  Network geometries are typically characterized by having a finite spectral
dimension (SD), $d_{s}$ that characterizes the return time distribution of a
random walk on a graph. The main purpose of this work is to determine the SD of
a variety of random graphs called random geometric graphs (RGGs) in the
thermodynamic regime, in which the average vertex degree is constant. The
spectral dimension depends on the eigenvalue density (ED) of the RGG normalized
Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior
of the ED in such a neighborhood characterizes the random walk. Therefore, we
first provide an analytical approximation for the eigenvalues of the
regularized normalized Laplacian matrix of RGGs in the thermodynamic regime.
Then, we show that the smallest non zero eigenvalue converges to zero in the
large graph limit. Based on the analytical expression of the eigenvalues, we
show that the eigenvalue distribution in a neighborhood of the minimum value
follows a power-law tail. Using this result, we find that the SD of RGGs is
approximated by the space dimension $d$ in the thermodynamic regime.
