An overview will be given of recent work on the solid state of random network media, focusing on the structure and implications of their thermal and architectural fluctuations. Perhaps surprisingly, given the randomness of their architectures, random network solids possess a number of simple, universal, properties, including the quantitative nature of their heterogeneity and rigidity. Near six dimensions – the upper critical dimension for the random solidification transition – a renormalization-group treatment of fluctuations reveals the extent to which ideas from percolation theory can and cannot capture the physical properties of the random solid state. By contrast, in and near two dimensions – the lower critical dimension – it is low-energy, long-wavelength, Goldstone-type fluctuations that play an essential role. These fluctuations, which amount to shear deformations, furnish an expression for the elastic shear modulus of random solids in terms of the order parameter, as well as giving rise to a constant shift in the distribution of localization lengths, relative to the mean-field distribution. In two dimensions this shift diverges, fluctuations restore the classically-broken translational symmetry, and the constituents of the random solid state are no longer truly localized. However, order-parameter correlations decay algebraically and the shear modulus remains nonzero, so that – as with crystalline solids – two-dimensional random solids exhibit quasi-long range, albeit random, order.

University of Illinois at Urbana-Champaign