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, $g(\omega )\sim {\omega }^{2d\mathrm{/}\beta -1}$ with $\beta =\min \{4,\alpha +2\}$, which implies a low-temperature specific heat $C(T)\sim T^{2d\mathrm{/}\beta }$ and a heat-kernel decay $Z(t)\sim t^{-d\mathrm{/}\beta }$, defining a spectral dimension $d_s=2d\mathrm{/}\beta$.} 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.