We develop a first-principles theory for the vibrational density of states (VDOS) and thermal properties of network materials built on stationary correlated disordered point configurations. For elastic networks with stretching and bending interactions, we show that the empirical spectral measure of the vector dynamical matrix converges in the thermodynamic limit to the pushforward of Lebesgue measure under longitudinal and transverse Fourier symbols, which depend only on the edge kernels and the two-point statistics of the underlying point set. For disordered hyperuniform systems in the stretching-dominant regime, we showed that hyperuniformity does not alter the leading Debye exponent but suppresses sample-to-sample fluctuations, yielding a smooth, crystal-like acoustic edge. For bending-dominant systems with hyperuniform scaling $S(k)\sim k^{\alpha }$, the VDOS exhibits an algebraic pseudogap at low frequency, $g(\omega )\sim {\omega }^{2d/\beta -1}$ with $\beta =\min \{4,\alpha +2\}$, which implies a low-temperature specific heat $C(T)\sim T^{2d/\beta }$ and a heat-kernel decay $Z(t)\sim t^{-d/\beta }$, defining a spectral dimension $d_s=2d/\beta$. This hyperuniformity-induced algebraic edge depletion could enable novel wave manipulation and low-temperature applications.

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• Classical statistical mechanics
• Density of states
• Network diffusion
• Network structure
• Phonons
• Specific heat
• Disordered systems