Exotic Acoustic-Edge and Thermal Scaling in Disordered Hyperuniform Networks

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 scalar (mass--spring) models whose dynamical matrix is a distance-weighted graph Laplacian, we prove that the limiting spectral measure is the pushforward of Lebesgue measure by a Fourier symbol that depends only on the edge kernel $f$ and the two-point statistics $g_2$ (equivalently the structure factor $S$). For hyperuniform systems with small-$k$ scaling $S(k)\sim k^\alpha$ and compensated kernels, {the VDOS exhibits an algebraic \emph{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. Generalization to vector mechanical models and implications on material design are also discussed.