Amorphous solid state: A locally stable thermodynamic phase of randomly constrained systems

The question of the local stability of the (replica-symmetric) amorphous so lid state is addressed for a class of systems undergoing a continuous liqui d to amorphous-solid phase transition driven by the effect of random constr aints. The Hessian matrix, associated with infinitesimal fluctuations aroun d the stationary point corresponding to the amorphous solid state, is obtai ned. The eigenvalues of this Hessian matrix are all shown to be strictly po sitive near the transition, except for one-the zero mode associated with th e spontaneously broken continuous translational symmetry of the system. Thu s the local stability of the amorphous solid state is established. [S0163-1 829(99)01631-0].