The past half-decade has seen an increasing interaction between combinatorialists, probabilists, computer scientists and theoretical physicists concerned broadly with the study of "probability theory on graphs" or "statistical mechanics on graphs". The programme will build on this cross-fertilisation. It is particularly timely for a number of reasons: * methods from mathematical physics are beginning to make their mark on previously intractable combinatorial problems; * increasing computer power, together with the wide availability of symbolic-algebra packages, has brought the possibility...