Fluctuation-stabilized marginal networks and anomalous entropic elasticity

We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work we show that thermal networks exhibit a non-zero shear modulus $G$ well below the isostatic point, and that this modulus exhibits an anomalous, sublinear dependence on temperature $T$. At the isostatic point, $G$ increases as the square-root of $T$, while we find $G \propto T^{\alpha}$ below the isostatic point, where ${\alpha} \simeq 0.8$. We show that this anomalous $T$ dependence is entropic in origin.