Random solids and random solidification: what can be learned by exploring systems obeying permanent random constraints?

In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state . In this random solid state, particles are permanently but randomly locali zed in space, and a rigidity to shear deformations emerges. Owing to the pe rmanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the con ventional glass transition) in the sense that it is amenable to treatment b y established techniques of equilibrium statistical mechanics. In this pape r I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities of and differences between such systems and t hose exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correla tions and susceptibilities that signal the approach of the random solidific ation transition, discussing the physical interpretation and values of thes e quantities both at the Gaussian level of approximation and, via a renorma lization-group approach, beyond.