Finite temperature mechanical instability in disordered lattices

Mechanical instability takes different forms in various ordered and disordered systems. We study the effect of thermal fluctuations in two disordered central-force lattice models near mechanical instability: randomly diluted triangular lattice and randomly braced square lattice. These two lattices exhibit different scalings for the emergence of rigidity at $T=0$ due to their different patterns of self stress at the transition. Using analytic theory we show that thermal fluctuations stabilize both lattices. In particular, the triangular lattice displays a critical regime in which the shear modulus scales as $G \sim T^{1/2}$, whereas the square lattice shows $G \sim T^{2/3}$.