Threshold phenomena in random structures

Abstract

The physical theory of phase transition explains sudden changes of phase in materials that undergo gradual changes of some parameter like temperature. There are analogs of phase transition in the theory of random graphs, initiated by Erdös and Rényi. This paper gives a nontechnical but precise account, without proofs, of some of the beautiful discoveries of Erdös and Rényl about threshold phenomena in graphs, describes an application of their methods to interval graphs, and gives some examples of threshold phenomena under other definitions of randomness and in combinatorial structures other than graphs. The paper offers some speculations on possible applications of random combinatorial structures to telecommunications, neurobiology, and the origin of life. A rich man commissioned three experts, a veterinarian, an engineer, and a theoretical physicist, to find out what made the best race horses. After a few years they reported their results. The vet concluded from genetic studies that brown horses were the fastest. The engineer found that thin legs were optimal for racing. The theoretical physicist asked for more time to study the question because the case of the spherical horse was proving extremely interesting. Aharon Katchalsky No one is exempt from talking nonsense; the only misfortune is to do it solemnly. Montaigne