  We investigate the Laplacian spectra of random geometric graphs (RGGs). The
spectra are found to consist of both a discrete and a continuous part. The
discrete part is a collection of Dirac delta peaks at integer values roughly
centered around the mean degree. The peaks are mainly due to the existence of
mesoscopic structures that occur far more abundantly in RGGs than in
non-spatial networks. The probability of certain mesoscopic structures is
analytically calculated for one-dimensional RGGs and they are shown to produce
integer-valued eigenvalues that comprise a significant fraction of the
spectrum, even in the large network limit. A phenomenon reminiscent of
Bose-Einstein condensation in the appearance of zero eigenvalues is also found.

Statistical Mechanics;Physics and Society