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For the critical exponents near the sol-gel phase transition, classical theories like those of Flory and Stockmayer predict one set of exponents, whereas scaling theories based on lattice percolation predict different exponents. The two groups of theories differ in their treatment of intramolecular loops, space dimensionality and excluded volume effects. In this article, the differences and similarities between the results of the competing theories are reviewed. For example, a gel fraction like (p-pc)β vanishes for conversion factors p very close to the gel point pc, the weight average molecular weight diverges as (pc-p)−γ for p very slightly below pc, and the radius of macromolecules at the gel point p=pc varies as the ϱ-th power of the number of monomers in that macromolecule. Classical theories predict β=γ=1 and ϱ=1/4 whereas the percolation theory gives β ≃ 0.45, γ ≃ 1.74 and ϱ ≃ 0.40. We also generalize the percolation concept to include interaction effects and concentration fluctuations; in this case the sol-gel phase transition may be connected with a phase separation.

We review recent results about the segregation process of a granular mixture of disks on an horizontally oscillating tray. In this condition an initially disordered mixture first segregates via the formation of stripes perpendicular to the driving direction; then stripes merge in a coarsening process. We discuss quantitatively both the short-time and the long-time dynamics of the system, and the dependence of the observed phenomenology on the frequency and amplitude of oscillation of the tray. The same system is also investigated when, instead of being disordered, is initially prepared in two stripes parallel to the driving direction. In this condition the interface between the two stripes manifests an instability which again leads to the formation of stripes perpendicular to the driving direction. Finally, we shortly review the mechanism which have been proposed in order to explain the observed segregation process.

The opening talk of an interdisciplinary meeting should ideally start at “square one”. In the present case this means I should assume no previous background in the field of equilibrium phase transitions. Although everyone in the audience has some background in this field, the background of no two people is identical. Hence I begin with a brief introduction to phase transitions. Accordingly, I shall organize this talk around three simple questions: (i) “What happens?” That is to say, “What are the basic phenomena under consideration?” (ii) “Why do we care?” (iii) “What do we actually do?”

Percolation theory is reviewed. Intuitive arguments are given to derive scaling and hyperscaling relations. Above six dimensions the breakdown of hyperscaling is related to the interpenetration of the critical large clusters, and to the appearence at p c of an infinite number of infinite clusters of zero density with fractal dimension d f = 4. The structure of the percolating cluster made of links and blobs is characterized by an infinite set of exponents related to the anomalous voltage distribution in a random resistor network at p c . The surface structure of critical clusters below pc, which is relevant to the study of random superconducting networks, is also discussed. In particular, an exact result is presented which shows that in any dimension the interface of two critical clusters diverge as (p c -p)-1 as the percolation threshold is approached.

A new relation is given in percolation theory from which follows that the backbone of the incipient infinite cluster is made of singly-connected links and “blobs”, the number of links diverge with an universal exponent 1. It is also shown that this exponent characterizes the crossover exponent of the dilute s-state Potts model while for the n-vector model is given by the low density resistivity exponent.