the geometrical phase transition on Earth, and it
also identifies the critical nodes that will be more exposed to global climate
change in the Earth's relief network.
