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

Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime