Displacement Correlations in Disordered Athermal Networks

We derive exact results for correlations in the displacement fields $\{ \delta \vec{r} \} \equiv \{ r_{\mu = x,y} \}$ near-crystalline athermal systems two dimensions. analyze produced by different types of microscopic disorder, and show that disorder at scale gives rise to long-range with a dependence on system size $L$ given $\langle r_{\mu} r_{\nu} \rangle \sim c_{\mu \nu}(r/L,\theta)$. In addition, we polydispersity constituent particle sizes random bond give logarithmic scaling, $c_{\mu \nu}(\rho,\theta) \text{const}_{\mu\nu} - \text{a}_{\mu\nu}(\theta)\log \rho + \text{b}_{\mu\nu}(\theta) \rho^{2} $ $\rho~(=r/L) \to 0$. This scaling is case external forces each vertex network, $c^{f}_{\mu \text{const}^{f}_{\mu \nu} -( \text{a}^{f}_{\mu \nu}(\theta) \text{b}^{f}_{\mu \log ) \rho^2 $. Additionally, find stiffness differ their symmetry properties. Finally, also predict model polydispersed soft disks subject pinning forces, involve disorder. verify our theoretical predictions using numerical simulations spring contacts