We consider the random variable: z = 1 + xl + xlxz + xlxzxj +. .. , where the x, are independent, identically distributed variables. We derive some asymptotic properties of the distribution of z, which are related e.g. to the low-temperature behaviour of the random field king chain. For a special class of distributions of the x,, exact solutions are presented. We also study the cases where the distribution function of z exhibits a power-law fall-off modulated by a 'periodic critical amplitude'.

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.